Google

This is a digital copy of a book lhal w;ls preserved for general ions on library shelves before il was carefully scanned by Google as pari of a project

to make the world's books discoverable online.

Il has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one thai was never subject

to copy right or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books

are our gateways to the past, representing a wealth of history, culture and knowledge that's often dillicull lo discover.

Marks, notations and other marginalia present in the original volume will appear in this file - a reminder of this book's long journey from the

publisher lo a library and linally lo you.

Usage guidelines

Google is proud lo partner with libraries lo digili/e public domain materials and make them widely accessible. Public domain books belong to the
public and we are merely their custodians. Nevertheless, this work is expensive, so in order lo keep providing this resource, we have taken steps to
prevent abuse by commercial panics, including placing Icchnical restrictions on automated querying.
We also ask that you:

+ Make n on -commercial use of the files We designed Google Book Search for use by individuals, and we request thai you use these files for
personal, non -commercial purposes.

+ Refrain from automated querying Do not send automated queries of any sort lo Google's system: If you are conducting research on machine
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the
use of public domain materials for these purposes and may be able to help.

+ Maintain attribution The Google "watermark" you see on each lile is essential for informing people about this project and helping them find
additional materials through Google Book Search. Please do not remove it.

+ Keep it legal Whatever your use. remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other

countries. Whether a book is slill in copyright varies from country lo country, and we can'l offer guidance on whether any specific use of
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner
anywhere in the world. Copyright infringement liability can be quite severe.

About Google Book Search

Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers
discover the world's books while helping authors and publishers reach new audiences. You can search through I lie lull lexl of 1 1 us book on I lie web
al |_-.:. :.-.-:: / / books . qooqle . com/|

I

Al.KXAMIKH Xl\

S'C.

vitf. \

S7€>

AERODONETICS

i

AERODONETICS

CONSTITUTING THE SECOND
VOLUME OF A COMPLETE

WORK on AERIAL FLIGHT

BY

F. W. LANCHESTER
7

With Appendices on the Theory
and Application of the Gyroscope, on
the Flight of Projectiles, etc.

LONDON

ARCHIBALD CONSTABLE & CO. LTD.
ORANGE STREET LEICESTER SQUARE

1908

^7

f'

f-a'H?**-

BRADBURY, AONKW, & CO. LD., PRINTERS
LONDON AND TONBR1DGE.

. '

I

I

1

I.

PKEFACE

The object and scope of the present work have been stated at
length in the preface to Vol. I., which appeared as recently as
December, 1907 ; there is bat little to add to this statement.

In conducting the investigations and otherwise in collecting
and arranging the matter included in the two volumes now
completed, the author has made an effort to do, in as thorough
a manner as possible, that preparatory work which, in his opinion,
should properly precede the serious and hitherto hazardous
undertaking of full-scale experiment. It is in other words the
author's intention to provide such foundation theory and data
as to bring the problem of mechanical flight into the legitimate
domain of the engineer, and to obviate in the future all need for
empiricism.

The author is aware that considerable work remains to be
done in the way of extension, both in the more complete eluci-
dation of many questions relating to the flight of birds, and in
the direction of the application of theory in the design of the
flying-machine for aerial navigation. It has not been found prac-
ticable to deal specifically with these extensions in the present
volumes, but much that is suggestive will be found embodied
appropriately in the text. The passive mode of bird flight, as
involved in gliding and soaring, has however been discussed at
length, for it is in the study of this mode of flight that the
greater part of that which is essential may be learned. 1

1 There is a false impression — one that is far too prevalent — that the
essence of flight consists in the flapping of wings. Nothing is further from
the truth.

So far is this error current that attempts have boon sometimes made to

406153

vi PREFACE

The present time is one of considerable importance in the
history of aerial flight : the motive power engine has, during the
last few years, reached a stage in its development at which its
weight has become sufficiently reduced to render mechanical
flight possible, and already several partially successful machines
have been built, and flights of several miles have been made.
Some of these machines are deficient in many important
respects; the propellers for example are of relatively small
diameter and are commonly of too quick a pitch and too high a
revolution speed for best efficiency. The longitudinal stability
also, instead of being automatic, is only maintained by the
watchful attention of the aeronaut and the dexterous manipulation
of a horizontal rudder. Further than this the cooling of the
motor cylinders is not altogether effective, and the duration and
range of flight is largely a matter of how long elapses before the
water is boiled away. The question of weight makes it difficult
to employ a thoroughly efficient "radiator" aB used on road
vehicles, and without doubt direct air cooling will sooner or later
come into vogue. For the above reasons the flights so far made
have been of comparatively brief duration.

In the future it is unlikely that the flying-machine will be

limit the term flyiny-machine to appliances of the wing-flapping kind. As
illustrative of this vein of thought the statement is sometimes made that an
aerodrome is not a flying -machine at all, that it does not fly, and that it is
only sustained (presumably like a satellite) by the speed at which it travels !

Critics who talk thus know little or nothing of the subject of flight as
taught by nature. It is true that many of the smaller birds and most
insects are capable of stationary flight, but the heavier birds, as for example
the eagle, the vulture, the stork, the albatros, the larger gulls, and nearer
home the swan, the goose, and even the wild duck, cannot fly without
considerable horizontal velocity. When leaving the ground (or water) these
birds require a run of many yards in order to attain the necessary velocity.
According to Mouillurd, who was an accurate observer, most birds of more
than a pound or two in weight may be effectively " caged" if confined in a
simple fenced enclosure (open to the sky) of appropriate size; they are
unable to acquire the velocity necessary to their flight.

Beyond the above very many of the larger birds rarely use the active
(wing-flapping) mode of flight at all, but nevertheless it is not said that they
no longer fly on that account.

PREFACE vii

limited in its performance to short flights over prepared ground
at a few metres height, ready to come to earth at a moment's
notice ; it will rather seek safety in altitude, probably flying in
most part at a height of at least two or three thousand feet,
where in the event of any minor mechanical failure or other
hitch the time of gliding descent will be at the disposal of the
aeronaut for adjustment, or, if it is found necessary to land, for
the selection of a suitable site. Thus, if an aerodrome be flying
one mile above the surface of the earth, there will be a circle of
some ten or twelve miles diameter available within which to
choose a place to alight ; or, if we assume the velocity of flight to be
forty miles per hour, the aeronaut will have at his disposal a period
of some eight or ten minutes before he need finally come to earth.
The manner in which the industries connected with locomotion
have proved mutually helpful is very striking ; the dependence of
the march of progress on such mutual assistance, nearly always
traceable in the evolution of mechanism, is perhaps nowhere
more apparent. Thus the modern automobile may almost be
said to owe its existence to the pneumatic tyre, an invention
whose utility was first established by the bicycle; and, in
turn, the flying-machine has only become possible through the
development of the internal combustion engine in the hands of
the automobile engineer. It is evident that the market for the
light weight petrol motor has grown up as a direct consequence
of the demand for high speed, a demand that could not have
arisen in the absence of the pneumatic tyre ; hence even the old
" hobby horse" may be regarded as one of the stepping stones to
the conquest of the air, and we are led to regard the flying-
machine as marking a great step in a vast evolutionary movement
in locomotion, rather than as constituting in itself an independent
invention.

The title of the present volume, " Aerodonetics," is one of
two alternatives suggested in the preface to Vol. I., chosen as
being from its derivation the more appropriate. The arrange-
ment is as follows : —

viii PREFACE

Chapter I. is an introductory account of the general pimciples
concerned in the equilibrium and stability of an aerodone in
flight, illustrated by actual examples, including an account of
the author's earlier experiments.

Chapters II. and III. consist of an analytical investigation of
the flight path on a basis restricted by a hypothesis excluding
the influence of the size and moment of inertia of the aerodone,
and assuming resistance to flight to be either absent or accounted
for by an equal and opposite force of propulsion. This investiga-
tion culminates with the plotting of the flight path from the
equation (as given in the frontispiece), and includes a discussion
of certain special cases, notably that of the flight path or
phugoid 1 of small amplitude. This investigation is the foundation
of the greater part of the subsequent work, where it is frequently
referred to under the title of the Phugoid Theory ; it constitutes
the key to the quantitative study of longitudinal stability, and to
the solution of many kindred problems in free flight.

Chapter IV. is devoted to the discussion of some of the more
immediate and self-evident consequences of the Phugoid Theory,
including the preliminary consideration of the influence of wind
fluctuations both as isolated gusts and as possessing a definite
periodicity.

Chapter V. deals with an important extension of the Phugoid
Theory, by which account is taken of influences excluded by the
initial hypothesis, i.e., resistance and moment of inertia. In
this chapter the theory is brought to a point at which it becomes
of very great value as bearing on the correct proportioning of an
aerodone or aerodrome ; the investigation culminates in an
equation, the equation of stability, by which the conditions of the
permanence of the flight path are clearly defined.

Chapter VI. is an account of the experimental verification of
the previous work (Chapters II., III., and V.). This account
comprises a brief resume of observations by earlier writers as
confirmatory of the Phugoid Theory, a series of experiments

1 From the Greek <j>vyrj and etSor (lit. flight-like) ; compare Glossary.

PREFACE ix

devised and carried out by the author as a direct test, and the
application of the theory to the investigation of the stability of
birds in flight, and to pre-existing flight models, including the
gliding machine of the late Herr Lilienthal. The results are
conclusive.

Chapter VII. consists of a series of investigations on lateral
and directional stability, and may be looked upon as an indepen-
dent section of the work. The method of treatment adopted is
the initial discussion of these two kinds of stability separately,
and their subsequent treatment in combination under the title
rotative stability; this investigation also culminates in an
equation which defines the limiting conditions of the stability of
the kind under consideration.

Chapter VIII. constitutes in part a resume and in part an
extension of that which has gone before. It comprises a review
of the groundwork of the theoretical investigations, with some
notes and discussion of the limitations and existing deficiencies
of the work ; also an extension of the investigation of Chapter V.
to cover the conditions that arise in the case of an aerodrome
propelled by prime mover and screw propeller, with a further
investigation on the rate of damping of the phugoid oscillation.
The chapter concludes with a discussion of the theory of
corresponding speed and its application to scale model experi-
ment, and with some remarks on forms of aerodone that may be
considered as departures from elementary type.

Chapter IX. is a distinct branch of the work and deals with
the phenomenon of soaring, both from the point of view of
observation and in the light of theory, much of the preceding
work being brought to bear on this complex subject with
considerable effect. The first portion of the work includes
quotations from observations by Darwin, J. A. Froude,
Mouillard, Langley, and others, and the theoretical investiga-
tions have for their starting point the well-known dictum of
Bayleigh, that when soaring is possible the motion of the wind
is either not horizontal or not uniform.

x PREFACE

Chapter X. is principally an account of experimental method,
and includes many notes, observations, and hints that will be
found of value to those who are tempted to take up the experi-
mental study of aerial flight.

The terminological innovations adopted in the present work
are given in the glossary following Chapter X., in which are
included new words or words bearing a special or restricted
meaning, as employed in both Vols. I. and II. The present
glossary is thus, in great part, a repetition of that previously
given.

Numerical work has been done by the aid of an ordinary
25 cm. slide rule, with a liability to error of about one-fifth of
one per cent., an amount which is quite unimportant.

It is again with very great pleasure that the author tenders
his thanks to Mr. P. L. Gray for his assistance in the reading
and correction of the proof sheets.

May, 1908.

CONTENTS

CHAPTER 1.

Fkee Fligut, General Principles and Phenomena.

§ 1. Introductory.

2. Introductory Remarks — continued.

3. The Ballasted Aeroplane. A Simple Case.

4. The Ballasted Aeroplane. Longitudinal Stability.

5. The Ballasted Aeroplane. Lateral Stability.

6. Ballasted Aeroplane. Directional Stability.

7. Ballasted Aeroplane. Interaction of Motions in and about the Co-ordi-

nate Axes.

8. Other Forms of Aerodone.

9. Some Successful Gliding and Flying Modols.

10. Author's Experiments, 1894.

1 1 . Author's Experiments — continued. The Aerodone.

12. The Author's Experiments— continued. The Catapult.

13. Site and Date of Experiments.

14. The Author's Experiments. Records.

15. The Author's Experiments. Discussion.

16. Author's Experiments. Remarks and Summary.

17. Author's Experiments. Remarks and Summary — continued.

CHAPTER II.
The Phugoid Theory.— The Equations of the Flight Path.

rn

§18. Introductory.

19. Initial Hypothesis.

20. The Phugoid Equation.

21. Substitution for the Constant ».

22. Radius of Curvature.

23. Some Special Cases of the Phugoid Curve.

CHAPTER III.
The Phugoid Theory. — The Flight Path Plotted.

§ 24. Preliminary Considerations.
25. Plotting the Curves. The Trommel.

xii CONTENTS

§ 26. Plotting the Curves. The Use of the Trammel.

27. Plotting the Inflected Carve.

28. Plotting the Inflected Curve. Example.

29. Phugoids of Small Amplitude.

30. Phugoids of Small Amplitude. Form of Orbit.

31. Phugoids of small Amplitude. Plotting.

32. The Phugoid Chart.

33. The Time Period of the Phugoid Path.

34. The Time Period of the Phugoid Path. Special Cases.

35. The Time Period and Form of the Phugoid Path. Special Cases—

continued.

36. Eolations of Time Period, Phase Length, and Velocity for Phugoids of

Small Amplitude.

37. Variations in the Value of C for Geometrically Similar Curves. The

Constant K.

CHAPTER IV.

Elementary Deductions from the Phugoid Theory.

§ 38. Permanence of Stability. <

39. Unstable Conditions. The Danger Zone on the Phugoid Chart.

40. Stability in Face of a Disturbing Cause.

41. The Constant C as an Index of Stability.

42. Wind Fluctuation. A Question of Relative Motion. *

43. Wind Gusts of Changing Velocity and Direction.

44. Stability in the Face of Wind Fluctuations.

45. Stability in the Face of Wind Fluctuations — continued.

46. The Practical Limit of Stability.

47. Tho Influence of a Periodic Disturbance.

48. Unaccounted Factors in relation to the Flight Path

49. Unaccounted Factors — continued.

CHAPTER V.

Stability of the Flight Path as Affected by Resistance and

Moment of Inertia.

§ 50. Introductory.

51. Influence of Resistance on Amplitude.

52. Influence of Moment of Inertia.

53. Influence of Moment of Inertia — continued.

54. Influence of Moment of Inertia— continued.

55. Tho Quantitative Problem.

56. The Form of tho Nearly Straight Phugoid.

57. On the Form of the Change in // duo to Resistance.

58. On the Form of the Change in // duo to Resistance — continued.

59. On the Changes in the Magnitude of H n due to Moment of Inertia.

60. Statement of Case.

61. The Equation of Stability.

62. The Equation of Stability. The Investigation continued.

63. The Equation of Stability. Tho Investigation concluded.

CONTENTS xiii

CHAPTER VI.

EXPERIMEJTTAL EVIDENCE AND VERIFICATION OF THE PHUQOID THEORY.

§ 64. Preliminary. The Importance of the Experimental Verification.

65. Penaud's Experiments.

66. Mouillard's Observations.

67. Marey's Experiments.

68. The Author's Experiments in Confirmation of the Phugoid Theory.

69. Confirmation of the Phugoid Theory as to Time, Velocity, and Phase

Length.

70. Verification of the Equation of Stability.

71. Experimental Verification of the Equation of Stability.

72. Experimental Verification of the Equation of Stability — continued.

73. Verification of the Equation of Stability — continued. Small Scale

Experiments.

74. Verification of the Equation of Stability. Small Scale Experiments —

continued.

75. The Stability of Birds in Flight.

76. The Stability of Birds in Flight— continued.

77. The Stability of Birds in Flight — continued.

78. Stability of the Hirundo Apus.

79. Stability of Diomedea Exulans.

80. The Ea nation of Stability applied to the Author's 1894 Models.

81. Lilienthare Machine. Stability Investigated.

82. Resumd.

CHAPTEB VII.
Lateral and Directional Stability.

§ 83. Introductory.

84. The Mutual Relationship of Lateral and Directional Stability.

85. lateral Stability.

86. Lateral Stability. Oscillations in the Transverse Plane.

87. Lateral Stability. Oscillations in the Transverse Plane — continued.

88. Oscillations in the Transverse Plane — continued.

89. Oscillations in the Transverse Plane. Damping Influences.

90. Oscillations in the Transverse Plane. Influence of Moment of Inertia.

91. Oscillations in the Transverse Plane. Form of the Oscillations.

92. Lateral Stability. Oscillations in the Transverse Plane in Practice.

93. Directional Stability.

94. Directional Stability — continued.

95. A Study in Directional Equilibrium and Maintenance.

96. A Study in Directional Equilibrium— contin ued.

97. Directional Stability — continued.

98. Directional Stability. Practical Considerations,

99. Fin Resolution.

100. Fin Resolution — continued.

101. Botative Stability.

102. Botative Stability — continued.

103. Botative Stability. Basis of Investigation Defined.

104. Botative Stability. Preliminary Investigation.

105. Botative Stability. Investigation.

106. On the Aerodynamic Radius.

107. On the Aerodynamic Radius. Theoretically Considered.

108. On the Aerodromic Radius.

xiv CONTENTS i

l

CHAPTER VIII. !

Review of Chapters I. — VII. and General Conclusions.

§ 109. Preliminary.

110. Aerodynamic Basis of the Phugoid Theory.

111. Basis of the Equation of Stability.

112. The Equation of Stability. Unaccounted Factors.

113. Basis of the Theory of Lateral and Directional Stability.

114. Aerodynamic Basis of the Investigation on Rotative Stability.

115. Limitations and Unaccounted Factors.

116. Limitations and Unaccounted Factors— continued.

117. Limitations and Unaccounted Factors — continued.

118. The Influence of the Mode of Propulsion.

1 19. The Influence of the Mode of Propulsion— continued.

120. The Influence of the Mode of Propulsion — continued.

121. Rate of Damping of the Phugoid Oscillation.

122. Damping of the Phugoid Oscillation. Examples.

123. Damping of the Phugoid Oscillation. Examples — continued.

124. Damping of the Phugoid Oscillation. Examples — continued.

125. Damping of the Phugoid Oscillation. Examples — continued.

126. The Law of Corresponding Speed.

127. Theory of Corresponding Speed.

128. The Theory of Corresponding Speed — continued.

129. The Law of Corresponding Speed in its Relation to the Phugoid

Theory.

130. The Theory of Corresponding Speed — continued.

131. Scale Model Experiments.

132. Scale Model Experiments — continued.

133. Scale Model Experiments — continued.

134. Scale Model Experiments. Allowances.

135. Scale Model Experiments. Summary.

136. Departures from Elementary Type.

137. The Theory of Lateral and Rotative Stability.

138. Application of the Theory of Lateral and Rotative Stability.

139. Conclusions.

140. Historical Note.

CHAPTER IX. '

Soaring. '

§ 141. Introductory.

142. Author's Observations on Soaring Flight.

143. Soaring Flight Described by other Observers.

144. The Different Modes of Soaring Flight.

145. The Vertical Component of the Wind. Meteorological Considerations. I

146. The Vertical Component of the Wind — continued. '

147. The Vertical Component of the Wind as dependent upon Geographical ;

Conditions. 7

148. The Up-Current in its Relation to Soaring Flight.

149. The Up-Current in its Relation to Soaring Flight — continued. i

150. Dynamic Soaring. *

151. Dynamic Soaring. Preliminary Investigation. 7

152. Dynamic Soaring. Historical Development of Modern Theory. j

153. Dynamic Soaring. Theory of the "Switch-back" Model. j

154. Theory of Dynamic Soaring. Quantitative Treatment.

CONTENTS xv

§ 155. Theory of Dynamic Soaring. Quantitative Investigation — continued,

156. Theory of Dynamic Soaring. Harmonic Wind Pulsation.

157. The Form of the Orbit in Dynamic Soaring.

158. Dynamic Soaring in its Relation to the Phugoid Flight Path.

159. The Relation between the Energy of Turbulence and the Velocity of

Flight.

160. The Efficiency of Dynamic Soaring. Energy Available in Terms of

Total Energy of Turbulence.

161. Efficiency of Dynamic Soaring— continued.

162. Dynamic Soaring as determined by Different Kinds of Aerial

Disturbance.

163. Dynamic Soaring as dependent on Dead- Water Region.

164. Dynamic Soaring. Mixed Conditions.

CHAPTER X.

Experimental Aehodonetics.

§ 165. Introductory.

166. Method of Experiment.

167. Construction. Materials Employed.

1 68. Materials — contin ued.

169. The Aerofoil.

170. The Aerofoil — continued.

171. The Fin-plan and Tail-plane.

1 72. The Measurement of Moment of Inertia.

173. Admissible Proportions of Models.

174. The Design of an Aerodone.

1 75. On the Angle of the Tail-plane and the Position of the Centre of

Gravity.

176. On the Angle of the Tail-plane and the Position of the Centre of

Gravity — continued.

177. Methods of Steering.

178. On the Ballasted Aeroplane.

179. Some Vagaries of the Flight Path.

180. Vagaries of the Flight Path — continued.

181. Vagaries of the Flight Path— continued.

Glossary.

Appendices.

Index.

VOLUME I.— ERRATA.*/

P. 4, line 14 from foot, for " plane the " read " the plane."

P. 10, line 6 from foot, for " icthyoid" read " ichthyoid."

P. 53, line 9 from foot, for " Poissuille " read " Poisouille."

P. 85, line 8 from top, the cavity of the labyrinth of the ear is given as an
example of a triply connected region ; this is in error, delete.

P. 113, line 4 below Fig. 50, for " were" read " was."

P. 199, line 7 from foot, for "4 X I" read "4 X 1."

P. 231, line 4 from foot (also in footnote), for " Moulliard " raid
" Mouillard."

P. 269, line 12 from foot, for "k" read " *."

P. 300, line 6 from top, for " tan (0 - 7) " read " tan ($ + y)," comp.
p. 299, line 3.

P. 431, line 3 from foot should read, " angle of the course relatively to the
apparent direction of the wind would be the sum of."

P. 432, lino 7 from top, delete words, — " about twice that stated even in the
most carefully designed craft. 1 '

]

AERODONETICS

CHAPTER I

FREE FLIGHT, GENERAL PRINCIPLES AND PHENOMENA

§ 1. Introductory. — The equilibrium and stability of a bird in
flight, or of an aerodrome or flying machine, has in the past
been the subject of considerable speculation, and no adequate
explanation of the principles involved has hitherto been given.

The question of stability is studied to the greatest advantage
in the case of gliding flight, for the problem is then presented in
its simplest form ; it is probable that the underlying principles
are identical, whether a bird is in active flight, or whether it be
poised on rigid pinions, as when soaring, or when merely gliding
from a point of greater to one of less altitude.

There are two methods employed by nature to secure stability
where animal life is concerned, i.e., mechanical means and nervous
control. As an illustration of the first may be cited the case of a
quadruped, whose equilibrium when standing is of the mechanical
order ; and of the second the biped, whose equilibrium is main-
tained by the action of the nervous system. 1

Mechanical stability does not necessarily involve an obvious
apparatus like the four legs of a table, as might be supposed
from the above illustration ; the stability of a spinning top for
instance, or of a common hoop, is entirely mechanical, but these

1 The equilibrium of a man standing erect is only maintained by a series
of reflex adjustments, although a rigid image or statue may be made
mechanically stable.

A. 1 B

§ 1 AERODONETICS

are cases of dynamic stability, in contradistinction to the static
stability of a table or a three-legged stool.

On the other hand, nervous control in the matter of stability
does not of the least necessity involve any conscious action of the
brain ; there are special nerve centres to which the function of
maintaining equilibrium is deputed.

It is always difficult in the case of any animal or bird to deter-
mine to what extent equilibrium is maintained by nervous
agency and to what extent it is automatic. In the case of a
bicycle rider we know that the equilibrium is to some extent
automatic, 1 but that it is not so entirely may be reasonably
inferred from the fact that it is necessary to leam to tide. In
the case of a bird in flight, we have no means of ascertaining
directly to what extent equilibrium is maintained by a continual
series of reflex adjustments, and to what extent by purely
dynamic action consequent upon its geometrical form considered
as an inanimate body.

It is almost certain that, for a bird in flight, in any sudden
evolution necessary to regain equilibrium after an unexpected
gust of wind, the nervous system is called into play, and perhaps
at times the whole conscious brain of the bird is involved in the
effort to restore the balance that has been momentarily lost.
These happenings may be said to be obvious to anyone who has
watched the flight of birds in stormy weather. It would seem
to follow as a matter of inference that a considerable number of
minor reflex adjustments are made and perhaps even a continuous
reflex adaptation may take place, invisible to an observer, but the
extent to which this is the case can only at the moment be a
matter of conjecture.

It is evident that the most promising way to examine the sub-
ject, both theoretically and experimentally is by an investigation
of the inanimate model or aerodone, and having ascertained all

1 It is well known that a cyclist can frequently ride his machine perfectly,
when either from cerebral inj uiy or inebriation he is quite unable to walk
without assistance.

f

FREE FLIGHT §2

that can.be done and accounted for on purely dynamical principles,
to revert to the subject of bird flight, equipped with the knowledge
so obtained.

§ 2. Introductory Remarks (continued). — There is an initial diffi-
culty associated with the study of equilibrium that perhaps has
been felt by others who have in the past attempted to investigate
the subject ; this difficulty is the indefiniteness of that which is to
be investigated.

Whatever kind of thing an aerodrome or aerodone may be, it
is evident that when not in flight it is absolutely unconstrained
in any one of its degi-ees of freedom ; it is, in fact, free to move
bodily in any of the three co-ordinate directions of space, or to
rotate about any of the three co-ordinate axes. In any other
locomotive appliance we are accustomed to start with some definite
limitations; thus a road vehicle is essentially constrained to
motions in one plane, and therefore has only three degrees of
freedom (i.e., motion in two co-ordinate directions and rotation
about one of the co-ordinate axes); a railway locomotive is
deprived of all but one of its degrees of freedom, longitudinal
motion alone being permitted.

When the aerodrome or aerodone is in free flight, its stability
depends upon the limitation that is imposed on its freedom by its
functional organs ; it must either be deprived of its undesirable
degrees of freedom altogether, or its motions permitted in these
undesirable degrees must continually react one on another so as
to in effect impose a safe limitation in every case ; the only
kind of continuous motion that-is permissible being translation
in the line of flight. It is the study of these actions and
reactions between the forces and couples in and about the
co-ordinate axes that forms the basis of the author's present
investigations.

The subject may be conveniently approached by a preliminary
discussion of the ballasted aeroplane and other simple forms of
aerodone; the present chapter is principally devoted to this

3 b 2

§ 2 AERODONETICS

discussion, including an account of the author's earlier (1894)
experiments.

§ 3. The Ballasted Aeroplane. A Simple Case. — The simplest
construction of flight model known is that described in §§ 162,
241, eiseq., of the author's " Aerodynamics," under the title of the
Ballasted Aeroplane. If an aeroplane of about the proportions
shown in Fig. 1 be loaded by appropriately disposed ballast, to
bring its centre of gravity to a point somewhere about one-quarter
of its width from the leading edge, it is found to be capable of
gliding flight and is stable both laterally and longitudinally within
certain limits.

For experimental purposes the author has found a mica plate
about 2 inches by 8 inches by about three to four thousandths of

Fig. 1.

an inch thick, gives the most satisfactory results, the ballast
taking the form of a split lead shot secured by closing on to the
leading edge of the mica plate at a point midway between its ends.
The ballasted aeroplane constructed as described is easily upset
by any sudden puff of wind, but in still air, as when experi-
menting inside a building, it may be employed to demonstrate
many important points relating to the equilibrium question.
A ballasted aeroplane may be launched directly by hand, the
mica plate being taken between the finger and thumb and
projected forward with as nearly a parallel motion as possible.
A better way is to employ a launching staff, Fig. 2, which
consists of a straight rod a few feet in length, capped with a
small rectangular platen on which the aeroplane is carried ; the

4

FREE FLIGHT

§4

staff is held in an approximately vertical position with the lower
end about shoulder high, and the launching is effected by giving
a swaying motion to the body.

By experimenting with a model constructed as
iibove described the following facts may be demon-
strated : —

(1) There is a critical velocity and angle at which
if the aeroplane be launched it will continue to
glide indefinitely. These may be termed the natural
velocity a nd na tural gliding angle .

(2) If the aeroplane be launched at other than
its natural velocity and gliding angle, it will perform
a wave-like trajectory, oscillating about the gliding
path of natural velocity, the oscillations gradually
diminishing in amplitude, and the path of the aero-
plane approximating more and more closely to a
uniform glide (Fig. 3).

(8) The natural velocity of a ballasted aeroplane
of given dimensions will depend upon its weight
and upon the position of its centre of gravity, being
greater when the weight is greater and when the
centre of gravity is nearer the front edge.

(4) JSPKat 4here is a particular position of the
centre of gravity that results in a least value of
natural gliding angle, and for planes of the form
stated the least value of the gliding angle is about
8 or 9 degrees.

§ 4. The Ballasted Aeroplane. Longitudinal Sta-
bility. — The subject of the longitudinal stability of the ballasted
aeroplane is dealt with briefly in the author's " Aerodynamics,"
§ 162, from which the following passage is quoted : —

" Let us suppose that the position of the centre of gravity be
such as will coincide with the centre of pressure when the plane
makes an angle = fii with its direction of motion. Now we

5

§ 4 AEROPONETICS

know (§ 184) that the position of the centre of pressure varies as
a function of /3 and that its distance from the front edge of the
plane diminishes the less the angle ; if then the angle /3 from
any accidental cause become less than ft, the centre of pressure
will move forward in advance of the centre of gravity, so that the
forces acting on the plane will form a couple tending to increase
the angle, and so restore the condition of equilibrium. Likewise
if the angle become too great the centre of pressure will recede,
and the resulting couple will tend to diminish the angle, and
again the equilibrium is restored; thus the conditions are those
of stable equilibrium, the plane tends to maintain its proper
inclination to its line of flight."

Fig. 3.

" There is not only equilibrium between the angle of the plane
and its direction of motion as above demonstrated, but also
between the gliding angle and the velocity of flight ; thus if the
velocity is deficient, so that the weight is insufficiently sustained,
the gliding angle and the component of gravity in the line of
flight automatically increase and the aerodrome (or aerodone)
undergoes acceleration. Conversely, if the velocity is excessive
the gliding angle (and so the propulsive component) diminishes,
and the velocity is thereby reduced." *

It would thus appear that there are two distinct kinds of
equilibrium involved in the longitudinal stability of the ballasted

1 The above explanation of the automatic stability of an aerodone or
aerodrome is, in a condensed form, that given by the author in his paper to
the Birmingham Natural History and Philosophical Society in 1894. A
more complete exposition is to be found in the author's specification of
patent 3608 of 1897, p. 6, the text of which is given in Appendix II. of
the present volume.

6

FREE FLIGHT

§5

aeroplane — (1) an equilibrium between the attitude 1 of the plane
and the direction of flight, that is to say, the plane tends to
preserve a predetermined angle to its line of travel ; (2) equili-
brium of a dynamical kind between the kinetic and potential
energy of the aeroplane, by which constancy of gliding angle
and velocity is secured.

It may be stated at once that the same principles apply quite
generally to aerodones of more fully developed type, in fact, so
far as the author is aware, there is no other way by which longi-
tudinal stability can be automatically secured, and any aerodone
to fly successfully must comply with the necessary conditions.

§ 5. The Ballasted Aeroplane. Lateral Stability. — The lateral
stability of the ballasted aeroplane is closely connected with the

Fig. 4.

square-cut form of its extremities ; it is at least certain that an
oval or an elliptical plan form is relatively deficient in this respect.
It is evident that so long as a rectangular plane is travelling
in a direction parallel to its end edges, the sustaining forces are
symmetrical, and there is no turning moment about the axis of
flight. If from any accidental cause, such as a slight motion of
the air, the plane finds itself moving so that its ends make an
angle with the line of flight (Fig. 4), the leading end L will, so

1 The word attitude is here used to denote the position of the aerodone
about a horizontal transverse axis ; it is the analogue of aspect as defining
the position of the plane about the vertical axis.

7

§5

AERODONETICS

to speak, be eating its own up-current, and consequently it will
experience an excess of pressure, and a turning moment will
arise tending to elevate the end L of the plane. When the plane
acting under the influence of this couple has assumed a laterally
inclined position (Fig. 5), the forces W (the weight of the plane)
and F (its pressure reaction) will have a tangential resultant, and
the plane will proceed to slide downhill, that is in the direction
indicated by the arrow ; this motion will result in an alteration
of course which will continue until the direction of flight is again
parallel to the end edges of the plane, when the turning couple
about the axis of flight will vanish.

If the initial disturbing cause be some want of symmetry of
the plane, such as a slight twist or "wind" (a defect to which

Fig. 5.

mica is extremely liable), then the final position of equilibrium
is such that the edges of the plane are slightly diagonal to its
line of flight, the couple due to the want of symmetry being
then balanced by an equal and opposite couple due to the added
reaction on the one or the other end of the plane. It is obvious
that the same causes that result in the plane setting itself
" square " to the line of flight in the case of a symmetrical plane
result in it automatically setting itself at the appropriate angle
in the case of a plane of defective symmetry.

It is evident that the motions of the plane after a disturbance
will not be dead-beat as has been suggested, but that owing to
the moment of inertia about the axis of flight, there will be a
period of oscillation while the plane is settling down to its
position of equilibrium. In practice the moment of inertia is

8

FEEE FLIGHT §6

diminished as much as possible by concentrating the ballast in
the middle of the front edge of the plane, when it is found that
the oscillations die out with great rapidity. If the ballast is
distributed along the front edge, or if the plane itself be of too
great weight in proportion to the total weight of the plane and
ballast, the oscillations do not die away, but tend to increase in
amplitude, so that instability results.

§ 6. Ballasted Aeroplane. Directional Stability. — The directional
stability of an aerodone may be defined as its stability about a
vertical axis. It is evident that if an aerodone were liable to
rotational changes of position about a vertical axis, its stability
in other respects would be impaired ; if, for example, in an
extreme case it were liable to slue completely round in a manner
analogous to the " side slip " of a motor car, it might at any
moment find itself travelling stern foremost, and its longitudinal
stability would have vanished.

Furthermore, we have already seen that lateral stability is
dependent upon freedom of lateral motion, and to some extent
the constancy of angular position (about a vertical axis) has
been assumed in the explanation given (§ 5).

In the ballasted aeroplane the directional stability is primarily
due to the influence of skin friction. It is manifest that if there
were no viscous connection between the plane and the air, if skin
friction were absent in fact, there would be nothing whatever to
restrain the plane from rotation about an axis at right angles to
its surfaces, and any accidental irregularity in its form, such as
a slight " wind " or twist, would result in it receiving a spin of
continually increasing angular velocity. The influence of skin
friction is to damp out any rotary motion that may be accidentally
acquired. This influence is an appreciable factor when a plane
is translationally at rest, but it is far more potent when the plane
is in motion. The reason for this is that the skin friction varies
approximately as the square of the velocity, and the difference
between this quantity for the right and left hand " wings " of

9

§ 6 AERODONETICS

the aeroplane, for a given rate of spin, is greater the greater
the velocity of flight.

If, as suggested in § 5, the mica plate have a slight twist or
wind, the pressure reaction on either the right or left hand
" wing " will be in excess, and, as we have already seen, this
results in the aeroplane setting itself diagonally to the line of
flight, so that the torque about the axis of flight due to the
defect of symmetry is balanced by that due to the end effect.

Under these conditions the permanence or otherwise of direc-
tion depends upon the alignment of the resistance resultant, and

Fig. 6.

the gravity component which forms the propulsive force. Thus
if (Fig. 6) these two forces are not in the same straight line they
constitute a couple, and give rise to a torque about an axis
normal to the aeroplane, and the flight path will not be straight.
The same considerations apply if the initial want of symmetry
is due to other causes, such as an unsymmetrical distribution of
ballast, an unsymmetrical variation of the smoothness, etc., etc.,
the most frequent cause of trouble, however, when mica is
employed, is that stated.

So long as the plane is carefully constructed from a plate of
mica that is a sufficient approximation to a true plane, the
turning moment that arises about a vertical axis is never of

10

FREE FLIGHT §7

sufficient magnitude to be of consequence ; if, however, the mica
plate is defective, the path of flight will be of spiral form, and if
the defect is serious the spiral may become a curve of decreasing
radius and instability may thus result.

The sense of direction of a ballasted aeroplane may be some-
times improved by the addition of a small fin near the front edge,
conveniently attached to the ballast (Fig. 7). With the same end
in view, the front corners of the plane may be "dog-eared"
slightly (Fig. 8), a procedure that is otherwise advantageous.
The simple ballasted plane has no " right side up," if properly

Fig. 7.

Fio. 8.

launched it will fly with either face topmost, but there is some
risk that in the act of launching the reaction will be downwards
instead of upwards, in which case the path of flight is entirely
modified, in a manner that will be explained later in the work.
In the case of the plane with the corners dog-eared (upwards),
this does not apply to the same extent, and the launching may
be effected with greater ease and certainty. Although the above
expedients may sometimes be found advantageous it is the
better plan to reject at the outset mica that is perceptibly on the
twist or otherwise contorted.

§ 7. Ballasted Aeroplane. Interaction of Motions in and about
the co-ordinate Axes. — It is of interest to summarise the inter-

11

>\

§ 7 AERODONETICS

action of motions in the different degrees of freedom (compare § 2)
involved in the maintenance of equilibrium as above expounded.

In the longitudinal stability three kinds of motion are involved
and interact : (1) rotation about a horizontal axis transverse to
the line of flight ; (2) motion of translation in a vertical direc-
tion, and (8) motion or change of motion in the direction of flight
(taken as horizontal for the present purpose). Employing the
usual axis symbols, and calling the direction of (mean) flight x,
and the axis at right angles thereto in a vertical plane y, the
transverse axis will be z ; and the three kinds of motion become
(1) rotation about z ; (2) translation along y, and (8) translation
along x. Thus if we were to take away the other three degrees
of freedom by arranging the flight model to slide between two
parallel vertical planes or walls, we should have the problem of
longitudinal stability left in its entirety and nothing else. This
is always a convenient supposition to make when the question of
longitudinal stability is under discussion.

In the maintenance of lateral stability two degrees of freedom
are primarily involved : (4) rotation about the axis of x, i.e., the
axis of flight, and (5) translation along the axis of z, i.e., the
transverse axis.

The sixth degree of freedom, i.e., rotation about a vertical
axis, primarily concerned in di rection maintenance , is also
closely associated with the question of lateral stability, and it
will be subsequently shown that motion of this kind requires to
be taken into account when dealing with that subject. There
is a peculiar kind of skew instability that is prone to arise if this
point is neglected.

§ 8. Other Forms of Aerodone. — The ballasted aeroplane has
been chosen for the purpose of initial discussion as being the
most elementary form of aerodone known, and as presenting an
object lesson in automatic stability of the simplest possible kind.
Several experimenters have developed gliding models or aerodones
of more elaborate form ; these, though generally inferior to the

12

FREE PLIGHT §9

simple mica aeroplanes above described, are worthy of mention
not only as constituting part of the history of the subject, but
also as forming a fitting introduction to the study of a more
highly developed type employed by the author for the purpose
of experimental study, and subsequently made the object of a
mathematical analysis.

In general the early experimenters appear to have taken their
inspiration from the models provided by nature, the aerodones
employed bearing, in most cases, a strong resemblance to a bird
in gliding flight.

In addition to the examples described in the following section,
a number of partially successful flying models or aerodromes have
been produced, notably by Hargraves (1885), an account of whose
experiments will be found in Engineering 1 ; the author (1894),
whose apparatus is described later in the present work, and
Langley (1896), of which particulars are given in a recent
publication. 9

There have also been a number of successful captive machines
and also a few effective attempts at actual flight; the former are
of but little interest to us from the present point of view, and the
latter (as in the gliding machine of Lilienthal) have been reported
as relying upon the skill of the aeronaut for their equilibrium,
either in whole or in part, and may therefore be dismissed from
the present stage of the discussion.

§ 9. Some Successful Gliding and Flying Models. — Some of the
earliest successful model experiments were made by Penaud about
the year 1870, an account of these is given in U Aeronaut. 3 Penaud
experimented with small paper-winged models, to some of which
he applied twisted rubber and a screw propeller as a means of
propulsion; a model so fitted (Figs. 9 and 10) is stated to
have crossed a pond 40 or 50 yards wide, and to have shown

1 Engineering, vol. xlix., p. 687; and vol. 1., p. 769.
* " Report Smithsonian Institution," 1900.
8 U Aeronaut, tome v., p. 2, 1872.

13

§9

AERODONETICS

complete stability. It would thus appear, unless some previous
records exist, that to M. Penaud belongs the credit of pro-
ducing the first stable flight model of a type that may be
regarded as an embryo flying machine. It is unfortunate that
the particulars given are not sufficient to permit of the exact
reproduction of the model and repetition of the experiments;
there is, however, no reason to question the accuracy of the
facts stated. Penaud also calls attention to the flight path

Fig. 9.

oscillation already referred to in connection with the ballasted
aeroplane. 1

Fig. 11 is a representation of a form of gliding model devised
by Mons. J. Pline, and figured in Marey's " Vol des Oiseaux " ;
this model appears to have been stable in flight, but from an
aerodynamic standpoint it is not a form of high efficiency, the

1 Compare § 65. Penaud says: . . . " et Ton observe alors assez sou vent
des oscillations dans le vol, comme nous en voyons decrire auz paasereaux
et principalement au pic- vert " (green woodpecker).

14

o

t-

ft
<

P-t

15

§9

AERODONETICS

gliding angle appears to have been excessive. According to the
particulars given the model consists of a folded sheet of paper
ballasted by a needle that may be adjusted as required, but
the one view given by Marey leaves the exact form in doubt. 1
Fig. 12 is a form due to Weiss, and derived from experiments
with models shaped like a bird; it shows complete stability,
and is prone to oscillate in its gliding path just as in the

Fig. II. 1

case of the ballasted aeroplane. In construction it consists
of a sheet of paper cut to the form shown, the front being
folded to give some stiffness to the leading edge, and ballasted
by a small piece of lead glued in position. The tips of the
" wings " are turned up slightly in the manner shown in the

1 Fig. 11 is from a model made by the author from Professor Marey's
description; a measurement of the gliding angle gave approximately 20
degrees.

16

FBEE FLIGHT

§10

line-of-flight elevation. The gliding angle of this model is good,
being approximately 9°.

Fig. 19 is a form due to Hele Shaw, and exhibited by him to
the Royal Society in May, 1907 ; it also obviously derives its
inspiration from nature, being a close representation of the
plan-form of a swallow ; it is quite stable, and also shows the
flight path oscillation to a pronounced degree ; this glider is
simply cut from a sheet of paper and ballasted by means of
split shot. The gliding angle of this form is also very good.

As far as the author has been able to judge, none of the forms
above given are superior to the simple ballasted rectangular plane,

- A- - n/

Fig. 12.

and the only obvious reason for the adoption of fancy shapes,
such as those of Weiss and Hele Shaw, appears to be found in
their common origin as imitations of forms presented by nature.

§ 10. Author's Experiments, 1894.— In 1894 the author com-
menced the practical investigation of stability in flight, some
preliminary theoretical work having shown that it should be
possible to obtain automatic equilibrium with a suitably designed
apparatus.

At the date in question, although Mouillard's " L'Empire de
l'Air" had been published several years, the author was not
aware of the automatic equilibrium possessed by the ballasted

a. 17 c

§10

AERODONETICS

30

6
M

18

FREE FLIGHT §11

aeroplane, or, indeed, of the existence of Moaillard's work
at all, and it is not surprising, therefore, that his models were
designed and constructed on entirely different lines.

In the author's original model (Figs. 14 to 20), the various
functions that are performed incidentally by the ballasted aero-
plane in the maintenance of equilibrium are carried on by
special organs, and this fact, which arises from the theoretical
origin of the appliance, renders it particularly well suited to
analytical study.

One of the deductions from the author's preliminary investi-
gations was in effect that the higher the velocity of flight the
greater the stability attainable ; the models employed were, in
pursuance of this conclusion, designed for velocities of some
40 miles per hour and upwards. It was estimated that such a
velocity would render the stability independent of any gusts of
wind such as would ordinarily be encountered, a result that was
fully substantiated by the subsequent experiments.

Owing to the high velocities employed the launching had to
be effected by means of a catapult, the construction of which is
illustrated and described in § 12 (Fig. 21).

§ 11. Author's Experiments (continued). The Aerodone. — The
form of aerodone employed by the author in these experiments
is drawn to ^ sc&le in Figs. 14, 15 and 16, and a photograph
reproduction is given in Fig. 17. Further details are given in
Figs. 18, 19 and 20.

The aerofoil sections D, E, F, O (Fig. 19) being of elliptical

plan-form 40 in. by 8 in., and the aspect ratio, n being lS a 8, the

actual area = '65 sq. ft. On the basis of a parabolic grading, 1

2 I*
the effective area, given by the expression ^ — , = '55 sq. ft.

The centre of gravity of the aerodone is arranged about 1 inch
from the front edge of the aerofoil, the latter being the unique
organ of sustentation.

1 "Aerial Flight," Vol. I., Aerodynamics, § 192.

19 C 2

»o

o

M

ft

I*

§11

AERODONETICS

«— 4

6

22

FREE FLIGHT

§U

The tail-plane is of triangular form, and has an area =
■20 sq. ft., its function is purely directive. The tail-plane has
the same function as is performed by the change in the position
of the centre of pressure of the ballasted aeroplane ; it preserves

1

Fio. 18.
the constant attitude of the aerodone in relation to ils line of
Right, in this respect its action being analogous to the feathers
on an arrow.

The distance between the centres of pressure of the aerofoil
and tail-plane is 8"9 feet approximate];.

Two fins, sections H and J (Fig. 20), are provided, of the

Fio. 19.
form shown in Fig. 15, whose area is approximately *2 sq. ft.
and '3 sq. ft. respectively. These have for their function the
maintenance both of lateral and directional equilibrium, and
their action is represented in the ballasted aeroplane by the

§ 11 AERODONETICS

end edge effect on the one hand, and the frictional equilibrium
on the other, assisted it may be by an auxiliary fin, or by
the upturned corners, as described in § 6.

The backbone (Fig. 18) is of triangular section, and serves for
the attachment of the various functional parts, also supporting
the ballast, by which the centre of gravity is brought to its
correct position.

The construction adopted was of a most substantial descrip-
tion, in no way suggestive of the lightness it is customary to
associate with a flying model, the backbone, tail-plane and
fins being of white pine, and the aerofoil of thoroughly seasoned
and dried pitch pine x ; the ballast being made from a strip of

H

Fig. 20.

sheet lead, wound spirally round the backbone, and nailed or
otherwise secured in position.

A small cross-piece serves the purpose of supporting the nose
of the aerodone on the runners of the catapult during launching.

§ 12. The Author's Experiments (continued). The Catapult. — The
catapult employed for launching the aerodone is illustrated in
Fig. 21 ; briefly this apparatus consists of two runners, on which
the aerodone is free to slide, separated by distance pieces
arranged at intervals along its length, leaving a groove about
1 inch in width by which the backbone of the aerodone is guided

1 Pitch pine is not a particularly suitable kind of timber, owing to its being
liable to develop a twist. Mahogany would be better in this respect, bnt it
is more easily injured by impact.

24

L.

t.

r
i

rl

r

L.

r*
L.

r

L

L

%

O

§ 12 AERODONETICS

during discharge. The " motive power " is furnished by an india-
rubber band, usually composed of about four strands of black
" elastic."

The india-rubber is secured to a projecting piece and is guided
by two pulleys or spools, one on each side of the runners. The
rubber is bound with twine at and about its middle point where
it engages with the notch in the after extremity of the backbone
of the aerodone. When loaded, the rubber is held in a state of
extension by notches near the rear end of the runners, and the
discharge is effected by prising it out of the said notches by
means of a forked lever.

The length of the free portion of the rubber ordinarily used
was 8 feet 4 inches, the weight being \ lb. and the maximum
range of extension employed 11 feet 6 inches.

A plotting of the extension diagram is given in Fig. 22, in
which abscissae represent extent of elongation and ordinates the
corresponding tensions in pounds. The figures given are loading
data, and the loading energy will be given by the area of the
curve for the extension employed. It is improbable that the
output energy is more than about 75 per cent, of the total.

§ 13. Site and Date of Experiments. — The site of the experiments
under discussion was the then residence of the author, " Fairview,"
St. Bernard's Road, Olton, Warwickshire ; the models being pro-
jected from a back first floor window facing west ; the point of
discharge was approximately 15 feet above the ground level,
where the general run of the land falls away at a slope of about
1 in 25.

The situation is represented diagrammatically in Figs. 25
and 26, in which are depicted some of the flights made. A
plan is given in Fig. 28 showing the general situation, on this
also some of the flights have been roughly laid out.

Experiments were made at different dates during June and
July, 1894, the models employed being in most part simple
aerodones as described in § 11, some half-dozen in all being made

26

§ 18 AEEODONETICS

4

si

V

3

o

6

ft

*

1

>

*

t

^

\

1 X

i

■ i

\

i

6

PH

§ 13 AERODONETICS

to the one design, and weighing approximately 1 lb. 7 oz. each.
Subsequently a more fully developed model or aerodone (Fig. 24),
elastic propelled with twin screws, was successfully flown. This
model is fully described in Appendix III.

§ 14. The Author's Experiments. Records. — A considerable
number of flights were made, probably some twenty or thirty
in all ; the records relate only to some half-dozen cases.

The object at the time being merely to demonstrate the stability
of a high velocity model, the records are not very complete, and
the actual velocities and times of flight were not in every case
fully noted. The forms of flight path given in the accompanying
figures are drawn as they appeared to observers present, and

*omo

J J

^^^^^^^^^^5^5^^^;

9 YAHM
i

Fig. 27.

cannot be regarded as more than rough approximations to the
actual curves ; the positions of the points of greatest altitude
are, however, fairly near the truth.

Referring to Fig. 28 (plan of site), we have Flight No. 1, date
not recorded, weight of aerodone entered as 1| lbs. ; very light
wind ; distance about 200 yards. This is the initial flight made
with first model.

Flight No. 2, June 24th (?), weight 1 lb. 7 oz. ; distance 280 or
290 yards; high wind with powerful gusts, probably about
80 m.h. direction W.S.W. ; time of flight 27 seconds. A magni-
ficent flight, remarkable " switch-back " flight path (Fig. 25),
distance, relative to ivind, probably over 600 yards.

Flight No. 8, June 24th (?), same aerodone as Flight No. 2 ;
distance 200 yards ; time of flight 1\ seconds ; very light wind.
Velocity = 55 m.h. (Fig. 26).

80

FREE FLIGHT §15

Flight No. 4, same as No. 3, distance 200 yards ; time not
recorded (Fig. 26).

Flight No. 5, June 24th (?), same aerodone as Flight No. 2;
distance 150 yards ; x time of flight 5£ seconds ; very light wind.
Velocity = 56 m.h.

Flight No. 6, July 3rd, twin screw aerodrome ; weight 2^ lbs. ;
distance 188 yards ; no appreciable wind ; time of flight 4 J
seconds 2 (Fig. 27).

§ 15. The Author's Experiments. Discussion. — The result of the
experiments described was to fully confirm the author's views as
to the possibility of securing automatic stability without the
employment of any equilibrium mechanism or " brain equivalent."
When the weather was calm the aerodones carved their way
through the air as if running on invisible metals, without the
smallest visible fluctuation or quiver. When on the other hand
the flight was made in a gale of wind, the flight path took the
form of a bold sweeping sinuous curve, without a momentary
suggestion of loss of equilibrium, but rather with the appearance
of some set and intelligent purpose.

In general the velocities of flight employed appeared to range
from about 50 to 60 miles per hour, and it is certain that at
velocities of this sort there is very little to fear from any
ordinarily bad weather conditions.

In general the launching velocity was very much higher than
the natural velocity, and consequently the gliding angle was
entirely masked and remained an unknown quantity ; the con-
siderable range of flight obtained was without doubt due to this
cause.

One remarkable fact in connection with the above series of
experiments is the exceptional time (27 seconds) the aerodone
remained in flight when launched in a gale of wind (Flight No. 2).
Taking the velocity relatively to the air as that measured in

1 Model collided with tree.

1 The accuracy of the timing in this flight is in doubt.

31

§ 15 AERODONETICS

other experiments with the same model, say 55 m.h., this length
of time corresponds to a flight of 725 yards.

There are two possible explanations of this anomalous flight.
(1) That the aerodone was actually soaring after the manner of
an albatros or gull. (2) That the initial kinetic energy, being
due to the velocity of the aerodone relatively to the wind, is much
greater when the aerodone is launched in the teeth of a gale,
and so the energy disposable in the flight will be greater in a
proportionate degree.

Both these explanations are possible, and either would, within
limits, account for an enormous increase in the observed range
of flight. At first sight alternative (1) seems most unlikely ; it
appears to imply an exercise of intelligence not possible for an
inanimate thing. Later experiments, 1 however, show that it is
by no means impossible for a simple aerodone to perform true
soaring evolutions, and that it is not at all improbable that this
was actually the case in the flight in question. On the other
hand alternative (2) alone is almost sufficient to account for the
additional energy as evidenced, so that the correct explanation
of the case in point must be regarded as uncertain.

§ 16. Author's Experiments. Remarks and Summary. — If we
define the line of flight of an aerodone as the path of its mass
centre, it is one feature of the designs employed by the author,
and in the models discussed in § 9, that the centre of resistance
is substantially coincident with the line of flight; the same is
approximately true of the soaring birds.

Attention is directed to this point on account of the fact that
many inventors have proposed acentric types of aerodrome, in
which the weights carried have been suspended some distance
beneath the supporting member ; this is typically the case in the
captive machine of Sir Hiram Maxim, a diagrammatic illustration 2

1 Compare Ch. IX. t § 158.

2 Fig. 28 is taken from a Patent Specification. It must not be supposed
that Sir Hiram Maxim actually constructed a machine in which the aeroplane
was trussed in the manner shown.

32

FREE FLIGHT

§16

of which is given in Fig. 28. In all probability this idea is
based on the superficial resemblance an aerodrome possesses to a
kite, the suspended weight being considered as the analogue
of the kite string ; or, perhaps it is from endeavouring to follow
the lines of an aerostat that this type has arisen, the aeroplane
or aerofoil being regarded as a simple substitute for the gas
bag.

The relative advantage of the centric and acentric types of
aerodone is a question that has not been thrashed out ; it is

Fio. 28.

probable that the latter without possessing any advantages
whatever, possesses certain grave disadvantages which will be
better understood in the light of the subsequent investigations.
The acentric arrangement also results in some considerable com-
plication from a theoretical standpoint apart from its demonstrable
failings.

The acentric type is not found in nature to any marked
degree, except perhaps in the case of some birds that are
inveterate rameurs, 1 when the gliding position is sometimes with

1 There is do convenient English equivalent for this word in the present
sense ; the term " wing-flappers" perhaps might be used.

A. 83 D

§ 16 AERODONETICS

the wings inclined upward at an angle instead of being fully
extended ; the pigeon is an example.

§ 17. Author's Experiments. Remarks and Summary (continued).
— It is of some interest to interpret the forms of flight path made
by the author's aerodones, in typical cases, in the light of the
ballasted aeroplane experiments described in § 8.

It is evident that the rising and falling curve of the trajectory
Fig. 25, and in other similar cases, constitutes a portion of the
wave-like flight path referred to in (2) § 3 ; and if we continue this
curve as though the launching had taken place from a greater
altitude, that is to say, as if the surface of the earth had not
intervened, we should have a curve of the form shown dotted in
Fig. 29, oscillating about a mean gliding path represented by the
line a a. In this figure the curve of flight has also been continued
retrospectively in order to show more clearly the sinuous flight
curve of which the actual trajectory forms but a fragment. It is
evident that the angle of the line of mean flight depends upon
the design of the aerodone, the less the total resistance of the
latter in proportion to its weight, the less the angle y : this is a
matter of aerodynamics. 1 It is further evident that the greater
the launching velocity the greater the distance at which the line
of mean flight path passes vertically over the point of discharge,
i.e., the greater the distance h in the figure.

In the particular flight depicted in Fig. 25, it is evident that,
owing to the existence of a head wind at the time of the experi-
ment, the velocity was considerably higher in effect than the
ordinary launching velocity; it would in fact under these circum-
stances be equal to the sum of the launching and wind velocities.
In consequence, the height of the mean flight path line is con-
siderably greater in this case as is represented in Fig. 80, and
as a result we have nearly three complete " periods " of the
flight path oscillation before the aerodone finally comes to earth.
The question of whether there was actual soaring taking place

1 Compare " Aerial Flight," Vol. I., Aerodynamics , Chapters VII. and VIII.

34

FREE FLIGHT

§17

■

o

<"/•"*

85

d 2

§ 17 AERODONETICS

during this flight, the alternative explanation of its abnormal
duration, is not for the moment under discussion.

The examination of the behaviour of an aerodone in flight has
in the foregoing demonstration been carried as far as has been
found possible without a mathematical analysis ; it now becomes
necessary to adopt a more rigid line of treatment and to examine
quantitatively the curves of flight and the conditions governing
their form and permanence.

86

CHAPTER II

THE PHUGOID 1 THEORY. THE EQUATIONS OF THE FLIGHT PATH

§ 18. Introductory. — The Phugoid theory deals with the longi-
tudinal stability, and the form and equations of the flight path
of an aerodone.

It has been proved by the experiments of the author and others
that a simple aerodone can be constructed possessing, within
certain practical limits, complete longitudinal stability, and the
general considerations on which this stability depends have been
stated. 2

We have seen firstly that there exists a stable state of equi-
librium between the attitude of the aerodone and the direction of
flight path in the vertical plane ; and, further, that there is a
complex system of equilibrium of a stable kind maintained
between the said direction of the flight path and the velocity of
flight, due to an interchange of kinetic and potential energy that
automatically takes place if the conditions of uniform gliding are
disturbed.

In the investigations that follow the character and form of
the flight path of an aerodone in free flight are deduced from
purely dynamical considerations, the argument being in the
main mathematical, aided by graphic methods, and reasoning of
other kinds. Constituting the foundation is a mathematical
analysis based on hypothetical conditions, the problem being
presented first in its simplest form ; the superstructure comprises
an extension of the initial theory to deal with the problem in its
more complete form, and the interpretation and application of the
results achieved.

1 From <t>uyTj and c«3os (lit. fliyhl-lih). 2 § 4.

37

§19

AERODONETICS

§ 19. Initial Hypothesis. — The first hypothetical condition
assumed is that the aerodone is only possessed of the three
degrees of freedom involved in its longitudinal stability. Thus
it is supposed free to move in any direction in a vertical plane
containing the line of flight or to rotate about an axis at right
angles thereto.

The above condition we may suppose fulfilled either by

X

i

I

Fig. 31.

imagining the aerodone to be constrained in its motion by two
parallel vertical guide planes (as suggested in § 7) or we may
suppose that lateral stability and constancy of direction are
otherwise accounted for, and maintained.

In the second place it will be assumed that the aerodone loses
no energy during its flight, either by considering the air as
frictionless and the supporting wave 1 as perfectly conserved, or
by supposing a force applied in the direction of motion precisely

1 " Aerial Flight," Vol. I., Aerodynamics, § 117.

38

THE PHUGOID THEORY §20

equal, at every instant of time, to the resistance experienced by
the aerodone in flight.

It is in the third place assumed that the whole mass of the
aerodone is concentrated at the mass centre, so that it shall
possess no moment of inertia about a transverse axis.

In the fourth place and lastly it is assumed that the size of
the aerodone is small in proportion to the minimum radius of
curvature of its flight path.

§ 20. The Phngoid Equation. — In Fig. 31, let pp be any portion
of the flight path of an aerodone, and let 7/ be the height of free
fall corresponding to the velocity of the aerodone at the point q ;
then the velocity at every point in the path of the aerodone will
be that corresponding to the ordinate measured from X y since
no energy is lost. Let this ordinate be reckoned positive
measured in a downward direction.
Let us take : —

II = distance of aerodone below horizontal datum line O X.
^r^lY?i F = the force per unit mass of the aerodone, due to the
^x?Z\kJt% reaction of the air on its supporting member

***Y * "f i / {always of positive sign).

7 • b?* m \sf = ^ ,e f° rce ^ ue I* ^ ne i nei *tia of unit mass of the

^ i& \ jL aerodone owing to its change of direction; that is

\^b*4r t the centrifugal force due to curvature of flight

path; measured as positive when in the same
direction as F and negative when contrary to F.
W = weight of aerodone per unit mass (= g).
V = velocity of aerodone in flight.
L = length of path.
= angle of path to horizon.
t = time of flight.
Between these quantities we have the following relations : —

where n is a constant.

' "Aerial Flight," Vol. I., Aerodynamics, § 159, also Chap. VIII.

89

§ 20 . AERODONETICS

W-t* r *

f = f r. (2)

Tlie total force at right angles to the path {i.e. the sum of F
and J ) must equal the resolved part of the weight, so that

F +/= Wcos = gcos® (3)

.\ gcoA@ = nX2gII + -®-V, (4)

but

dt =

(IL
V

•
• •

d& v ■

dt '

and

since

sin

© =

JdH
T dV

dh V ~ 2 9 J/ dlJ

F 2 = 2 g II '-£, (5)

./_ and since sin = y- ,. , dL = -f- -

sin

, d<d d& sin TnI cos ....

1 .'. (4) becomes <j cos = 2 // // ( « / ''7^^ ) (7)

rf COS CO* „ ....

~ ■ -37T = 27T - «• ^ (8)

This is the differential equation to the curve of flight in terms
of H, 0, and n.

Integrating the solution is —

cos Q = ^u7f+^ ^ (9)

where C is a constant.

This is the general equation to the curves of fliglit or the
jJmgoid equation.

§ 21. Substitution for the Constant n. — The value of the con-
stant n is defined by (1) —

F = n P = n X 2 g H.

Now, we know that there is a certain value of V at which the
weight of the aerodone is exactly balanced by its pressure
reaction, this is the natural rehcity of § 3. Let this natural

40

THE PHUGOID THEORY §22

v elocity be denoted by tlie symbol V n , and let the corresponding
value of H (i.e. the height of fall that would give rise to a
velocity = V H ) be denoted by the symbol Z/ n and termed the
natural height Under the conditions of hypothesis, if the
aerodone be launched horizontally with a velocity = V n , its path
will continue indefinitely as a horizontal straight line, and in this
case we have F = W = g and (1) becomes —

g = 2g H n n, or n = ,— . (10)

2//

n

Equation (9) then becomes —

~ e = sir. + v-,r < u >

The author has not been able to reduce this expression to a
form suitable for co-ordinate plotting, but the difficulty of laying
out the curves for different values of the constant C is sur-
mounted in an indirect way. 1 In the first place we will find an
expression for the radius of curvature, and examine a few special
cases.

§ 22. Radius of Curvature. — We may express the radius of
curvature r for any curve in the form : —

_ dL
r ~~ d&

Now by (10) n = -^ = -^ (for V* = 2 g 1I H by the

» r n

ordinary law? of a falling body),
and by (4) and (5)

g cos = 2 g 11 ( n + -^ J

1 At the time of making the plottings of the curves of flight or phugindt
given in the present volume (1897), the author imagined the method adopted,
described in the subsequent chapter, to be entirely new. It would appear,
however, to have been no more than an application of a method originated
by Professor C. V. Boys in 1893 to facilitate the plotting of the generating
curve of a capillary surface. Vide "Phil. Mag." vol. xxxvi., p. 75, " On
the Drawing of Curves by their Curvature."

41

§ 22 AERODONETICS

substituting for n

or

C08

2>r =

r 2

+ dL

d&

cos

<l

' H

2 ro* — 2/7

dL '

2/7

2 // i v

V n

= 2 j H H

r :

2 7/

cos —

77'

but IV = 2 « // w /. r = " '— T> - , (12)

H C

and since by (11) cos = -- ^ -f- ,

«* -"it V 7/

_ 2 7/ — 1 ] J

r " 77 C __ 77 ~~ JC __ 2 // '

8H n + ^H H* V'JI 3i/ n

r = ~C ^ 2 • < 18 >

II s/H 3 7/ M

Thus in addition to the equation for the slope of the curve at
every point (11), we have now the expression for the radius of
curvature. With these two equations we are in a position to
investigate the general characteristics and particular forms of
the curves of flight or Phugoids, as they may be appropriately
termed.

We may note firstly, that the value of 77 n is a constant for
any given aerodone, and secondly, that for any aerodone whose
II n is known, and for which simultaneous values of and 77 are
given, the constant C may be calculated, and the course of flight
is determined. Thus if 77i be the value of 77 when 0i is the
value of we have, by equation (11),

C = V// 1 (coa0 1 - 3 ^-).

(14)

Before proceeding to apply the equations generally to the
plotting of curves for arbitrary values, we may consider three
special cases in which the results are of particular interest.

42

THE PHUGOID THEORY

§22

CO

o

4a

§ 23 AERODONETICS

§ 23. Some Special Cases of the Phugoid Curve. — (1) Let an
aerodone be dropped from rest, head foremost, then, II\ = 0,
0! = 90° or cos 0i = 0.

Whence by either (11) or (14) C = 0. (Note: the value of

is in this case immaterial.)

And by (13)

2

r =

= - 8 II

H>

that is to say, r is constant ; the path of the aerodone is semi-
circular, with a radius three times the natural height (Fig. 32).

It is worthy of remark that in this special case the curve is
discontinuous, and it is necessary to suppose that the aerodone is
artificially inverted on its arrival at a cusp before it can proceed
to describe another semicircle. The fact that the radius is of
minus sign denotes that the centre of curvature is on the rare-
faction side of the aerofoil ; th* results from the convention
adopted in respect of F and /.

Case (2).— Let = 0, then cos = 1, and by (12) we have

2 Ih 2 Ih 2

2glh- Ih" 1 1"

F n 2 H n Ih H n

Under these conditions, if Hi = H n the radius becomes
infinite, and the path straight ; this is the condition of uniform
gliding. If Hi be less than II n the radius is of positive sign,
which signifies that the centre of curvature is on the compression
side of the aerofoil. If Hi be greater than II n , r becomes of
minus sign, and the centre of curvature is on the rarefaction
side of the aerofoil.

Case (3). — In the preceding case cos 0i was taken as = 1.
We will now suppose the aerodone to be travelling in the opposite
direction, so that cos becomes = — 1 ; w r e have

2 Hi 2

r =

-i-»" -2+ x

44

THE PHUGOID THEORY §23

The radius of curvature is now negative for all values of Hi. The
signification of this fact will be better understood in the light of
the subsequent chapter when the actual curves have been
plotted.

It may be noted that in order that cos should be = — 1
the aerodone must be launched upside down, the same "hand "
must always be kept towards the supposed observer. If the
aerodone were launched in the reverse direction the right nay up
this latter condition would be infringed.

45

CHAPTER III

THE PHUGOID THEORY. THE FLIGHT PATH PLOTTED

§24. Preliminary Considerations. — The form of equations (11)
and (13) is such as to indicate that so long as a value of C is
chosen proportional to the square root of ll n (the natural heujht),
the form of curves for all values of H n will be geemetrically similar,
though of different linear scale. This follows from the form of
the equation and from dimensional theory, for the constant C
is of the dimensions of the square root of a linear quantity. It
is thus of no consequence what particular value of H n be chosen
for the purpose of plotting the curves ; a series of curves plotted
for any one value of H n applies equally to any other value if
read to an appropriate scale.

It has already been pointed out that the phugoid equation does
not lend itself to plotting in the ordinary way ; the form of the

TT Q

expression cos = ■ __ -\ — j=~ is one that, so far as the author

8 JI n V 11

is aware, is not susceptible of being reduced to co-ordinate form.
It is consequently necessary to employ some other method of
plotting, thus taking the equation for the radius of curvature (13),

. __ 2

and laying the curve off step by step by means of a trammel the
difficulty is overcome.

§ 25. Plotting the Curves. The Trammel. — Having selected any
convenient value 1 for H n > or employing that proper to some

1 H n =61 in tho plottings given. There is no real reason why this par-
ticular value of JI n should have been chosen ; it could equally have been
made = 1.

46

THE FLIGHT PATH PLOTTED §25

particular experimental data, first take any simultaneous known
values of II and for the curve it is desired to plot. These two
values may be those given by the launching data, since at the
instant of launching the velocity and angle may be presumed
to be known.

Next calculate C from equation (14),

Hi

c = V-i/ 1 („0 1 -^)

where H\ and 0i are the simultaneous known values as above.
Now calculate r from equation (13) thus : —

2

and repeat this calculation for a series of small increments of //,
which must be set out as horizontal lines on the chart.

A trammel is next prepared (Fig. 83), on which these calcu-
lated radii are marked off from a fixed tracing point, the
extremity of each radius, as marked off, is pierced and indexed
with the value of II to which it relates. Thus in Fig. 33 the
fixed tracing point is that marked with a cross at the left-hand
end of the trammel, and the distances measured off to the points
marked 100, 150, etc., are the calculated values for the radius
of curvature for the values of H = 100 feet, 150 feet, etc.,
respectively.

A numerical example of the manner in which a trammel is
prepared for any particular curve is as follows : —

Data, Il n = 64, Hi = 36, cos 0! = — 1.

Calculation of constant : —

Hi

C = VIT^ ( cos 0! - -£± j=6x- 1 h = - 7-125.

Radius of curvature : —

2 2

2 - 7-125 _ ^

H V U 3 U n II v '11

47

§25

AERODONETICS

346
300

Z50

200

ISO

roo

CO

6

CM

48

THE FLIGHT PATH PLOTTED

§26

Solving this for progressive values of H we have

If.

// s Hh.

7-125

— ' L— 01042.
" si 11

36

216

•08300

—•04342

88

284

•03045

-•04087

40

258

•02814

—•03856

42

272

•02620

-•08662

44

292

02442

—•03484

etc.

etc.

etc.

etc.

r.

46-1
48-9
51'9
54-6
57-4
etc.

Fig. 34 is the trammel plotted from the above data.

NV10.

<0| o

12 £ © © e © e © 9 p *>!

I* "> 4 »* oc 9 o £ ^JQ55|

1 I I I I I I I 1 I I I I I I I I I l I l I I I i i ilniil I

cos® = -1 H = 36 H n = 64.

Fig. 34.

Transparent celluloid about *}% inch thick is a suitable
material for making these trammels.

§26. Plotting the Curves. The Use of the Trammel. — The
employment of the trammel in the plotting of the curves is
illustrated in Fig. 85, in which the trammel shown in Fig. 83
is being used to set out or plot the curve to which it relates.

The tracing point is first set to the datum value of 11 (Hi =
100 in the present example), and the radius line on the trammel
is set in a direction at right angles to the flight path (the flight
path is horizontal, that is, 0i = 180° in the example chosen,
the aerodone being inverted), in the case given this is vertical.
The trammel is now transfixed by a needle at the radius point
corresponding to the value of II, that is, where marked 100, and
the appropriate element of the curve is marked off. The needle
is then moved to the adjacent radius point, and another element
of the curve is drawn, and so on. In Fig. 35 the process of

49 e

A.

§26

AERODONETICS

plotting is shown in operation, and Fig. 36 gives the form of the
completed curve in the case of the numerical example, of which
the trammel is given in Fig. 34*

The method above described is one of astonishing accuracy;
the curves, such as that plotted, in which, when the aerodone is
at the highest point in the flight path, cos = — 1, may be
easily laid off with an error of not more than one part in 1,000
of their linear measure. It is in fact a question whether the
process is not really more convenient than co-ordinate plotting.

Fig. 35.

Curves such as that plotted, in which cos passes through a
value = — 1, may not inaptly be termed tumbler muxes, from
their resemblance to the flight of a tumbler pigeon. These
tumbler curves are easy to plot owing to the fact that the radius
r never changes sign. There is another type of curve which
results from the equation, but with a different value of the
constant C, in which there is a point of inflection at which the
radius changes sign ; at this point the value of r runs through
infinity. A curve of this type is shown plotted in Fig. 37.

50

THE FLIGHT PATH PLOTTED

§27

It is evident that curves of the inflected type cannot be plotted
continuously, in fact in practice it is not possible to approach
the point of inflection within less than a certain distance owing

Fig. 36.

to the unwieldy length of the trammel required. A modification
of the procedure becomes necessary.

§ 27. Plotting the Inflected Carve. — The first step to be taken
in the plotting of an inflected curve is to calculate its limiting
values of //, in order that the two parts of the curve in which r
is plus and minus respectively may be independently plotted.
This done, two trammels are made for the respective portions of
the curve of as great length as found workable, and the two
portions are then plotted independently to within as short a

51 e 2

I-

6

H
tm

•wfH

Z

rir

OS

6

««

- M«

n

m

THE FLIGHT PATH PLOTTED §28

distance from the point of inflection as possible, Fig. 38, (a) and
(c). Next the angle, that is, the value of 0, at the point of
inflection is calculated, and this angle is laid off on the chart,
Fig. 88 (6). Finally the three parts of the curve (a), (b), and
(c), are adjusted in their relative positions, as nearly as the eye
can appreciate, and continuity is given to the three components
by drawing a smooth curve (Fig. 87).

It may be remarked that as a means of drawing the actual
form of the curve with a sufficient degree of approximation to
serve every useful purpose, the above method is quite satisfactory.
From the theoretical point of view, however, it would be better
if we could find some means of actually plotting the missing
portions of the curve round about the point of inflection. It has
been suggested to the author that it may be found possible to
deduce an approximate co-ordinate expression for this portion of
the curve, and to employ the same to fill in the gap left by the
radius method ; the matter however is scarcely of sufficient
importance to justify the additional mathematical work.

§ 28. Plotting the Inflected Curve. Example. — The following
example, plotted in Fig. 87, to which reference has already been
made, illustrates the manner in which the plotting data are
calculated.

Example.

II n = 64 Jh = 16 cos 0j = 1.

By (14),

C = V-Hl (cose - *-) =4x(l-^) = 3-666.

Substituting for C in equation (14), also for cos for highest
and lowest points, that is, = 0, cos = 1, we have

8-666

-"*(>-£)

If , 8-666 ,

or iiw + v //" = lf

53

§28

AERODONETICS

the solution of this equation 1 gives two values of Hi —
II = 16, highest point on curve (already known).
H = 129*9, lowest point on curve (value required).
We now require the inflection data, i.e., the values of II and
at the point of inflection. Let us denote these values by the
symbols Hi and ®i respectively. We have

2

r =

2

= x

ii t V 1^ 8 ii n

(15)

(16)

*• HiVHi 8H n
Further, by equation (11)

008 < = 374 + 771;

Thus in the present example,

H< = ( ™™™ )» = 49-8

and

~ 49-8 , 8*666 _„
cos 0i = _ + _ = -779,

whence 0< = 38° 50'.

The data for the laying off of the two scales on the trammel,
Fig. 89, are as follows : —

//.

ir *Jli.

3 *(WW 1

3-600

jrr - "01042.

r.

16
18
20
22-5

64
74-2
89-4
etc.

05740 i
04950 1
0411

04698
08908
•03068

42*55
51-18
65*24

and

L299

120

110

105

100

1480
1315
1158
1076
etc.

00247
•002788
00318
008415
etc.

— 00795

— 007632

— 00724

— 007005

251-8

262

276-0

285-5

1 See Appendix,

IV.

•

54

J

THE FLIGHT PATH PLOTTED §29

It will be understood from the preceding section that the two
portions of the curve are plotted separately by means of the
trammels as above, beginning at the highest and lowest points
respectively and continuing until they can be carried no further,
when their path angles' will mutually approach the inflection
value ®i (Fig. 88).

§ 29. Phugoids of Small Amplitude. — In the case of the flight
path in which the launching velocity of the aerodone differs but
little from the natural velocity (0i being zero), the curves
become so flat that the foregoing method of plotting must be
considered impracticable; for such cases some other process
must be devised. We will investigate the conditions that obtain
when the variations in are very small, that is to say when
cos only differs from unity by a small quantity of the second
order.

Starting from the initial hypothesis we have by (1)
F = 2g Hn where by (10) n = irir

n n

When H = H n we have as before (§ 21), F = g ; let H differ
from H H by a small amount = A, so that H = II H + K and let
F become F + /, that is, g + /. Then,

g+f^FXyjzsgX. — g— -9 + w

n -"•* -"»

or /■=■—. or t~ = -tf~ which is constant.
H% h H n9

Hence /varies as h, that is to say, the force varies with the
displacement. Therefore for small amplitude the vertical com-
jwnent of the motion of the aerodone is harmonic. The time period
of this harmonic motion is given by the equation,

t = 2 * V^ (17}

55

§ 29 AERODONETICS

Now, dealing with the horizontal component of the disturbance,
that is, with the excess or deficit of velocity above or below the
mean, let V n + v be the velocity of the aerodone when its
ordinate is H n + //, then since

r 2 = 2 g H

(J » + vf = 2g (H n + h) (a)

but TV = 2g IT n (/>)

subtracting (b) from (a) we have

2V n v + i* = 2gh,

but for small amplitude r 2 is negligible, hence

V n v = gh (c)

or j = v , which is constant, therefore the excess or deficiency

of horizontal velocity v is at all times proportional to the vertical
displacement h, and it at once follows that, since the latter has
been proved harmonic, the velocity variation, or superposed
motion in the line of flight, is also harmonic.

§ 30. Phugoids of Small Amplitude. Form of Orbit. — The
simplest view to take of the problem, at the present stage, is
that the motions of the aerodone are plotted relatively to itine-
rant ordinates, the origin being supposed to travel with a uniform
velocity = V n along the axis of flight.

We have so far proved that the displacement in the direction
of the axis of y (that is, the variation of H) t is proportional to the
force by which it is produced, and is therefore harmonic. We
have further shown that the velocity along the axis of x (the
axis of flight), measured from the itinerant origin, is propor-
tional at every instant to the displacement on the axis of ?/, and
that this also is consequently harmonic ; it also follows from the
latter result that the two harmonic motions differ by an interval
of one quarter phase. It remains for us to determine the ampli-
tude ratio of the two harmonic motions, and the orbit and
consequently the course of the aerodone is known,

56

THE FLTGHT PATH PLOTTED §31

Let ri be the maximum velocity of the horizontal component
of the orbit motion, and let t be the time of one complete orbit.

Then by equation (17) t = 2* \J — -. Let x x be the half ampli-

tude of the horizontal component and let lt\ be the similar
measure (the half amplitude) of the vertical component.

We have x x = ?-- = i\ yj — n ,

2 it v q

and by §29, equation (c)

T„ ri = g h u
but T w = Vff H n ; substituting, we obtain

l fl = n •' - n = V 2 X ri sj --*

j «*

or hi = \/"^ X a-i.

That is to say, the vertical amplitude of tlie orbit is V2 times
the horizontal amplitude. And we already know that both the
horizontal and vertical components of the orbit are harmonic.
Consequently the form of the orbit is an ellipse whose major
axis is vertical and is greater than the minor axis in the ratio

V~2 is to 1.

§ 31. Phugoids of Small Amplitude. Plotting. — In the foregoing
sections we have proved that the " evanescent " form of the
phugoid, that is, the form of the phugoid curve when almost
merging into a straight line, is of trochoid- like form, in which
the orbit of the tracing point is elliptical instead of being circular,
and that the axes of this ellipse are in the relation one to the

other, the vertical to the horizontal, as V 2 is to 1.

The plottings of two such trochoidal approximations are given
in Fig. 40, and a convenient method of plotting (which is in any
case but a matter of simple geometry) is illustrated in Fig. 41.
In plotting these trochoid-like approximations the length of a
complete period, or "phase length," is first calculated from

57

w

THE FLIGHT PATH PLOTTED §32

equation (17) and from the known value of F M , this is laid off
and divided into a convenient number of equal parts. A celluloid
dial (Fig. 41), is next prepared with as many angular divisions
as there are linear divisions in the phase length, and on this dial
two circles are scribed having radii as 1 : V 2, and a small
tracing circle is drawn, touching both. This small circle is
divided into half as many divisions as there are in the dial, the
plotting is then performed in the following manner : —

Presuming that we begin with the highest point on the curve,
the dial is placed with the diameter passing through the centre
of the tracing circle vertical, the latter being at the top of the
dial. The point A is then pricked off, and the dial is moved bodily
from left to right through one of the linear divisions, and rotated
through one angular division counter-clochvise ,• the next point
reckoned clockwise, B, is taken on the tracing circle and pricked
off; the process being repeated until the curve is completed.

It is evident from geometrical considerations that the tracing
point derives its location along the horizontal axis from the
smaller circle, and vertically from the larger one, and so the
required compound motion is obtained.

§ 32. The Phugoid Chart. — The plotting of the phugoid curves
forming a complete series, from the straight line representing the
path of uniform gliding, to the tumbler type of curve with con-
stants varying to any desired degree, may be termed a phugoid
chart. In Fig. 42 such a series is given, including in all some
twelve plottings 1 ; these, beginning with the straight flight path
No. 1, include two trochoidal approximations, Nos. 2 and 3,
plotted by the dial method ; three true phugoids of the inflected
type, Nos. 4, 5, and 6 ; the special case of the semicircular flight
path No. 7 (this may be looked upon either as an inflected curve
or as a tumbler curve according to which way the aerodone
is turned over at the cusp), and five tumbler curves Nos. 8 to 12
inclusive. The calculations and data of these plottings are given
in Appendix V.

1 Also given to reduced scale in frontispiecp.

59

§32 AERODONETICS

The evolutes of the curves, as developed in plotting, are given
so far as the size of the plate permits, and the scale shown is
that employed in the calculations, though any other scale
may be employed without affecting the geometrical shape of
the curves.

The value of H n has been taken as 64 feet, and is, of course,
represented by the distance of the straight flight path below
the datum line ; the appropriate scale for any other value of
TT n is obtained by simply substituting the new value on the
chart.

The whole of the curves are plotted to one value of TT n from
one horizontal datum line, and the point of the curve forming
the starting point of the plotting has in every case been referred
to a single vertical axis y, at which the value of cos has been
taken either + 1 or — 1, and the value of It as minimum.

Each curve has its position of mid-phase marked (the point at
which its velocity is greatest), and a curve has been drawn
through the points so obtained, which is called the phase curve ;
this evidently should be a smooth curve, and it should depart
from the path of uniform gliding (curve No. 1) normally and
approach the axis of y asymptotically.

The phase curve shows well the manner in which thetrochoidal
approximations become inaccurate as the amplitude increases ; it
also serves as a check on the proper joining of the inflected
curves. In the present case the form of the curve suggests that
these inflected curves have been slightly " telescoped " in the
process of plotting. The phase curve, once its form is carefully
determined, will enable inflected phugoids to be plotted with
greater certainty; there will no longer be any necessity for the
process of adjustment.

§ 33. The Time Period of the Phugoid Path. — The complete
period of an aerodone, that is to say, the time taken by it
to travel from crest to crest of its path, is a quantity for which
the author has so far been unable to obtain a general expression;

60

THE FLIGHT PATH PLOTTED §34

in all probability the mathematical integration of the phugoid
curve in respect of time is a matter of very considerable
difficulty.

As a general solution the author has, for want of a better
method, employed a step by step graphic process, the working of
which is as follows. An inverse velocity curve (see Phugoid Chart,
Fig. 42) is first laid out, this gives by its abscissae the time taken
to traverse an arbitrary small unit of length at a velocity cor-
responding to the height II of the ordinate. The phugoid, whose
period it is required to measure, is then divided into a number of
linear elements of the size of the unit chosen, and an ordinate is
drawn on the inverse velocity curve, corresponding with the
mean position of each unit ; these ordinates are then measured,
and their sum gives the total period required.

The above method has been applied very carefully to curves 5,
7, 8, 9, 10, 11, and 12, and the results plotted as the " Time
Curve " shown. The degree of accuracy may be judged from the
deviations of the plottings from the averaging line drawn as
a smooth curve through them. It may be noted as a remarkable
feature of this curve, and one that is of the greatest possible
service in the experimental verification of the theory, that the
time period for all curves lying between the straight line and the
semicircle is approximately independent of the amplitude ; that
is to say, for a given value of II n the time period for the inflected
phugoids is sensibly constant.

§ 34. The Time Period of the Phugoid Path. Special Cases. —
In certain special cases, for example, in the almost straight
flight path, considered in §§ 29, 80 and 81, the time period may
be calculated, and in this case is given by equation (17).

The special case of the semicircular path also is one that may
be calculated, the time period being that of a pendulum of length

61

§ 34 AEBODONETICS

3 1I H swinging through an arc of 180 degrees. The expression
in this case is

n

'2 = if v — X X,

where I = 8 //„, and where the value of x is a function of the
arc of swing defined by the expression,

x

_ t . / 1 \ a • * A . / 1 X 8 \ 2 . .A

,/lX8x5\ a . „ A . / I x 8 X 5 X 7Y . a A . .

+ I a x 4 x ) 8tn a + (axlx-e-xs) * m a + etc "

where 2 A is the total arc of swing.

A
For a value of 2 ^1 = 180° we have ~ = 45° and calculation

gives x = 1*1795, so that the time period will be

/ 8 77,

*a = v V ^— n X 11795.

We may thus express the relation as

1-1795 7T V 8 — /h _

h v .<; __ 1-179 5 Va

J,^' ~ 2

= 2J0429_5 = VQ2U1>

that is to say, the time period of the semicircular path is greater
than that of the nearly straight path, to an extent only of about
2 per cent. These two special cases constitute the limits of the
inflected series, for which the graphic method gives the time
period as sensibly constant, and thus we have in substance a
confirmation of the previous conclusion.

The actual time periods for the two special cases under
discussion are as follows : —

(1) The nearly straight path,

/ 64

h = 2 7T >y/ -- = 8*85 sees.

62

THE FLIGHT PATH PLOTTED §35

(2) The semicircular path,

h = M795 X 1 7T J *5| = 9-04 sees.

▼ 32*2

As figured on the chart the time curve has not been corrected
for these calculated results.

§ 35. The Time Period and Form of the Phugoid Path.
Special Cases (continued). — As the amplitude of the phugoid is
increased beyond the limits of the semicircle, there is a rapid
shortening of the time period as shown by the manner in which
the time curve falls away almost immediately after that critical
form of flight path is passed. The time period of the tumbler
curves is in every case less than that of the semicircle, and the
time period diminishes continuously as the value of the constant
C gets less and less. It becomes of interest to examine the fate
of the time curve when outside the range of our plotting.

Let the velocity of the aerodone become very great so that W
is a negligible quantity, then the orbit will, in the limit, become
circular, since in this case V becomes sensibly constant, and F
is therefore also constant. Under these conditions we write

V*
W = zero, so that/ = F = — .

T

But by (1) F=n P.

but by (10)

V 2 . 1

.\ n V 2 = — , that is, r = — ,

r n

n = 2jj-. .'. r = 2 H n .

That is to say, if we select phugoids of greater and greater
velocity, for a given aerodone the form of the tumbler curve
approximates more and more closely to a circle of radius
= 2ff n .

If we suppose that we are investigating depths at which the

63

*

§ 35 AERODONETICS

phugoids are sensibly circular in form, that is, down where the
phase curve approximates very closely to the axis of y, then

V = V 2 g~II,
and we know length of path = 4 tt II n .

4 7T 7/ (it II ) 2

Hence t = -7 .-'or t 2 II = 8 — , which is constant,

V2gII 9

thus giving the form of curve to which the time curve con-
tinually approximates as the depth becomes greater and greater.

§ 36. Relations of Time Period, Phase Length, and Velocity
for Phugoids of Small Amplitude. — We may now return to
consider the simple relationship that exists between time period,
phase length, and velocity, in phugoids of small amplitude. A
clear statement of this relationship is desirable in view of the
subsequent experimental investigations, that are presented later
in the work, in confirmation of the phugoid theory.
Let L\ = complete phase length in feet.
t\ = time period as before.
V n = natural velocity.
We have

v 9

and V n = V 2 g II~

Whence L x = h V n = 2 tt ///« x ^~%y ji~

= 2 V~2 X tt II n
or Li = 8-88 II n .

Taking g = 82*2, h may be expressed in terms of II H thus,

h = 1105 VTf n .
The experimental verification is given in Chap. VI.

§ 37. Variations in the Value of C for Geometrically Similar
Curves. The Constant K.— It has been already remarked that

64

THE FLIGHT PATH PLOTTED §37

systems of phugoids based on different values of H n are
geometrically similar, and consequently a phugoid chart (such
as Fig. 42), may be read to any scale appropriate to the value
of H n chosen.

Then, for any particular curve and point on that curve as
defining the position of the aerodone, we have for different values
assigned to H n the following relations: —

TT

5— jj~ ■= constant; also cos = constant.

.'. cos — llTr = constant.

But we know from equation (14),

— - = cos —

V7£ 3 H n '

hence / — is constant for the point chosen, whatever the value

of H n9 that is to say, whatever the scale assigned to the curve,

and * — may be taken as an invariable constant relating to
v ±i n

the particular curve and is independent of the scale to which the

curve is read. We will represent this constant by the symbol K.

Th^s constant K will be termed the amplitude constant^ and all

curves for a given value of K are geometrically similar.

2

Also for the straight path {H = H n ) we have K = „ and

C= q H n , this is the condition of the maximum value of K.

The variations of K with H and cos for any given value of
II n follow the same law, and curve, as variations in C. K is in
fact the constant C deprived of its dimensionality.

A.

65

I

CHAPTER IV

ELEMENTARY DEDUCTIONS FROM THE PHUGOID THEORY

§ 38. Permanence of Stability. — The theoretical investigations
of the preceding two chapters constitute the proof, under the
restricted conditions of hypothesis, of the longitudinal stability
of an aerodone in flight, the whole demonstration being a
quantitative analytical version of the theory of stability
enunciated in §§ 3, 4 (Chap. I.).

The permanence of stability, according to the present proof,
rests definitely on the fact that the phugoid curve consists of a
succession of phases repeated, without change of form, an
indefinite number of times. If the substitution of real con-
ditions for those of hypothesis be found to result in any
progressive change in the form of the curve, that is, if the
constant C be found to undergo a change (for a given value
of JJ n ), or more broadly, if the constant K suffer any progressive
change of value, then the question of permanence of stability is
greatly complicated, and further investigation will be required. 1

There is one particular case of the phugoid curve, even under
the supposed conditions, in which the stability must be regarded
as indeterminate ; this is the semicircle when C = zero. The
fourth condition of hypothesis, § 19, is " that the size of the
aerodone is small in proportion to the minimum radius of
curvature of its flight path." Now, for the purposes of the
theoretical investigation, this condition may be supposed fulfilled
by the aerodone being assumed to have no size at all, but in the
case of the semicircular flight path this is not sufficient, for the
" cusp " must be considered to be a portion of the flight path

1 This further investigation forms the subject of Chap. Y.

66

ELEMENTARY DEDUCTIONS §39

of infinitely small radius ; we have to suppose the aerodone
artificially turned over after each semicircular flight before it can
begin the next semicircle.

If the aerodone be a thing of substance, then not only do we
meet with difficulty in the particular case of the semicircle, but
also in the case of all curves approximating thereto, whether of
the inflected or the tumbler variety ; a point is reached sooner or
later when the aerodone is sensibly embarrassed in turning over
by its fore-and-aft dimension, in other words by the length of
its tail, and the theoretical form of the curve ceases to hold
good.

§ 39. Unstable Conditions. The Danger Zone on the Phugoid
Chart — From the foregoing considerations we must regard the
environment of the cusp of the semicircular phugoid as con-
stituting a danger zone, on entering which an aerodone is liable
to complete loss of stability. The manner in which this loss of
stability takes place becomes apparent if we examine the case of
an aerodone of some considerable tail length on a flight approxi-
mating to the semicircle ; as the aerodone approaches nearer
and nearer to the cusp it gradually loses its velocity until, as it
arrives at the datum level, it finally comes to rest, and then
commences to run backwards. Now the flight of an aerodone in
the contrary direction to that for which it is designed must be
regarded as denoting that the limits of stability have been
exceeded, for this condition is quite inadmissible either from a
theoretical or practical standpoint. We consequently appreciate
that any cumulative change in the constant, either C or K,
tending towards zero value, is fatal to the permanence of
longitudinal stability.

So long as the aerodone is of reasonably small size in com-
parison to the phase length Li, the whole of the inflected phugoid s
of moderate amplitude, including those which may be approxi-
mated by the elliptical trochoids, and perhaps such as numbered
4 and 5 on the chart, may be regarded as curves of stable flight

07 p 2

L

§ 39 AERODONETICS

path, but as the semicircle is approached the conditions of
instability supervene.

If we go beyond the semicircle, we again find stability in the
tumbler form of flight path; this state, however, is more a
matter of scientific than of practical interest ; from the point of
view of the problem of flight we may regard any approach to
the semicircle as nearing the limit of stability ; the danger zone
in the neighbourhood of the cusp prescribes the limit permissible
to the amplitude of the flight path.

§ 40. Stability in Face of a Disturbing Cause. — One immediate
- consequence of the phugoid theory is that the stability of an
aerodone, in face of a given disturbing cause, increases with its
velocity of flight. Thus it is evident that if an aerodone,
travelling at its natural velocity in still air, were to meet with
a gust of wind travelling in the opposite direction, it would
immediately begin to rise, just as if its value of II had suddenly
been increased, which in effect it has. Its subsequent path,
evidently, then depends upon the velocity of the gust in relation

to the value of V n ; should the velocity of the gust reach (VH — 1)
V n the aerodone will find itself in the middle of a semicircular
swoop, and on rising will pass into the danger zone and lose its
equilibrium. Consequently it is the relation of the velocity of
the gust of wind to the natural velocity of the aerodone that
constitutes the criterion, and the higher the natural velocity the
higher (in the same proportion) is the velocity of the gust of
wind that can be encountered with impunity.

The same naturally applies in passing from an adverse gust
into still air, or, that which is the same thing, if the gust of
wind take the aerodone from behind. In either case if the
velocity of the gust be equal to that of the aerodone the latter
will find itself at rest relatively to the air, and consequently its
stability is in jeopardy ; we may regard an aerodone under these
conditions as at the cusp of a semicircular phugoid in the act of
being turned over.

68

ELEMENTARY DEDUCTIONS §42

Thus, a change iu the velocity of the wind taking effect as a
gust either with or against the direction of flight, may cause
instability, the magnitude of the wind change required to bring
the constant C to zero being *73 V n in the case of an adverse
gust, or V n in the case of a gust in the direction of flight.

§ 41. The Constant C as an Index of Stability. — We have seen
that in the particular case of C = zero, the semicircular phugoid
constitutes the worst possible case of unstable flight path, and
that as the constant C increases to its maximum value (when
the path of the aerodone becomes a straight line) the flight path
is further and further removed from the danger zone. We may
consequently regard the constant C as an index of the stability.

Stability is not a thing that we have learnt to measure, so
that it is impossible to assert that the stability is in any way
proportional to C ; we can merely learn to associate an increase
in C with an increase in stability and vice versa.

When we are concerned with the tumbler curves, we find that
the converse is the case, for the further removed the tumbler
curve becomes from the semicircle, the lower is the value of C.
We can, since the values of C are here of minus sign, rectify the
anomaly by making C 2 the index of stability, when the rule
applies for tumbler and inflected curves alike.

As a gauge of stability C 2 is more appropriate than K a , for K
undergoes no alteration in the face of a change in the value of
7/ w , and we know from the preceding section that stability is
improved by an increase in H H . The importance of K in dealing
with the maintenance of stability will appear from its mode of
employment in the subsequent chapter.

§ 42. Wind Fluctuation. A Question of Relative Motion. — The
supposititious case of wind fluctuation dealt with in § 40 is
merely one example of this kind of disturbance. The turbulent
component of wind motion is a matter of great complexity at
present but little understood ; in all probability a wind of any

69

§42 AERODONETICS

considerable velocity consists very largely of vortex motion, and
an aerodone, in passing from one part of such a "vortex pack "
into another part, experiences many and varied changes both as
to velocity and direction. It has been demonstrated experimen-
tally by Prof. Langley tbat the higher the wind velocity, .the
higher in proportion becomes the velocity of turbulence reckoned
either as the maximum or the mean velocity difference. Thus
the variations are not confined to horizontal fluctuations as
discussed in § 40, but include vertical components of motion
distributed in an irregular manner.

The question from the point of view of an aerodone in flight
is entirely one of relative motion. As the aerodone passes from
one mass of air into another, possessed, of a different velocity
and direction of motion, it is the difference of the two velocities
with which the aerodone is concerned, and not the absolute
velocity of either. Thus the main motion of translation of the
wind has no influence whatever except in the case of an aerodone
at the time of launching from a fixed point.

The above fact simplifies the consideration of the problem
materially. We have already dealt with the case of a change of
wind velocity in the direction of flight, we will now, on similar
lines, deal with the more general case of a change of velocity
and direction, and show how the use of a fully plotted phugoid
chart enables any case, however complex, whose data are defined,
to be dealt with to any desired degree of approximation.

§ 43. Wind Gusts of changing Velocity and Direction. — Owing to
the fact that we have assumed the whole mass of the aerodone
to be concentrated at its centre of gravity, and that it conse-
quently possesses no moment of inertia, we can suppose that its
directive organ acts instantaneously ; thus, when an aerodone
passes from one mass of air into another, having a different
relative direction of motion, it will instantly adapt itself to the
new conditions in respect of its attitude to the line of flight.

The influence of distributed mass is more fully discussed

70

ELEMENTARY DEDUCTIONS

§43

Inter, but it may be remarked parenthetically that the general
effect of moment of inertia about a transverse axis is to hinder
and delay the adaptation of the attitude to the change of relative

FLICHT PATH

' «^"

T * e ±

&*-"&-

c

Fig. 43.

direction, and the greater the moment of inertia the larger the
directive organ or tail plane required ; beyond this the present
argument is not seriously affected.

Let us suppose, as a general proposition, that the motion of

Fig. 44.

the aerodone through (relatively to) the air at any instant is
represented both as to direction and magnitude by the line A B
(Fig. 48). Let now a change take place in the velocity and
direction of the wind, which we will represent by the line li D.

71

r

§ 43 AERODONETICS

It is evident that the new relative velocity and direction may he
obtained by a parallelogram of velocities, and will he given by
the line C B ; for, if we suppose the path of the aerodone repre-
sented by C D, a particle of air from B and the aerodone
from C will, if the velocities and directions be maintained for a
short period of time, reach the point D simultaneously.

Now, referring to Fig. 44, since we know the velocity of the
aerodone at the point A\ 9 we know its value oi H ; let this be
the height measured from datum line No. 1 and = H n . And
since we know the angle of the path we can plot (or lay off from
a fully plotted chart), the phugoid which constitutes the path of
flight for the time being; this is marked "Path 1," and under
the given conditions is a uniform glide.

Let A i C\ represent the change of wind at the instant the
aerodone arrives at the point A\ 9 then draw the parallelogram
and obtain the new line of direction and velocity D\ A\. Next
from the value of V so obtained calculate the new value of H
and draw datum line No. 2, and, as before, lay off flight path (2)
from phugoid chart.

Let A* C* be a further wind fluctuation encountered at the
point i4 2 , then, proceeding as before, we obtain the new relative
flight path (3), and so on.

§ 44. Stability in the Face of Wind Fluctuations. — In the actual
changes that constitute wind fluctuation the gusts do not arise
with absolute suddenness as depicted ; it is evident, however,
that we may make the individual steps as small as we please,
and so approximate, as closely as we wish, to any known set of
conditions.

The absolute flight path is not in any case a jagged curve, as
shown in Fig. 44, which represents the curve plotted relatively
to the changing gusts of wind. If plotted to some fixed co-
ordinates the sudden changes of direction and velocity would not
exist ; this kind of plotting would, however, be very tedious, and,
fortunately, would serve no useful purpose.

72

ELEMENTARY DEDUCTIONS §45

It is evident that observations made at a fixed observatory of
the curves of wind fluctuation would not of necessity apply to an
aerodone in flight, under identical wind conditions. If, however,
the fluctuations are due to some kind of organised turbulence, as
would be the case if the wind consists of a kind of vortex pack,
we could, from time observations made at a stationary point, plot
a typical sample on a linear basis, and it would be fair to assume
that the changes so plotted represent the average experience of
the aerodone in its flight path.

§ 45. Stability in the Face of Wind Fluctuations (continued). —
The method of the foregoing sections enables us to plot the
relative flight path of an aerodone in a known fluctuating aerial
field to whatever degree of approximation may be desired, and
this plotting may be taken as prescribing the stability or other-
wise of the flight path. If the aerodone, in the course of its
career, find itself at the cusp of a semicircular phugoid, or if in
practice it be even in the near vicinity of the cusp, it must be
regarded as having lost its equilibrium, and the wind conditions
must be deemed dangerous to flight. Likewise, if at any time
the value of C falls below zero, so that the path of flight is a
tumbler curve, the conditions of stability have, by our definition,
been violated.

It has been shown in § 40 that for a single gust of defined
magnitude the stability of an aerodone may be assured by
increasing to a sufficient degree its natural velocity, this in-
volving a structural alteration of the aerodone or an increase of
its weight in accordance with established aerodynamic principles.

With a certain reservation, that will be discussed later, it is
evident that the same principle applies where the disturbing
cause comprises a number of lesser fluctuations, and that the
greater the magnitude of the wind turbulence the greater will be
the value of V n necessary to ensure stability. In other words
safety in flight depends upon the employment of a sufficiently
high velocity for the conditions that obtain.

78

§ 46 AERODONETICS

§ 46. The Practical Limit of Stability. — A question arises that
is very difficult to approach from a theoretical standpoint;
namely the extent of the danger zone in the vicinity of the cusp.
This is, it has already been pointed out, dependent upon the size,
principally the tail length of the aerodone, and the subject is not
one that can at present be dealt with on rigid lines.

Referring to the phugoid chart, it is evident that an aerodone
travelling on the curve entitled No. 6, would be getting perilously
near the region of the cusp, whilst curves Nos. 2, 8, and 4 do
not give rise to any anxiety.

Dealing with the question on the basis of § 40, we know that
it is an adverse gust of any given magnitude that has the greater
influence on the value of C ; if the aerodone is safe in face of
an adverse gust of given magnitude, a greater gust may come
up from behind without causing instability. We also know
that the velocity of an adverse gust, that will change an
aerodone from the straight line gliding path to the semicircle
is (V 8 — 1) F n , or = '78 V w and we can express the values of the
gust (relative) velocity for the other curves plotted as follows.

Curve No.

2
8
4
5
G

Giwt Velocity, in Terms of J'n.

103 V n

•200 V n

•885 r M

•425 V u

•520 r n

Thus it is evident that if v be the gust velocity relatively to the
air in which an aerodone is gliding tranquilly, the relation v =
•78 V n is definitely dangerous ; v = *5 V n is uncomfortably near
the limit ; and v = *88 V has every appearance of being
reasonably safe.

When we later pass on to consider the damping influences
that are at work we shall see that it is possible to design an
aerodone whose stability shall be immune to considerably greater

74

ELEMENTAEY DEDUCTIONS §47

disturbances than above stated. When any influence is at work
to continuously damp out the amplitude, that is, to raise the
value of C, the change that takes place during one half-phase
may be quite sufficient to carry the aerodone from an unstable to
a stable path, and it may consequently be necessary for it to be
on a tumbler curve in the trough of its flight path in order that
it should arrive on the semicircular phugoid at the moment of
cresting, and so experience instability.

The practical limit of stable flight path is one that in all
probability may be best investigated by model experiments ; the
rules relating to scale models, and the laws of corresponding
speed and similarity, are dealt with later in the work. 1

§ 47. The Influence of a Periodic Disturbance. — Wind fluctuations
are sometimes met with having a definite periodicity. There is
perhaps no evidence that wind fluctuation in general is periodic
in character; on the contrary it may frequently consist of an
irregular turbulent motion. In certain cases, however, especially
in the vicinity of deep sea waves, there is unquestionably at
times a definite period both in the frequency and in the form of
the fluctuations.

The condition of an aerodone in the presence of regular aerial
waves or pulsations is closely analogous to that of a vessel rolling
in a seaway ; so long as the two periods, that of the vessel or
aerodone and that of the wave motion, do not approximately
synchronise, the conditions call for no special comment ; once,
however, synchrony is established, the stability of the aerodone
will be in jeopardy.

In the case of the rolling ship we have a damping factor in the
skin friction and other resistances the hull offers to rolling
motion to prevent the oscillation from becoming excessive. It is
evident that we must look for some equivalent means of exercising
a similar influence on the aerodone in its phugoid path.

The analogy between the motions of an aerodone and the

1 S§ 126, 127, efat?.

75

§ 47 AERODONETICS

rolling of a ship is strictly limited to tlie sense of its appli-
cation. In the case of the aerodone the oscillatory disturbance
is neither pitching nor rolling in the nautical sense, both these
being dynamically rotational in character. The phugoid oscilla-
tion cannot be strictly analogous to either of these since the
aerodone as at present defined has no moment of inertia.

The form of the phugoid oscillation for small amplitude
consists, as has been shown, in motion in an elliptical orbit.
This motion is dynamically the same as could be produced if
the aerodone were suspended in space by a spring system having
equal elasticity in all directions ; as such the motion in question
is one form of a possible infinite number ; it is evident that in
such a system anything in the nature of a series of impulses
planted periodically, with about the required frequency, may
eventuate in the amplitude exceeding the limit permissible.
This dynamic picture of the condition of the aerodone, which is
founded on the phugoid theory, gives the most vivid impression
of the susceptibility of an aerodone in flight to vibrational dis-
turbance, that is, to a decrease in the value of the phugoid
constant. The problem of meeting disturbances having a
synchronising period, resolves itself into finding a means of
applying a " dash-pot " to damp out the oscillations as fast as
they are generated.

§ 48. Unaccounted Factors in relation to the Flight Path. — There
are certain factors excluded by the hypothesis of Chapter II.
which have a very far-reaching influence on the behaviour of an
aerodone in flight. Referring to the conditions laid down in §19,
we have firstly the exclusion of all motions relating to lateral
stability ; this we will allow to stand. We have secondly the
supposition that the aerodone loses no energy, and thirdly
that it possesses no moment of inertia about a transverse axis ;
these two conditions involve the "unaccounted factors" of which
it is now necessary that we should take stock.

It has been set forth in the statement of hypothesis that

76

ELEMENTAKY DEDUCTIONS §48

condition (2) may be supposed complied with in two distinct ways,
namely, either by supposing the air as frictionless and the sup-
porting wave as perfectly conserved, or by supposing a force of pro-
pulsion in the direction of motion precisely equal, at every instant
of time, to the resistance experienced by the aerodone in flight.

It has further been shown 1 that the law of the phugoid path,
so far as weight supported and resistance are concerned, is that
all variations are as the velocity squared, that is to say, both
the aerodynamic resistance and the frictional resistance vary as
I' 2 , the latter to the usual degree of approximation. It is evident
that the influence of resistance is indissolubly associated with
the question of propulsion, for if we could supply propelling
mechanism whose thrust varied as the velocity squared the
conditions of hypothesis in this respect would be complied
with. When we have finally to consider the full aerodromic
problem of the flying machine, we shall have occasion to discuss
the idiosyncrasies of various means and machinery of propulsion,
but for the purposes of aerodonetics pure and simple, we are
concerned essentially with the behaviour of the gliding model,
in which the propulsion is supplied as a component of gravity
and is therefore to be regarded as constant ; if we supply a con-
stant horizontal force acting on the aerodone, this force and
that of gravity will give a resultant invariable both in magnitude
and direction.

It is evident, however, that the gravity component is not the
exact equivalent of a constant force of propulsion, for, from the
above argument, the component of gravity must be regarded
as acting in the constant direction of the mean gliding path,
whereas a force of propulsion must act at every instant in the
changing direction of the line of flight. In the further develop-
ment of the subject, owing to the extreme difficulty of the problem
in its entirety, the flight path is assumed to be of small ampli-
tude, so that the distinction under discussion becomes unnecessary,
the difference being negligible.

1 §§ 20, 110, and 111 ; also "Aerial Flight," Vol. I., Aerodynamics, § 159.

77

§ 49 AERODONETICS

§ 49. Unaccounted Factors (continued). — It is shown in the
subsequent chapter that, under the conditions of natural gliding
or uniform propulsion, the effect of resistance is to exercise a
damping influence on the phugoid oscillations, tending to a con-
tinued increase in the value of C, and, unless adverse influences
are at work, the flight path continually approximates more and
more closely to the straight line or path of uniform gliding.
This result is one that might well be anticipated from the action
of the supposititious dynamic model of § 47. If we imagine a
constant force applied to the aerodone parallel to the minor
axis of its orbit in one direction, the supposed direction of flight,
and a variable force in the opposite direction whose value is
greater when opposed to the orbit motion, and less when in its
favour ; then it is evident that the mass in oscillation will do
work in respect of the variable applied force, and therefore its
amplitude will diminish.

The effect of taking into account the moment of inertia of the
aerodone about a transverse axis is also dealt with in the chapter
that follows, which constitutes an extension of the phugoid
theory, carrying it to a point at which it becomes of the greatest
practical utility. The influence of moment of inertia is the
reverse to that of resistance, for it tends to an increase in the
amplitude of the flight path oscillation. The result of this
further investigation is quantitative and culminates in an
equation which correlates every factor and constant essentially
involved in the flight of a real material aerodone for phugoids of
small amplitude.

78

CHAPTER V

STABILITY OF THE FLIGHT PATH AS AFFECTED BY RESISTANCE AND

MOMENT OF INERTIA

§ 50. Introductory. — In the present chapter we shall discuss
the influence of departures from the original hypothesis in respect
of resistance and moment of inertia, both of which were initially
assumed to be absent.

It has not been found possible to achieve any results by
including these quantities in the original investigation of
Chapter II. ; the form of expression is far too unwieldy, and the
integration and plotting become impossible.

It is consequently necessary to deal with the matter as a kind
of correction, and to confine ourselves more especially to the
investigation from a less general point of view, especially
directing our attention to the practical aspect of the problem, as
concerned in flight under normal conditions.

In the particular case of the uniform straight line phugoid,
supposing a constant propelling force, or that which amounts to
the same thing, an appropriately inclined path, 1 it is evident
that the conditions are those of equilibrium, for the resistance is
at every instant balanced by a propulsive force, and since there
is no rotary movement of the aerodone about a transverse axis,
the existence or otherwise of moment of inertia can make no
difference. Although the condition is one of equilibrium, it does
not follow that the path of flight is stable, it is possible that it
may be a condition of unstable equilibrium so that on the least

1 A simple resolution of forces proves the identity of these alternative
suppositions, except so far that the applied propelling force results in the
total reaction being slightly greater than the weight

79

§50

AERODONETICS

disturbance taking place, the resulting phugoid tends to a
continual increase of amplitude.

The conclusions of the present chapter have been to some
extent anticipated by the statements made in §§ 48 and 49 ; in
short, it will be shown, firstly, that the effect of resistance is to
damp out oscillations, whereas that of moment of inertia is the
converse ; and, secondly, it will be shown that for small ampli-
tude these two influences may be pitted one against the other so
as to leave the amplitude of the phugoid path unaffected. This
condition marks the limit of stability of the gliding flight path.

§ 51. Influence of Resistance on Amplitude. — It is, as before,
assumed that the total resistance varies as the square of the
velocity of flight, and that the propelling force is constant.

Let li be total resistance.

F, V w II and H n stand as before.
Q = uniform applied thrust.

h = II n — II =. height of aerodone above path of mean

gliding.
= difference of thrust and resistance repre-
senting net force of propulsion acting
on aerodone.

ft

ft

tt

tt

q =Q - R

Then—

or

or

R/H = Q/H n
.: Q- It a H n - H

q oc /i.

Eef erring to Pig. 45, we have the path of mean gliding p p,
and the phugoid path p x jh, separated by a varying distance h to
which we have shown q proportional. Now, when q is positive
as when the aerodone is " cresting " (in the vicinity of the crest
of its wave-like path), it is receiving energy, and its II value is
consequently increasing; this is equivalent to the aerodone being

80

STABILITY OP THE FLIGHT PATH §51

»o

5

|\

\

\

I I 3
/ *

/
/

/

Y

i

\
\

»v.

\

A.

81

o

§51

AERODONETICS

transferred downward on the phugoid chart as it progresses
(Fig. 46). On the other hand, when the aerodone is " troughing"
(in the trough of a flight path undulation), it is losing energy,
for q is negative, and consequently the value of H is decreasing ;
this is equivalent to the aerodone being transferred upward on
the phugoid chart from one curve to another as illustrated by
Fig. 47.

There is, of course, no discontinuity of path such as that shown
in the figures ; the correct procedure would be to conceive the
phugoid chart to move in space, and that the air and aerodone
retain their existing relationship. Thus, in Fig. 48, the varia-
tion in II is provided for by an undulating datum line : this

?

Fig. 48.

represents the locus of a datum point laid off from instant to
instant.

Referring now to Figs. 46 and 47, we see that the transference
that takes place in the position of the aerodone on the phugoid
chart, due to the action of the force q, has the effect, both during
" cresting " and "troughing," of diminishing the amplitude of
the phugoid path. This is particularly clear at the points of
phase and mid-phase where = 0; any transference at either
of these points in the direction indicated gives rise to an obvious
diminution of amplitude that, figuratively speaking, could be
measured with a foot rule.

We thus see that the effect of resistance is to damp out or

82

STABILITY OF THE FLIGHT PATH §52

diminish the amplitude of the phugoid undulation, and so results
in an increase in the value of C. Before following the matter
further, we will pass on to a preliminary survey of the influence
of moment of inertia.

§ 52. Influence of Moment of Inertia. — An aerodone in flight
pitches through an angle that increases with any increase of the
path amplitude. If the angle at the point of inflection be, as
before, represented by the symbol 0i, then the total angle ranged
by the aerodone is 2 0*. Knowing the law of the change of angle
in respect of time, the total angular amplitude (= 2 0$), the time
period — that is, the phase time of the phugoid — and the moment
of inertia of the aerodone, we could calculate the resulting torque
or couple for all values of 0.

Now this couple is supplied by some reaction received from the
air on the supporting and directive surfaces ; for it is a couple
applied to the aerodone from without, and the material
atmosphere is the only external agent which we are prepared
to recognise.

Further, the organ of the aerodone whose duty it is to pre-
serve the functional aspect or attitude of the aerodone to its line
of flight, is the tail, and since it is the preservation of the
functional aspect that necessitates the angular oscillations, we
may look to the tail to provide one of the forces by which the
couple is constituted. In other words, as the aerodone "pitches "
in its phugoid path, the tail plane supplies one of the varying
forces necessary to impart to, and take away from, the aerodone,
the rotational motion involved.

Now this varying force on the tail plane involves that its upper
and under surfaces are alternately under compression and rare-
faction, which means that the tail plane moves obliquely to the
line of flight, with an obliquity and sense varying with the
magnitude and direction of the torque.

Moreover, since this force on the tail must be sustained by
some equal and opposite reaction, there will be an equal and

83 o 2

§52

AERODONETICS

oppositely varying force on the aerofoil itself. 1 It is these two
forces that constitute the couple.

§ 53. Influence of Moment of Inertia (continued). — In the phugoid
theory it arises from the initial hypothesis that there is absolute
constancy of functional aspect or attitude, and the condition that
the reaction varies strictly as J 72 , that is the application of the
V 2 law, rests on this fact.

We now find that when an aerodone possesses moment of
inertia, such perfect constancy does not exist. Now to vary the

Fig. 49.

functional aspect of the supporting member of an aerodone (the
aerofoil) means varying the fundamental constant, the n of the
differential equation (8), § 20, or the II rt of equation (11); for
any change of functional aspect means in effect a variation of
the aerodynamic angle /3, and our knowledge of aerodynamics
tells us that this means an increase or reduction of the supporting
power of the aerofoil for any given speed.

Now a variation in the value of H n means an alteration of

1 This does not mean that the aerofoil is at any time sustaining a negative
(i.e., downward) reaction, but rather that the upward reaction undergoes
variation as due to the added downward or upward pressure consequent on
the oscillations that 3 re taking place.

84

STABILITY OF THE FLIGHT PATH §54

scale of the phugoid chart; it therefore remains for us to
determine in what way such an alteration of scale will
operate. We have seen that the swing of the aerodone
terminates at each point of inflection, that is to say, at a
point of inflection the aerodone has no rotational motion, the
condition being that of greatest angular displacement. Con-
sequently, it is at or about such points that the torque has
its greatest value, and the value of H n is at one of its
extremes.

Let us examine the condition on the downward inflection;
at this point the inertia torque is clockwise (the direction of
flight being from left to right), and the equal and opposite applied
torque is counter-clock. This signifies that the tail of the aero-
done is moving obliquely as in Fig. 49, and the effective angle /3
is less than its normal value, and therefore H n is increased.
Beyond this the pressure developed on the upper surface, and
negative pressure on the under surface, of the tail, the two con-
stituting the tail plane reaction, is experienced by the aerofoil as
additional load ; this being the equal and opposite force con-
stituting the couple. There is thus a further cause tending to
an increase in the value of //„.

Likewise it can be shown that in the region of the upward
inflection lf n will be diminished.

Consequently, we find that during tronghing the value of H n is
diminishing, that is to say, the phugoid chart is contracting ; whilst
on the other hand, during cresting, the value of II is increasing ,
and the phugoid chart is expanding.

§ 54. Influence of Moment of Inertia (continued). — Thus we find
that whereas resistance takes effect on the value of the variable
71, the influence of moment of inertia manifests itself by a change
in the magnitude of the constant II H . Moreover, the changes
due to moment of inertia have no influence whatever on the value
of II for the time being of the aerodone, since there is no energy
being given or taken away, so that the conditions introduced by

85

§ 54 AEEODONETICS

moment of inertia are that // remains unaffected in the face of
changes in H n , that is, changes of scale of the phugoid chart ;
therefore if the chart contracts during troughing, as has been
proved to be the case, the amplitude is increasing ; likewise when
the chart expands during cresting the amplitude is again increased.
Thus the influence of the moment of inertia of the aerodone about
a transverse axis, if not counteracted in some way, is to render
the gliding flight path unstable, any disturbance results in the
aerodone describing a phugoid path of continually increasing
amplitude that must sooner or later bring the flight path into
the danger zone when loss of equilibrium must occur.

The word amplitude as used in the preceding section does not
of necessity mean the amplitude as absolute linear quantity,
although as such the argument applies equally, but rather the
relative amplitude, that is, in relation to the scale of the phugoid
chart. There is a relationship between the relative amplitude
and the absolute amplitude, analogous to that between the
constants K and C, comp. § 87.

When we have to deal with changes in the value of H, as
where we are concerned with the effects of resistance, it is unim-
portant whether we regard the changes in amplitude as involving
changes in C or K, these two constants being in constant propor-
tion. When, however, we are concerned with changes in H n , and
therefore in the scale of the phugoid chart, there is an essential
difference, for C is of the dimensions s/ L and its value is affected
by the change of scale as well as by the alteration of amplitude.
It is therefore necessary to renounce the employment of C in
favour of K, when the effects of moment of inertia are under
discussion.

The results of the preliminary investigation may be presented
as follows. We have ascertained that resistance and moment of
inertia have opposite effects on the value of the amplitude con-
stant K, and whereas resistance has a benign influence involving
an increase of K and a damping down of the oscillation, the effect
of moment of inertia, on the contrary, is detrimental, and if

80

STABILITY OP THE FLIGHT PATH §55

unchecked results in a decrease of K and a progress towards
instability.

The experimental demonstration of these results will be given
in a later chapter.

§ 55. The Quantitative Problem. — The quantitative aspect of the
present branch of the subject is one of considerable difficulty. It
would appear possible to deal with the influence of resistance or of
moment of inertia, either separately or together, by plotting the
curve step by step, making the corrections due to these two dis-
turbing causes stage by stage, and so approximating as closely as
found practicable to the continuous variation that in reality takes
place. It is improbable, however, that the labour involved in such
a process would be justified by the results to be obtained, unless
perhaps, for the purpose of comparison with curves obtained
photographically, as a master check on the whole theory.

If we take the special case of the nearly straight flight path,
that is, with the amplitude bordering on the evanescent, and
suppose firstly that there is no resistance to flight, but that the
aerodone possess moment of inertia, then from the foregoing
sections the amplitude will gradually increase, that is, the
constant K will diminish. If now, when the amplitude has
reached some definite value, we suppose the aerodone deprived
of its moment of inertia, and that it experience resistance to
flight, 1 the amplitude will forthwith begin to diminish, that is, K
will increase, and we may suppose this to go on until the original
condition is restored.

Let us now suppose the aerodone to possess moment of inertia,
and also to experience resistance to flight, then it is evident that
the proportions may be such that K will diminish from the one
cause faster than it will increase from the other, in which case
the conditions tend to instability, or the reverse may be the
case when the gliding flight path will become definitely
stable.

1 Following the V 2 law.

87

§ 55 AERODONETICS

The investigation that follows has for its object to ascertain
the circumstances under which the one effect will exactly balance
the other, so that we shall be able to determine when the path is
on the point of becoming unstable, and specify in such a form
as will permit of the theory being checked by means of models of
the critical proportion.

§ 56. The Form of the Nearly Straight Phugoid. — It lias been
shown in Chap. III., § 30, that the form of the phugoid of small
amplitude is a curve of trochoidal type, it differing from a true

trochoid only in respect of its elliptical 1 : V 2 orbit instead of
the circular orbit of the regular curve. This curve of trochoidal
type may be regarded as taking a position intermediate between
a true trochoid and a sine curve; it is the projection of a trochoid
on a plane at 45°.

It is a property of the trochoid, and one may say of this whole
family of trochoid-like curves, that as the amplitude diminishes
the distinction between one and another or between a trochoid
and a sine curve tends to vanish. Thus referring to Fig. 50, in
which the curve p p is a trochoid, and pi j>i is a sine curve of the
same amplitude and phase length, the maximum horizontal
distance separating the two curves A B is equal to half the radius
of the trochoidal orbit, and if 0i be the path inclination at this

point we have , n = tan X . Let h x be the maximum dis-
placement, or half amplitude of the curve, then hi = A B, but 0i
varies directly as the half amplitude h u or for small values of
0i, tan 0i <r h u hence

A C

- 4 - T1 a hi, or A C a: hi*.
A B

Consequently when h becomes small A C becomes negligible.
In the case of the actual phugoid approximation we can arrive
at the same result from another point of view.

It has been shown, § 29, that for phugoid s of very small

88

6

•®

©

►H

8<)

§ 56 AERODONETICS

amplitude the vertical component of the motion of the aerodone
is harmonic ; consequently if the rise and fall of the aerodone be
plotted against time, that is with time as abscissae, the result is
a sine curve. The reason that the actual path is not a sine
curve is that absciss© = distance.

If the velocity were uniform instead of fluctuating then the
curve of h by L would be the same as the curve h by T, and the
velocity becomes more and more nearly uniform as the ampli-
tude is diminished. Therefore, for amplitudes of very small
magnitude the h by L and h by T curves may be taken as
identical.

Hence for phugoids bordering on the limiting case of the
straight line, we may regard the flight path as a sine curve.

§ 57. On the Form of the Change in II due to Resistance. — The

assumption involved when dealing with the influence of resist-
ance is that the aerodone is propelled by a uniform force, and
that the net resistance is due to the differences that arise from
point to point in its path between this uniform force and a gross
resistance proportional to Y 2 .

It has been shown (§ 51) that the net resistance under these
conditions is proportional to the quantity //, measured positive
in a downward direction. 1

Now the value of II is continually undergoing variation, in
accordance with the phugoid equation. Any changes due to
resistance may be looked upon as superposed on the equational
changes.

Resistance changes may be conceived to take effect on the
value of II at the ojrposite end of the ordinate, so to speak, to the
equational changes, that is to say, we may imagine the said
changes to be provided for by a vertical movement of the
phugoid chart.

The simplest conception is to suppose the datum line to

1 This is the same thing as taking the net propulsive force -proportional to
h measured positive upward.

DO

STABILITY OF THE FLIGHT PATH §58

undergo vertical displacement, and, as in § 51, to abandon the
datum line in favour of a datum point (Fig. 48), this datum
point being supposed to travel with the ordinate H, marking its
zero from instant to instant. The ordinate II will then be
measured by the amount cut off between two cwves, the upper, or
datum, curve giving the variations due to resistance, and the
lower, or flight path curve, variations due to the phugoid
equation.

Let //i be the departure of the datum point from the axis
of x, due to the resistance, measured positive in a downward
direction.

Let L be the linear measure of the flight path. Owing to the
assumption of small amplitude L is also the linear measure
along the horizontal axis.

Then W L -— is the rate at which the aerodone (of weight

= W) is doing work against resistance, as measured by its loss
of " head." And the work done against resistance is propor-
tional to the net resistance X L, that is, h L where h is the

ordinate of the flight path as in Fig. 45 (§ 51). Or, -tj^ a A.

That is to say, the tangent of the angle of the datum curve is
proportional to the ordinate of the flight path. But the latter is
a sine curve. Therefore the datum curve is a sine curve differing
by one quarter-phase. 1

§ 58. On the Form of the Change in II due to Resistance
(continued). — There is a point of some subtlety to which it is
necessary to devote attention. The relationship shown in the
preceding section between the flight path curve and datum
curve, if postulated as correct at any stated point, will become
less and less accurate as time goes on, owing to the reaction of
the one curve on the other. Thus we know that the displace-
ment of the datum point results in a reduction in the amplitude

1 The well-known result of the differentiation of a sine curve.

91

§ 58 AERODONETICS

of the flight path, whereas we have assumed the flight path
amplitude to be unaffected. Given, however, that the relative
positions of the aerodone, flight direction, and datum point, are
correct at any particular instant, then the datum curve for that
instant is correctly indicated by the sine curve, as set forth.

At the present stage it is perhaps desirable to make clear the
reason for allowing a result so transparently slipshod to pass.
Let us suppose that we have at command means of some kind
by which the value of 1/ can be varied in the reverse manner to
that just demonstrated, and suppose that, laying down a sine
curve flight path, we are able to prove that our new means of
acting on the value of H acts on the datum point in just the
reverse way to that in which resistance acts, that is, by causing
a harmonic variation of opposite phase, then we can so adjust
the relative magnitude of the two disturbing causes as to have
no effect on the datum point whatever.

Under these conditions it is of not the least consequence
whether or no either of the individual disturbances is liable to
an accumulated error, for this error can only arise when the
flight path is disturbed by datum variations, and if the two
disturbing causes act simultaneously no datum variation will
arise. Thus, if we examine any small increment of the flight
path, since no change takes place in the datum level while the
aerodone traverses that increment, the relation of flight path to
datum will hold good at the end of the increment as if neither
disturbing cause existed, and the same will apply to the next
increment and so on indefinitely.

In the present investigation, the two disturbing causes do not
act in precisely the manner set forth above, but the same con-
siderations apply.

§ 59. On the Changes in the Magnitude of H n due to Moment of
Inertia. — The present problem resolves itself into two portions ;
firstly, we have to establish the relationship between a torque or
couple acting on the aerodone, and the resulting change in the

92

STABILITY OF THE PLIGHT PATH §59

value of H H ; secondly, we have to investigate the nature of the
changes in the torque that arise from the rotational inertia of
the aerodone in flight.

We know that the forces that constitute the components of the
couple consist of a reaction on the tail plane on the one hand,
and a reaction added to the load on the aerofoil on the other.
We consequently require to employ aerodynamic data, the expres-
sion II 7 oc 1^/3 being for the moment all that is necessary. 1

IV
Since P or H n we have, W <x H n $ or //„ x — con-

p

sequently,

dH n = dW^- tf/3 3f,
or approximately for small increments,

But A IF is the change in the load on the aerofoil due to the
torque, and is therefore proportional to the torque.

And A /3, the change in the angle |3, is the angle made by the
tail plane to its line of flight ; this is proportional to its pressure
reaction, and therefore proportional to the torque.

Consequently,

*"■ - ' (i - f )•

and for small variations the latter term is sensibly constant,
hence, Aff n a r,

thus the change in the magnitude of H n is proportional to the
torque which produces it.

Now as to the variations of the torque as related to the flight
path. Since we are assuming that the flight path consists of a
sine curve, the variations of tan will be harmonic. But the
values of concerned are well within the definition of a small
angle, hence the variation of will also be harmonic. Now for

1 "Aerial Flight," Vol. I., Aerodynamics, § 159.

93

§ 59 AERODONETICS

a harmonic angular motion we know that the torque at every
instant is proportional to the angle, therefore the torque r and
the variation of H n are harmonic.

Since the variations of H n from the above reasoning follow
tan 0, the curve of variation of H n is a differentiation of the
flight path, and therefore differs from it by one quarter phase
(Fig. 51).

§ 60. Statement of Case. — It has now been demonstrated that
the two disturbing causes under discussion, resistance and moment
of inertia, result in variations of II on the one hand, and H n on
the other ; and, further, that for small amplitude the variations
of both quantities are of the same form, i.e., harmonic. The
object of the further investigation is to show in what manner
the two kinds of variation can be made to cancel out.

Now the quantities H and H n are not directly related to one
another in the phugoid equation in such a way as to render
their interdependence in the present connection quite obvious ;
in fact, in the equation, where if is a variable, H n is a constant.

We cannot destroy a variation of H n by any conceivable
variation in H, or vice versa; neither can we eliminate the
effect, for any variation of H n results definitely in a change of
scale of the phugoid chart, a matter in which variations of //
have no influence whatever.

The key to the position is found in the fact that if H and II n
be made to vary simultaneously in like ratio, the amplitude
constant K undergoes no change, the flight path being repre-
sented as a single phugoid curve, from which the aerodone never
deviates, but whose scale undergoes a regular periodic change.

Now, since we are dealing with curves of very small amplitude,

H is sensibly equal to H n , and we can express the condition of

K = constant in the form,

dll _ .
dll n ~ l '

And the variation of both quantities is harmonic in form,

94

STABILITY OF THE FLIGHT PATH §61

and in effect alike in phase, hence if the conditions are fulfilled
at any one point in the curve, they will be fulfilled for all
points.

§ 61. The Equation of Stability. — Employing British absolute
units,

Let W = weight of aerodone (poundals).
V = velocity of flight (ft./sec).
K = aerofoil constant (W= K P).
I = moment of inertia of aerodone about transverse

axis (lbs. X ft. 2 ).
a = tail area (sq. ft.).
C and c be aerodynamic constants for tail plane, as

defined in Vol. I., § 177.
= aerofoil trail angle (as in Vol. I.).
y = natural gliding angle (radians).
0i = maximum path angle (radians).
I = length of tail, i.e., from centre of pressure of aerofoil

to that of tail plane (ft.),
r = turning moment about a transverse axis (poundals

X ft.).
T X = maximum turning moment about a transverse axis

(poundals X ft.).
L = distance in the line of flight.
L\ = complete phase length (ft.).
t\ = complete phase time (sees.).
P = density of air (lbs. per cu. ft.).
If, H n9 h, etc., stand as before.

Let us deal firstly with the effect of resistance, and express the
maximum rate of variation of H in terms of flight path as

= -T7- ; the quantity thus expressed being thus defined as

the rate of variation at a particular point on the flight
path.

95

§ 61 AERODONETICS

We must first relate 0i to the path amplitude. Let 7*i be the
maximum value of h, then on the harmonic basis,

, tan 0i Tji

2 IT

But when 0i is small, tan 0i = b

i 0i ^i „\

or, h = ~t J. (1)

Now the rate of loss of " head " per foot of horizontal flight
path at V n velocity, on the basis of no propulsive force is = tan y;
or conversely, the steady rate in terms of L at which an increase
of H must be provided by a propulsive force, in order that it
should remain constant = tan y. And we know that the loss of

II per unit flight path varies as H itself ; therefore — ^ -- 1 X

tan y represents the loss of head per foot if no propulsive force is
applied ; hence with uniform propulsion we have,

dll _ H n + 7n .

-77 = — jj — tan y — tan y,

til j ri n

n

dll h

or = — f lin y.

dlj H n
Substituting (1) for h\ the expression becomes,

dH A U ±
-jT = o— . r- tan y.
dls 2tt H n

But by § 86 L x = 2^2^ #«,

d !j = V k 20i«a«y. (2)

§ 62. The Equation of Stability. The Investigation continued.
We have next to deal with variations of II n > due to moment of
inertia.

Let us denote the increment of variation of H n by the symbol
hH n , and in accordance with § 59 let us regard it as consisting
of two parts bill„ due to the change of attitude of the aerofoil,

96

STABILITY OF THE FLIGHT PATH §62

and b*H n due to the change of effective load on the aerofoil.
Let A/7 n , Aiff n , and A a /7 A be the maximum values of these
quantities respectively. Likewise let 5/3 be the increment of
variation of the angle of trail of the aerofoil, (/3), of which the
maximum value is A/3; A/3 therefore is also the angle of
inclination of the tail plane with respect to the incident stream.
From aerodynamic considerations we may write, pressure
reaction on tail plane

= C p a V n * X e »/3

and torque corresponding to this reaction,

r = C p a V n * c 6/3 I

or ' ifi = CpJvjcl

% TT -H* T

- *lHn = 77

C P aV n *cip
or for maximum value,

~ Al "" = CpaVSclp (8)

Now as to 5a//„, i.e. the variation due to change of the effective
load on aerofoil, due to component of couple, we have

T = lbW,

and since H» oc W, -,,— = — ~-,

H

n

Let AT represent the aerofoil constant, so that W = K V n 2 , then
substituting t = j. - — ,

or, b*H n =

»
I K VJ'

n

a. 97

§ 62 AERODONETICS

Writing this for maximum value we have

Now,

AaZ7 » = ntr? (4)

AH n = AxH, + A a ff„ = cJ^cTfi + KtrV

AH n = ?fj (^1^ + i), (5)

and this variation takes place in one quarter-phase length of the
flight path.
Now on the harmonic basis, maximum rate of change, i.e.,

n is x times the mean rate ; hence

dL 2

dL "~ 2 ~ Li '

but Li = 2\/2"irff tt .

Substituting (5) for A H n and for Li, we have

*H« - I y 4g w Tx / 1_ 1 \

= VIC * U + Cpacfi} (6)

§ 63. The Equation of Stability. The Investigation concluded. —
The next step is to express the value of the maximum torque r x
in terms of the moment of inertia I and the angular amplitude
of the flight path.

The magnitude of n is the maximum moment of the couple
generated in causing the aerodone to oscillate harmonically about
its transverse axis through a total angle equal to twice the
maximum inclination of its path.

Now the total angle of oscillation is 2 0i, hence r x will be
given by the expression,

Ti - ~ir x 2 ° x - -«!«—•

98

STABILITY OF THE FLIGHT PATH §63

Substituting in (6) we have

dH n _ 4 0x1^ / 1 , 1 \ m

dL^ ^2lV n *h* X \K + Cpaefi )' ' (i>

And the conditions of stability are denned by the equation

riff dH H
dL^ dL'

.-. by (2) and (7),

, - 4 0! I * 3 / 1 , 1 \

V 2 0i tan y > -^ ^JyJ]^ \k+ C p a c fi )'

or

2lir» / 1 . 1 \

ton y > j-pr^ ^ + c ^ J-

But F. = ^-\ .'. by § 86, F B » tf = 8 * 3 fl, 3 .

Hence we may write the expression —

, h 4 I g n * tan y t

*-fn + ZX~) >1 '

* \K^ cC pap)

which is the equation of stablity, <I> being termed the coefficient
of stability , which, as the equation shows, must be greater than
unity.

There is one factor that has not been taken into account in
the foregoing analysis — the influence of the " wash " of the
aerofoil on the angle of the tail plane. It is evident that since
the aerofoil communicates downward motion to the air coming
within its sweep, the tail plane will be situated in a down current,
and consequently its attitude of zero pressure reaction will be
inclined to the line of flight. The extent of this inclination will
depend upon the distance of the tail plane astern of the aerofoil.
If the two are in close proximity the inclination of the tail for
zero reaction will be but little less than itself. If, on the other
hand, the tail plane be arranged far astern it will only require to

99 h 2

§ 63 AERODONETICS

accommodate the residuary down current of the peripteral system,
which will be given by the expression

which may be otherwise expressed as /3 (1 — c) where c is the

constant ratio g proper to the aerofoil. 1

Now so long as there is no change in the value of /3 the present
considerations have no influence on the problem, for the tail
plane is initially designed to occupy its position of zero pressure
reaction, and there the matter ends. When, however, any
change takes place in j8, as due to the moment of inertia, this
reacts on the effective tail angle, so that if 5/3 be the initial
variation, the relative direction of the tail current or "wash"
will change through an angle 5/3 (1 — *), and the consequent
displacement of the tail will be through an equal angle. This
will result in a further increment 5)8 (1 — e) being added to the
effective value of j3. This in turn will give rise to a further
change in the tail angle = 5)8 (1 — c) a , and so on, the total
resulting addition to the increment 5/3 being the sum of a
series

5/3 (1 - c) + 5/3 (1 - e) 2 + 5/3 (1 - e) 8 , etc.,
which is given by the expression,

c
consequently the increment as given in the foregoing analysis

1 — € 1

requires to be multiplied by a factor 1 H , that is x -,

in order to correct for the influence of "wash."

On applying this correction, equation (8) of § 62 becomes,

* lMn -C P aV»*cip X V
1 " Aerial Flight," Vol. I., Aerodynamics, Oh. VIII.

100

STABILITY OF THE FLIGHT PATH §63

and, carrying the correction through, we reach finally,

d> - 4 I H n * tan y

X \K^ cC pea?)

which is the complete equation oj stability. It will be noted that
the correction has been based on the supposition that the tail
is long, that is to say that the tail plane is well out of the
immediate influence of the aerofoil, and is only affected by the
residual peripteral motion.

101

CHAPTER VI

EXPERIMENTAL EVIDENCE AND VERIFICATION OF THE PHUOOID THEORY

§ 64. Preliminary. The Importance of the Experimental Verifica-
tion. — The experimental verification of the conclusions of the
preceding chapters is evidently a matter of very great moment,
for the results obtained from theoretical considerations, if experi-
mentally established, form a definite proof, not only of the
aerodonetic portion of the work, but also of the aerodynamic
fabric in which the phugoid theory has its foundation.

The checks that have so far been applied to the theory are of
various kinds. In the first place there is the general qualitative
resemblance of the curves of flight, both as determined and
illustrated by other workers and by the author himself, to the
curves as plotted from the phugoid equation. Secondly, there is
the direct measurement of the phase length, phase time, and
velocity relationship as a check on the results given in § 36.
Thirdly, there is the experimental determination of the condition
of neutral stability for the phugoid of small amplitude, as a check
on the investigation given in Chapter V., and the demonstration
of the conditions of flight path stability and instability by means
of model experiments. Lastly, there is the post-mortem examina-
tion of birds as throwing some light on the extent of the regard
paid by nature to the requirements of theory ; in this category
may also be included the examination of gliding machines and
models, made empirically in the absence of any knowledge of the
equation of stability.

It will be shown in the present chapter that the theory is able
to withstand every test that has so far been applied.

It is important to point out that although the model experi-
ments employed for the purpose of demonstration are on a small

102

EXPERIMENTAL VERIFICATION §65

scale, on a very small scale in some cases, the theory makes
no discrimination in the direction of a superior limit of size, and
that therefore the results will apply also, within determinate
limits of error, to full-sized models or even flying machines of
many tons weight. There is a difference between reasoning from
empirical data from the small to the large, and in the applica-
tion of a generalised quantitative theory, proved true in respect of
any one sized model, to any other sized model that may be desired.

§ 65. Penaud's Experiments. — Some successful experiments
were made by Penaud about the year 1870, with small paper
winged models, propelled by twisted rubber and a screw
propeller. Mention has already been made of these experiments
in § 9, and a Penaud model has been illustrated in Figs. 9 and
10. One of these models is stated to have crossed a pond 40 or
50 yards wide, and to have shown complete stability.

The theory of longitudinal stability held by Penaud as set
forth is not satisfactory, but in spite of this fact the author feels
that the defective argument, in the form presented, may be
largely a matter of faulty description, and that perhaps Penaud
had actually in his mind some idea, even if of rather an indefinite
kind, of the theory subsequently propounded by the author in
1894, mentioned in Patent Specification 8608 of 1897, and given
in § 3 of the present volume. In order that this point may be
judged on its merits, the text of Penaud's argument is given in
Appendix I.

Only slight mention is made of the phugoid oscillation ; a
note occurs as follows calling attention to the point. Penaud
says : — " On observe alors assez souvent des oscillations dans le
vol, comme nous en voyons decrire aux passereaux et principale-
ment au pic-vert." l

1 The Oreen Woodpecker. The author has remarked aud commented on
the undulating flight of this bird in Vol. I., Appendix IV. The resemblance
traced by Mons. Penaud is more apparent than real, the undulations in the
flight path of the Green Woodpecker are not those of the phugoid theory as
is the case with an inanimate model.

108

§66

AERODONETICS

§ 66. Mouillard's Observations. — M. Mouillard, in his " L'Empire
de l'Air " (1881), describes at some length the behaviour of a
rectangular aeroplane both ballasted and otherwise, and gives
diagrams representing the forms of motion observed. Un-
fortunately no information is given from which the relative scale
of the curves can be estimated, and the representations are at
best but sketches.

Fig. 52 is from a diagram given by Mouillard as typical of
the motions of the ballasted aeroplane, but the corresponding

Fig. 52.

reference in the text is insufficient to make clear the nature
of the motions that are depicted. The author was at one time
inclined to think that Mouillard had actually seen and endea-
voured to record the phugoid oscillations, but it is by no means
certain that such is the case. The whole of the description is
obscured by theories of an impossible kind, and it is only here
and there that passages occur that are really relevant.

It would appear from the description that Mouillard's card-
board aeroplanes were very inferior to the mica planes described
in Chapter I. and in Vol. I., § 162, and it is quite possible that

104

EXPERIMENTAL VERIFICATION

§66

the oscillations . recorded by Mouillard were merely vibrations
about the attitude of equilibrium, that is to say, oscillations of
much quicker pitch than those of the flight path, due to some
initial disturbance of the balance between the attitude of the
plane and the position of its centre of pressure. This possibility

Fio. 53.

is suggested by the following passage, " Tres souvent un aeroplane
produit, a part ces monrements imperceptibles 9 on pourrait dire
theoriques, d'autres grands mouvements de chutes et de releve-
ments : cela tient a ce que la correction du gouvernail n'est pas
parfaite." This passage occurs where Mouillard discusses the
effect of a fold (pli) made in the rear portion of the aeroplane

105

§ 66 AERODONETICS

to minimise the extent of the oscillations : this fold he refers to
as acting as a rudder (gouvematf).

The author has noticed quick pitch vibrations occasionally
when experimenting with mica models, which are in no way
related to the phugoid oscillation.

It would appear from the foregoing that Mouillard's observa-
tions must be considered a doubtful quantity in the present
connection.

§ 67. Haroy's Experiments. — Mons. E. J. Marey, in his " Vol
des Oiseaux," describes an experiment made with a paper

Fra. 54.

gliding model of a pattern devised by Mons. J. Pline in which
a photochronograpbic record is made of the flight path when
the model being suspended from its tail is let fall from rest.

The form of the model employed is illustrated in Fig. 53,

drawn from the representation given in Marey 's work. It is

described as made from a single sheet of paper, folded in some

manner not made clear, and ballasted by means of a needle or

106

EXPERIMENTAL VERIFICATION §67

§ 67 AERODONETICS

wire that may be slid forward or backward in order to shift the
centre of gravity as may be desired. 1

The flight path described on being released head foremost is
given in Fig. 54, the point indicated in successive positions
being that marked "A" in Fig. 58; in the original in Marey's
work the motion of the model is given as photographically
recorded.

According to the phugoid theory we should have to deal here
with the special case of the semicircle, but though the first
portion of the curve is approximately the arc of a circle, the
latter portion tends to become a straight line. It may also be
noted that the model undergoes very little acceleration during
the latter portion of the flight recorded, the " pitch " of the
photographic images being nearly constant.

It is evident that in the latter part of the flight the model has
practically settled down to its path of uniform gliding, in which
case the damping of the phugoid oscillation must be very rapid
indeed, so much so that the aerodone is to all intents and pur-
poses " dead beat." This is quite in harmony with the fact that
the gliding angle y (taking the latter portion of the flight as the
gliding angle) appears lo be about 1 in 2, which betokens a
resistance nearly 50 per cent, of the weight, and consequently
a very rapid decrease of the phugoid amplitude.

Marey does not state the number of images per second
employed in the case in question, but he gives the scale of the
diagram by division into 10 cm. squares. Taking the radius of
the circular path as 80 cm., the value of H n will be 27 cm.
approximately or = "9 ft., the corresponding velocity is V n =
7*5 ft./sec, and the pitch of the images on the steady gliding
portion of the curve is 5*5 cm. By division we obtain 40 as the
number of images per second. If any record exists of this
experiment it would be interesting to know the actual number of
exposures employed.

* A drawing of a model made by the author from Mons. Marey's descrip-
tion has already been given, Fig. 11.

108

EXPERIMENTAL VERIFICATION

§68

( E

s

'B

•22

US

o

M

§68

AEEODONETICS

§ 68. The Author's Experiments in Confirmation of the Phugoid
Theory. — By the use of thin lamin© of mica in combination with
other improvements, the author has succeeded in bringing flight
models of small size and low velocity to a degree of perfection
not previously obtainable, and during the last few years, 1905-7,
a considerable amount of work has been accomplished by means
of these mica aerodones in the confirmation and extension of the
Phugoid Theory.

A photograph of a mica aerodone is given in Fig. 55, and
scale drawings of some of the models used for the purposes of
quantitative investigation are given in Figs. 56, 57, 58, etc.

Fia. 57 (§ full size).

The method of construction is as follows. A mica lamina of
a few thousandths of an inch in thickness, or even less than
one one-thousandth of an inch in some cases, is cut to the shape
required to form the aerofoil and fastened to a " bolster," which
is a piece of wood of the necessary form to give to the aerofoil
the curvature desired. The tail plane and the fins are also of
mica, and are attached by fish glue or other adhesive to a
wooden backbone, which also carries the necessary ballast, in
the form of lead foil, at its forward extremity.

With an aerodone constructed as above described the whole
of the properties of the phugoid curve can be readily demon-
strated in a large room or lecture theatre, from the simple case

110

EXPERIMENTAL VERIFICATION §69

of the steady glide to the curves of tumbler type. With a
certain amount of practice with any individual model it becomes
quite easy, by varying the initial velocity imparted, to get any
recognised type of flight path at will, and by varying the pro-
portions of the aerodone in accordance with the requirements
of theory, the flight path may be rendered stable or otherwise
as desired.

§ 69. Confirmation of the Phugoid Theory as to Time, Velocity
and Phase Length. — By means of these mica models, the author
has been able to demonstrate experimentally the relations
established in Chapters II. and III. as summarised in § 36.

In these experiments, the first series of which took place in
1905, soon after the method of making these low speed models
had been developed, the largest model employed did not exceed
five grams, and the smallest was about one-tenth this size. In
spite of this fact, the agreement with theory was found to be
almost perfect.

Now since if there is a departure from the conditions of the
hypothesis, this departure will be greatest for small models and
low velocities where the F 2 law of resistance begins to break
down, it is evident that for birds of any size, and still more for
aeronautical flying machines, the main results of the phugoid
theory may be taken as rigidly applicable.

Example.

Mica aerodone. Series A. Model No. 2. Weight, 4'7 grams
(Figs. 56 and 57).
Observer, author, assisted by Mr. A. J. Hill, July 3rd, 1905.
Velocity. A flight of 85 ft. occupied 6 sees., or

V n = 14 ft. sec.

Phase length. The distance occupied by five complete oscilla-

129
tions measured 129 ft., or (§ 86) L\ = --- = 26 ft. approxi-
mately.

Ill

§ 69 AERODONETICS

Time period. On semicircular flight path (aerodone let fall
and time taken to first crest 1 of flight path), three observations 2 ,

1-9
1-9
20

8) 6-8

1-98

On undulating flight path (aerodone launched at slightly over
its natural velocity),

20

20

1-8

8*6 (double period)

8'6 (double period)

4'0 (double period)

20

10 ) 19-0

1*9 sees, average.
Thus, h = 1-9.

In order that the above experimental determinations may be
used as a check on the theoretical investigation, they require to
be compared on the basis of § 86. This may conveniently be
done by the calculation of IT^ from the three different sources ;
the substantial identity of the resulting values constitutes the
proof.

(1) From the measured velocity, by the ordinary equation to

the falling body,

196
H n = i^ = 8-04 feet.
n 64*4

1 Owing to the damping factor the flight path does not form a cusp, but
rather a crest, the curve becoming inflected

9 The time being taken by an ordinary stop watch, the division into legs
than J second is probably not justifiable.

112

00

«

o

A.

118

§69

AERODONETICS

3

114

EXPERIMENTAL VERIFICATION §70

H n = t, L = 298 feet.

(2) From the phase length (by § 36),

26

8-88

(8) From the time period (by § 86),

7/ » = ( H56 )' = 2 ' 95 -

The whole of the above results are within 4 per cent., which
does not exceed the probable magnitude of experimental error.

Again. Series B. No. 2. Fig. 58.
V n determined as = 14 ft. sec.
h „ „ = T83 (average of 8 trials).

From V n we obtain i/ = 8'04 feet.
„ ti „ „ /f=274 „

Again. Series C. No. 1. Fig. 59.
V n determined as 10 ft. sec.
h „ „ T37 (average of twelve trials).

From V n we obtain H n = 1*56 feet.

99 h 99 99 1*11 == 1*54 „

§ 70. Verification of the Equation of Stability. — In the verifica-
tion of the equation of stability, the considerations discussed
in § 64 still apply, so that the range of the equation should
be restricted only by some inferior limit of size; hence the
importance of small scale models and low velocities. It is
at the same time desirable to extend the range of the experi-
mental verification as far as possible in the direction of larger
sizes and higher velocities as representing more nearly the
actual conditions of practical aeronautics.

Up to the present the author's quantitative experiments have
been conducted indoors, with models ranging up to about one
half ounce weight, but the application of the equation to the
author's 1894 models, and to the gliding machine of the late
Herr Lilienthal, forms an effective continuation.

115 i 2

§ 70 AER0D0NETICS

The sizes and velocities suitable for employment vary with
the circumstances under which experiments can be conducted ;
thus, experimenting indoors in a room 20 ft. by 80 ft, an
aerodone having a natural velocity of about 5 or 6 ft./sec. is
found to be most suitable ; the short phase length corresponding
to these low velocities — some 4 or 5 feet — renders it possible
to observe six or eight undulations of the phugoid path, and so
to determine whether the amplitude is subject to change, or
whether it is sensibly constant. When experiments are con-
ducted out of doors a natural velocity of about 14 ft./sec. is found
to be appropriate ; this, whilst not excessive from the point of
view of ease of observation or of convenience of launching by
hand, is sufficient to render the aerodone consistent in its
behaviour on a calm day.

Models of more than a few ounces weight require to have a
higher natural velocity, and it becomes desirable to employ a
launching device such as the catapult described in § 12 ; it is
then no longer any advantage to keep the natural velocity
down to the minimum, and velocities of 25 or 30 ft./sec. may
appropriately be employed. In this case the phase length will
amount to something like 100 to 140 feet and a flight of two
or three hundred yards is desirable, involving a drop of about
a hundred feet. Thus the experiments should be conducted
from the edge of a cliff or other elevated point.

§ 71. Experimental Verification of the Equation of Stability.
— For the purposes of the present investigation four models
have in all been employed. These, in order of size, are
nominally as follows: —

12 gram model : Figs. 55, 60, and 61.
4 „ „ Fig. 62.
£ „ „ Fig. 68.
" One grain " model : Fig. 64.

The above aerodones were in the first instance designed in

116

EXPERIMENTAL VERIFICATION

§71

\

A

O

/

§ 71 AEEODONETICS

accordance with the principles set forth in Vol. I. for minimum
resistance, and rough calculations were made in advance in order
to ensure that the stability factor <f> should be approximately
equal to unity. The models were then constructed and adjusted
to the critical condition by observation of the flight path, the
final measurements being then made, and so the data obtained
for the final calculations.

In the case of the larger two models the adjustment was made
from the unstable to the stable, by adding ballast in the immediate
vicinity of the centre of gravity; in the emaller models the
adjustment was made in the opposite direction, i.e., from the
stable to the unstable, by the reduction of the tail area.

Incidentally the present experiments demonstrate that,
generally speaking, an aerodone that is just stable for paths
of small amplitude is unstable if the amplitude is great, though
whether this is universally the case is not certain.

The whole subject of model construction and adjustment is
dealt with in Chapter X., where the necessary detail instructions
will be found to enable a repetition of these experiments to be
made.

§ 72. Experimental Verification of the Equation of Stability
(continued). — The data relating to the 12 gram model, Figs. 55,
60, and 61, are as follows : —

Mass = 12-3 grams = -0267 lbs.
.-. W = -86 poundal.

V n = 9'8 ft./sec. (observed value)

/. H = |i = 1-85 ft.

— V 2 — 8( J. 5 — Ul.

= -24.
Tan y = # 18 (observed value).

118

EXPERIMENTAL VERIFICATION

§72

Tail data,

I = -28 ft.

a = -0625 sq. ft.

Constants,

p = -078.

C = -7

c = 227

(tail data)

n = 4.
(aerofoil)
n = 12.
Moment of inertia,

I ='00085 lb. ft. 2 (by measurement and calculation).

= -75

Fio. 61 (£ full size).

Condition of flight path observed for three successive oscilla-
tions, recorded ^ajttst stable for small amplitude.

Calculation of 4> —

4 I H n * tan y

(1 + I )

<*> =

c C p c a j3

_ 4 X '28 X 1'82 X '13
~" 00035 ( 100 + 716 )

- *** - 03
~ 285 - m '

119

§ 72 AERODONETICS

Now the observed condition of flight path corresponds to
4> = 1, so that the agreement is within 7 per cent, which is as
close as is to be expected. The discrepancy, such as it is, may
be due to some imperfection in the equation, for undoubtedly
imperfections exist; there are factors of secondary importance
of which the theory takes no account. It is equally possible,
however, that it is due to inaccuracies of observation and
measurement, the possible error in the determination of the
natural velocity, V n , alone might easily account for a difference
of greater magnitude than that recorded.

Again, 4 gram model, Fig. 62, the data are as follows : —
Mass = 4'52 grams = *01 lbs.
.-. W = -32 poundal.
' V n = 9-5 ft./sec.

' n ~ 64-4

= 1-4 ft.

W '82
K = V* = 90 = - 0086 -

= -2.

Tan y = -15.

Tail data,

I = -25 ft.

a = -028 sq. ft.

Lonstants,

P = -078.

C =-7 1

(tail data)

c = 2*27 .

n = 4.

« = -71 (a

erofoil) n = 9*8.

Moment of inertia,

I = '000138 lb. ft. 2 (by measurement and calculation).

120

.1

0)

^2

d

§72

AERODONETICS

Condition of flight path observed for three successive oscilla-
tions, recorded as nearly stable for small amplitude.

Calculation of 4> —

4 X -25 X 1-96 X -15

* =

•000133 ( 285 + 2040 )
•294

•809

= -95

which is substantially in agreement with the observed condition.

§ 73. Verification of the Equation of Stability (continued).
Small Scale Experiments. — In order to ascertain if possible the
extent to which the theory may be limited in its application to
models of small size, measurements were made of an aerodone
of nominally J gram weight, Fig. 63, with the following
results : —

Mass = '245 gram = '00054 lb.

.-. W = '0174 poundal.

V H = 5 ft/sec.

25

H. =

■»

= -89.

K =

fi =

Tan y =

Tail data,

I =

a =

Constants,

9 =

C =

= -000695.

64-4

0174
25

25.

265 (mean of two observations).

105 ft.
00184 sq. ft.

078.
67.

= 2*0 (Tail data; n = 2 approx.).
= # 7 (Aerofoil data ; n = 9) .

122

.a

00

CO
CO

o

§ 73 AERODONETICS

Moment of Inertia,

I = -000,000,885 lb. ft. 2 (by measurement and
calculation).
Condition of flight path recorded as just stable for small
amplitude.

Calculation of 4> —

4 X '105 X -152 X -265

4> =

•000,000,885 (1,440 + 29,700)
•° 169 = -65.

•026

In this case the agreement is not so close as in the preceding
examples, the stability according to direct observation being
apparently 50 per cent, better than as computed from the
equation. Further trials of the model failed to show any error
in the observation data. An inaccuracy of 10 per cent, in the
velocity measurement would be required to account for the dis-
crepancy ; it is unlikely that an error of one quarter this amount
would have escaped notice.

It is most probable that the estimate of the moment of inertia
is in error. The measurement of so small a model is a matter
of some difficulty without special apparatus; but it is also
possible that the viscosity of the air has a steadying influence
on models as small as the present example.

§ 74. Verification of the Equation of Stability. Small Scale
Experiments (continued). — Having thus ascertained that the laws
on which the aerodonetic theory is based hold good, as a rough
approximation, down to a model weighing but one quarter of a
gram, it was decided to carry the investigation to a still finer
point. To this end an aerodone was designed to weigh only one
grain ('065 gram) and to have the same velocity as the previous
model, 1 namely, 5 ft./sec. (Fig. 61).

On completion, and after adjustment, the actual weighing gave

1 It was thought preferable to vary only the size of the aerodone and not
its velocity, to make sure that the conditions are other than those of
corresponding speed, § 126.

124

EXPERIMENTAL VERIFICATION

§74

'062 grams, or a trifle under that intended, but the natural
velocity was exact to within a possible 5 per cent, error. The
adjustment made in the first instance gave a degree of stability
sufficient to render the flight path steady enough for the purpose
of observation, the author estimating, from cxpeiience with other
models, that the stability factor = 2 or thereabouts. The actual
value was then calculated from the equation, as below, the result
being 1*67. The tail plane was then altered to reduce the stability
factor, and the further flight trial served as a final confirmation

Fig. 64 (full size).

of the truth of the equation ; with 4> = -84 the flight path had
become capricious.

The measurement of the moment of inertia of so small a model
presented some difficulty with the rather primitive apparatus at
the author's command, and in the present case the value of this
quantity was arrived at by calculation, the component parts
being weighed separately, and their moment of inertia individually
and collectively assessed.

One grain model (Fig. 64). Data as follows : —
Mass = -062 gram = '000187 lb.
.\ W = '0044 poundal.

V n = 5 ft./sec.
.'. H n = -89 ft.

0044

A' =

25

= -000176.

j3 = '24 (mean value).
Tan y = '8.

125

■

§ 74 AERODONETICS

Tail data,

I = '052.
a = -0019.

(Tail data).

Constants,

= -078.

C = -7
c = 28

e = -68 (Aerofoil data).

1 = -000,000,17 lb. ft. 2 (computed from weight of

components).

Observed flight path quite stable, 4> estimated about = 2.

Calculation of 4> —

- _ 4 X '052 X '152 X '3
- -000,000,17 (5,700 + 27,800)

= '°° 9 - 5 = 1-67
-0057 '•

The alteration made with a view of reducing the flight path to
the "just stable " condition consisted in cutting the tail down to
approximately half its lateral breadth, the portions cut off being
made to adhere by capillarity to the remaining portion so that
the balance should be unaffected. The alteration involved new
data as follows : —

a= -001

c = 2-0.

Re-calculating we have,

4 X 052 X '152 X '8
•000,000,17 (5,700 + 61,000)

__ '0095 _

"" '0118 "~ tt

The actual flight path of the altered model was found to be
scarcely stable ; it was impossible to obtain flights of a suffi-
ciently consistent character to make a redetermination of either
V n or y. Unfortunately, this model was on the point of developing

126

EXPERIMENTAL VERIFICATION §75

a lateral oscillation (see § 90), which shows itself first in the
region of the crest of the phugoid flight path. For this reason
the instability that arose was of a mixed kind, but the main
conclusion was not in doubt.

We are thus able to conclude that, even in the case of a model
of but one grain weight, the departure from the ordinary laws of
flight is comparatively unimportant, and that the influence of
viscosity does not give rise to a new regime till the size is still very
much further reduced. This conclusion can only be, in reality, an
approximation, for the author has failed, in spite of repeated
endeavour, to obtain gliding angles (i.e., values of y) for these
very small models to approach those easily obtainable with
models of say even a few grams weight. Thus for a one grain
model, n = 6, the best result so far obtained is that given in the
example tan y = *8 ; whereas for the model of '25 grams weight
tan y = *26 ; for a model of "6 gram weight (aspect ratio = 6),
tan y = % and for a model weighing 80 grams, y = *17 is a
value easily obtainable. It is evident, therefore, that in -these
very small sizes we are approaching the point when the influence
of viscosity begins to materially affect the law of resistance, in
spite of the agreement with the equations.

§ 75. The Stability of Birds in Flight— The application of the
stability equation to birds in flight is not to be undertaken
without some preliminary examination of the conditions. The
circumstances surrounding the problem as presented by nature
are such as to obscure to some extent the more essential factors.
Thus the directive organ, the tail, is not, as is the case in the
gliding models employed by the author, a clearly defined plane
whose constants are known, but rather a prolongation of the
streamline form constituted by the body of the bird. In addition
to this the problem is complicated by the question of active flight.
We do not know precisely what may be the effect on stability of
the mobility of the aerofoil, that is to say, the flapping of the
wings practised for the purpose of propulsion. Beyond these

127

§75

AEEODONETIGS

two points there is also the question of the flexibility of the after
edge of the aerofoil, which evidently renders the angle /3 variable

Fit). 63.

under changes of effective load such as arise in the course of the
phugoid path.

Dealing with these points in the order raised, we have first to
discuss the influence of the position of the tail on its effective

EXPERIMENTAL VERIFICATION §76

value. It is evident that as a generalisation of the conditions it
would be necessary to abandon the notion of treating the tail-
plane as a separate entity altogether, in favour of an expression
denoting the directive value of the whole streamline body, tail
and all.

It would seem, however, that a restriction exists that renders
a generalised treatment unnecessary. It is believed from the con-
siderations discussed in Gh. VII., § 101, that the centre of lateral
resistance for small inclination to the line of flight must approxi-
mate very closely to the centre of gravity, in order that the lateral
and directional equilibrium should be stable. 1 It follows from this
that the streamline form constituting the body must be almost
in neutral equilibrium about a vertical axis. But with the
exception of the tail, an organ which may usually be clearly
defined, the form of the body of a bird is, so far as it is possible
to estimate, a solid of revolution ; hence it will, if divested of the
tail plumage, be also in neutral equilibrium about a horizontal
axis. Therefore we may regard the whole directive value of
the combination as due to the tail itself, taking due account
of its position in relation to the flow of the stream in its
vicinity. 9

§ 76. The Stability of Birds in Flight (continued).— As illus-
trating the method of treatment under discussion, diagrams are
given of the body and tail of the common swift (hirundo apus),
Fig. 65, and the wandering albatros (diomedta exulans), Fig. 71.

Referring to Fig. 65, which is a § life-sized representation, it
is evident that the tail may be regarded as consisting of two tri-
angular fins projecting from the surface of a streamline solid of
revolution into the surrounding stream. Most birds have the
power, which they exercise to a very great extent, of expanding
or contracting the tail area at will; this capacity is well
developed in the swallow family, and is indicated in the figure

1 Compare §§ 95 et seq,

• See "Aerial Flight," Vol. L, Aeroilynamics, Chaps. I. and III.

A. 129 K

to
o

M

EXPERIMENTAL VERIFICATION

§76

by the two extremes, the solid line representing the extent to
which the tail is developed under ordinary conditions, and the
dotted line showing the full expanse. It may be remarked as a
matter of observation that the swift varies its speed of flight

I

Fig. 67.

within wide limits, probably from about 20 feet per second to
something over 50 feet per second, 1 and it does not close its

1 The speed of flight of the swift, though perhaps at times greater than
here given, has been grossly exaggerated by many writers ; it has even been
stated to exceed 200 miles an hour !

181

k2

§76

AEBODONETICS

wings in flight, as is common with many of the smaller birds
when exceeding their normal speed of flight. 1 The tail area is
noticed to be greatest at low velocities, and the wing area may
also be observed to undergo a similar variation, the silhouette at
high and low flight velocity being represented respectively at (a)
and (b) in Fig. 67.

In the case of the albatros, Fig. 71 (J full size), the tail
proper is of very small area, but the legs are carried extended,
and the area of the webbed feet forms a substantial addition to
the effective tail spread. 2

The conditions under which the tail member or members have

Fig. 68.

to perform their function are illustrated diagrammatically in
Fig. 68. Owing to the fact that the region of the tail of a
streamline body is one of increased pressure* the velocity of the
stream (relatively to the body) is less than the velocity of flight ;
this follows from hydrodynamic principles. Under actual con-

1 Some of the smaller members of the swallow family partially fold their
wings when in rapid flight, the common house martin for example, Fig. 66.

3 In the official reports of the South Polar Expedition, Mr. E. A. Wilson
says: •' When on the wing the feet are held folded together at full length
under the tail and extending well beyond its longest feathers, giving the
effect of a markedly wedge-shaped tail with a white terminal border.
This, of course, is not the case, for the tail is bordered by black at the
extremity, and the appearance of white beyond the black is due to the
whitish feet."

3 Compare " Aerial Flight,*' Vol. I., Aerodynamics, § 11, also Chap. III.

182

EXPERIMENTAL VERIFICATION § 77

ditions this difference is added to by the fact that the relative
velocity of the stream has been further reduced by the skin-
friction, the tail being surrounded by the f fictional wake. 1 Under
these conditions the directive efficiency of a given area must be
materially less than would otherwise be the case.

The form of the areas employed by Nature in the tail member
is one of comparatively low aerodynamic value, the value of n,
the aspect ratio, being generally less than unity. 2 Now for
n = 1 we know that the value of the constant c = 1*8 and
probably for the forms actually employed c will have a value
considerably less than this amount. 8

Taking into account both the low value of n and the adverse
conditions under which the tail member has to perform its duty,
it would appear that in round numbers we may take the effective
value of c equal unity without being far from the truth.

§ 77. Stability of Birds in Flight (continued). — The influence of
active flight as affecting the conditions of stability is one of which
it is very difficult to take account.

Inasmuch as the wings perform their function of support with
the same effect, whether they are in motion, as in active flight,
or whether they are rigidly extended, as in gliding or soaring,
and since the manner in which they act, so far as the support of
tho load is concerned, is the same in both cases, it is improbable
that there is any fundamental difference that will invalidate the
application of the theory. It is, however, likely that there may
be some quantitative difference, owing to the intermittent nature
of the supporting reaction, that would have to be met by a special
coefficient.

For the time being it has been deemed inadvisable to attempt
to deal with any such addition to the theory. In the calculations
that follow it will be assumed that we are studying the conditions

1 "Aerial Flight," Vol. L, Aerodynamics, § 17.
* " Aerial Flight," Vol. L, Aerodynamic , § 150.
8 •• Atrial Flight," Vol. L, Aerodynamic § 159

183

CO

fa

CO

o

EXPERIMENTAL VERIFICATION §78

that obtain when the bird is gliding or soaring, and our results,
although strictly speaking limited to these conditions, may be
taken as being of the right order of magnitude when the more
complex conditions supervene. In the flight of the two birds
under consideration, the swift and the albatros, the gliding or
soaring mode is very largely employed.

The influence of wing flexion under changes of load is one of
which no account has been taken in the Phugoid Theory. From
an examination of the conditions it would seem that if this be
the only addition imported into the hypothesis of Chapter II., the
change in the form of the phugoid curve will be of a symmetrical
character, and to some extent the form of the curve for a given
aerodone will, if the aerofoil be made flexible after the manner
of the wings of a bird, undergo a change in the direction of
resembling a phugoid of greater H n value.

It is difficult to follow the matter much further, but it appears
certain that an appropriate degree of flexibility may add very
appreciably to the stability, both in face of an adverse gust of
wind, and as influencing the effective value of the stability co-
efficient <P. Thus if it can be shown that a given bird or
aerodone possess stability on the assumption that the aerofoil is
rigid, then it will probably possess stability in a greater degree
if the aerofoil is flexible. We may therefore treat the wing
spread as rigid for the purposes of the investigation.

§ 78. Stability of the Hirnndo Apus. — Fig. 69 represents, half
life size, an example of the common swift shot and measured by
the author. The data are as follows : —

Mass = 40*6 grams = '09 lbs.

.-. W =2*9 poundals.

V n = Natural velocity, taken as = 32 ft./sec.

H n = 16 ft.

K =-0028.

= -15.

Tan y = '16.

185

o
I-

o

4

EXPERIMENTAL VERIFICATION § 79

Tail data,

I

= 3.

«

a

= -007.

Constants,

P

= -078.

C

= -66.

c

= 1-0.

€

= -68.

Moment of

inertia,

Radius of gyration =

= '11 ft. (measured). 1

1 =

•0121 X

•08 =

•00097 lb. ft 2 .

Calculation of * —

<f> =

4 X *8 X 256 X 16
•00097 (857 + 27,500)

^^

49

27 ~~

1-8.

Thus we have the result that the stability factor is nearly
double the critical value. This multiple may be looked upon
as the " factor of safety," by which a sufficient rate of damping
of the phugoid amplitude is secured.

§ 79. Stability of Diomedea Exulans. — In the preceding example
the moment of inertia was determined by actual measurement,
in the case now under consideration the author has been under the
necessity of taking his measurements from a " set up " specimen
of a young bird (Fig. 70), of probably about 14 or 15 lbs. weight
when killed. The degree of approximation to which we are at
present able to work, renders slight inaccuracies of data
unimportant.

1 Measured by the method of double suspension described in Appendix VI.
The influence of the outstretched wings on the moment of inertia is
negligible; hence the radius of gyration was measured with the wings
removed, and the mass employed in the calculation of the moment of inertia
as that of the body of the bird under like conditions.

187

§ 79 AERODONETICS

The body forms of birds capable of flight are of exceedingly
uniform design, and it is evident that the relation of the radius
of gyration to the length of the body differs but little in
different species. The problem is, therefore, similar to that
of the moment of inertia of an ellipsoid or other settled
geometrical form. Owing to the preponderating influence of
the length of the body in the direction of the axis of flight,
the radius of gyration may be expressed approximately in terms
of this axis ; thus, —

Radius of gyration = Length^ streamliners

The length of the streamline axis is measured from the point
where the body form merges into the tail plumage, to the point
where the beak may be estimated to finish, assuming the stream-
line form to be completed symmetrically, as is actually the case
in the example of the preceding section.

Example. Albatros : Figs. 70 and 71. Data as follows : —
Mass = 15 lbs. (assumed value).
.*. W = 480 poundals.

V n = 50 ft/sec. (= 34 m./h.).
/. H n = 89 ft.

/3 = -25 (value assumed from tables). 1
Tan y = '14 (probable value).

Tail data,

I = 1-7 ft.

a — '82 sq. ft. (including added area due to feet).

Constants,

P = "078.

~~ .. ~ !• Tail constants.
c =1-0 (

€ = *75 Aerofoil.

1 Aerial Flight, Vol. I., Aerodynamics, § 181.

138

EXPEEIMENTAL VERIFICATION §79

Moment of inertia,

2*5
Radius of gyration = — =- = '5 ft.

I = -25 X 15 = 875.
Calculation of * —

_ 4 X 1-7 X 1,520 X -14

<t> =

3-75 X (5*2 + 823)

_ 1,450

"~ 1,230 ""

Thus in the albatros we find a lower "factor of safety" than
in the case of the swift. This may plausibly be attributed to
the more constant conditions under which this bird finds itself,
and the greater difficulty that Nature must have in providing a
large stability factor in the case of the heavier bird without
an undue increase of speed.

We must at present regard the application of the equation
of stability to birds in flight more as an illustration than as
even an approximate demonstration; for the data are con-
fessedly inadequate. Thus an error in the computation of the
natural velocity of but 10 per cent, will affect the value of * in
the ratio of 2 : 3.

In the case of the swift where the tail area is a variable
quantity, we have a further possibility of error. In the calcula-
tion of the preceding section the tail area has been assumed as
of approximately minimum spread; the maximum spread is
about double the area given, so that the stability factor, <J\ will,
when the tail is fully spread, be approximately double that
calculated, or say = 36. We may on this basis assess the
minimum velocity of flight at which the swift will possess
automatic stability; we know that 4> varies as P*, hence the
velocity at which with a full spread of tail <i> becomes unity
will be

- 82 = 28 ft./sec.
189

§ 79 AERODONETICS

It is convenient in many ways to employ the equation of
stability in a form that will give the critical value of H n (and
thus F n ), from the other known data, both when applying the
equation to birds and in some cases to gliding models. This
form of the equation is as follows : —

^ 4 I tan y

Thus in the case of the hir-undo apus we have when a = "014 : —

^ " 4 X '8 X ; 1()

= j&* = 8-4

V -192

or, V n = 8 V8-4 = 28-2 ft./sec.

which is the result already obtained.

A most gratifying feature in the application of theory to the
investigation of bird flight is the consistent nature of the
results ; the agreement between aerodynamic and aerodonetic
theory, and the agreement of both with the results of ordinary
observation is quite as close as, in the absence of more accurate
measurement, can be expected. Thus 20 ft./sec. was the author's
own estimate of the lowest velocity of flight employed by the
swift, before the theoretical computation had been made ; and, in
the case of the albatros, the velocity, 50 ft./sec, calculated
from the aerodynamic tables (Vol. I., § 187), is in satisfactory
agreement with the equation of stability.

As evidence of the soundness of the deductive method applied
to the study of flight, a single experience may be cited, The
stability of the albatros as first calculated by the author was
unsatisfactory, and for a time the reason for this was not clear.
It was when casting about for an explanation that the possible
employment of the feet as an auxiliary tail area suggested itself,
and after reading almost the whole of the albatros literature

140

EXPERIMENTAL VERIFICATION

§79

extant, the author found confirmation of this view firstly in a
photo-print in which the position of the feet could never have
been located as a matter of ordinary observation, and later in

Fig. 71.
the report of the South Polar Expedition by Mr E. A Wilson,
to which reference lias already heen made. 1

§ 80 AER0D0NETICS

§ 80. The Equation of Stability applied to the Author's 1894
Models. — There is considerable interest attached to the retrospec
tive application of the equation of stability, as showing whether
the early flight models of the author and others were of stable
flight path, or whether their stability was only apparent and due
to the comparative shortness of the flights made.

It is evident that, since the loss of equilibrium that takes
place is due to the cumulative increase in the amplitude of the
path, non-compliance with the equation of stability will not
result in an immediate loss of equilibrium, and a flight path of
several phase lengths is necessary for an experimental demon-
stration to be convincing.

Now that the equation of stability has been fully established,
we may, in the absence of such prolonged flights, fall back on
the equation itself to tell us whether the flight path was stable
or otherwise.

Example. 1894 aerodone, Figs. 14, 15, 16, and 17.

Mass = 1*48 lbs.

.•. W = 46 poundals.

V n = 70 ft./sec.

.\ H n = 76 ft.

46
70 J

ft = *1 (mean).
Tan y = '25 (probable value).

Tail data,

I = -39 ft.

a = *2 sq. ft.

Constants t

p - -078.

C = -7.
c = 2*0.
6 = -75.

142

K = wL = -0094.

EXPERIMENTAL VERIFICATION

§80

6

M

143

§ 80 AERODONETICS

Moment of inertia,

I = 5 lb. ft. 2
Calculation of <I> —

^ = 4 X 39 X 5,800 X '25
5 X (106 + 601)

_ 22,600 _
~~ 3,535 " ° '

Example. 1894 twin screw aerodrome, Fig. 24.

Mass = 2*5 lbs.

.-. W = 80 poundals.

V n = 80 ft./sec.

H„ = 100 ft.

. *

QA

if = g = -0125.

/3 = '1 (mean).
Tan y = -25.

Ta£Z data,

I = 8-3 ft.

a = '25 sq. ft.

Constants,

P = '078.

C = -7.

c = 2-3

€ = -7.

Moment of inertia,

I = 18 lb. ft. 2

Calculation of 4> —

<I> =

4 X 3-3 X 10,000 X ^5
13 X (80 + 455)

33,000

= — = 4*7

7,000 * ' *

144

EXPERIMENTAL VERIFICATION §81

§ 81. Lilientbal's Machine. Stability Investigated. — The late
Herr Lilienthal achieved a considerable number of flights by the
aid of an appliance represented in Figs. 72 and 78. In an
account of these experiments given in Nature, 1 the following
passage occurs, and is of interest not only as a statement of
the manner in which Lilienthal sought to maintain equilibrium,
but also as showing the views that were current at the date in
question as to the stability of birds, and on the problem of
flight generally.

" Lilienthal depends for the success of his apparatus on him-
self, trusting to his powers of instinct to keep his equilibrium by
corresponding movements of his centre of gravity. Man, in this
case, is the main flyer, the apparatus being only an adjunct,
and it is from the ability of the former that he expects to obtain
positive results. His apparatus is simple, cheap, and easily
constructed; these are great points, as experiments can be
carried on, even at the expense of the loss of a few machines.

" The whole success of aerial flight can be summed up in the
word equilibrium, and it is here that the difficulty lies. Given a
perfectly quiet or very nearly still air, there is no doubt that
machines can be constructed so as to soar and travel through
the air. This state of atmosphere is very rare; but, on the
other hand, there are all sorts of disturbances, currents and
wave motions, which render aerial navigation a far greater
difficulty than is usually imagined.

" One often envies a bird which, with perfect ease, soars above
us ; but it must be recollected that it is endowed with a delicate
system of nerves, which are always on the alert, to answer to
any call made on them to sustain equilibrium. These move-
ments are made quite unconsciously, and with the loss of a
minimum amount of energy. To construct an apparatus that
would accomplish this in an efficient manner would be simply
impossible; but there seems no reason why man should not

1 Nature, December, 1894, p. 177.
A. 145 L

§ 81 AERODONETICS

approximate to it to a certain extent by the help of an appro-
priate framework. With perseverance and many trials he should
be able to master at least some of the rudiments, and eventually
make short flights."

The author has collected the data necessary for the calculation
of the stability factor of the Lilienthal machine from various
reliable sources, the data for the drawings and most of the
particulars having been obtained from an actual Lilienthal
machine the property of Mr. T. J. Bennett of Oxford. The total
weight with aeronaut, and the velocity are taken as given by
Ghanute in an article in the " Encyclopaedia Britannica." The
gliding angle is taken as approximately 10 degrees, which agrees
closely with various accounts published, and the moment of
inertia has been computed from the weights and disposition of
the components.

The quantity /3 has been calculated aerodynamically, and has
also been estimated from the curvature of the fore and aft ribs
(Fig. 72), the results being respectively '35 and "82; the value
'33 has been taken as probably not far from the truth.

Tabulation of data : —

Mass = 220 lbs.

.-. W = 7,100 poundals.

V n = 28 m./li. = 34 ft./sec.
.*. H n = 18 ft.

x = Lioo = 6>1

84*

/3 = -88.
Tan y = -18.

Tail data,

I = 9 ft.

a — 12*5 sq. ft.

146

§ 81 AERODONETICS

Constants,
' p = '078.
C = -66.
c = 1-8.
€ = -5.

Moment of inertia,

Rati, of
Mass. Gyration 9 . 1.

Aerofoil ... 48 X 1-75 2 147

Tail ... 6 X 9* 486

Man ... 166 X 1-88 2 294

Total ... 927

Calculation of <I> —

^ = 4 X 9 X 324 X ^18
"927 X ('164 + 5-2)

= i^sK = '424.
4,960

Thus the Lilienthal machine did not conform to the equation
of stability, and it is literally true that the aeronaut trusted to
his instinct to maintain his equilibrium. It is a most lament-
able circumstance that Herr Lilienthal paid for his temerity
with his life. 1

1 According to an account contributed to Nature, September 3rd, 1896,
by Prof. Karl Runge, Herr Lilienthal met with his fatal accident on
August 6th, 1896, when experimenting with a machine of the double deck
type, that is to say, with an aerofoil consisting of two superposed members,
otherwise closely resembling that of the illustration. It would thus appear
that the actual machine from which the accident took place was not identical
with that to which the above calculation applies. Further than this, according
to the account, the loss of equilibrium was due to a sudden gust rather than
an accumulated oscillation. In spite of these facts it is probable that a suf-
ficient factor of stability would have been the means of saving Lilienthal's
life ; to appreciate this fact it is only necessary to experiment with models
whose stability factor is less than unity, both indoors in still air and out of
doors when there is the slightest movement of the air. The tendency of an
aerodone whoso <I> value is low (materially less than 1), to indulge in fatal

148

EXPERIMENTAL VERIFICATION §82

§ 82. Resumi. — The experimental evidence offered in the
present chapter fully demonstrates the Phugoid Theory with the
extension leading up to the equation of stability to be established
fact. We are entitled to regard the main results achieved as
proved beyond dispute.

Before passing on to the next branch of the subject it is
desirable to review the position and point out some of the
immediate consequences of the preceding work.

In the first place it may be remarked that the Phugoid Theory,
besides providing us with a new point of vantage from which to
study the phenomenon of flight as presented in various forms by
Nature, constitutes a key, if not the key, to the problem of
mechanical flight. By its means we can make all the calcula-
tions necessary to ensure the longitudinal stability of an aerodone
or aerodrome. Besides being enabled to realise at a glance the
essentials as to the general disposition of weights and supporting
surface, etc., we can calculate the necessary tail length, tail area,
and flight velocity, that will render the machine automatically
stable in its flight path. The application of the theory in this
respect and its extension to meet the conditions of power pro-
pulsion will be discussed in a later chapter.

One of the salient consequences of the phugoid theory is the
necessity of speed. There is a relation between the size of an
aerodone and the minimum velocity at which it is stable, thus
the larger the model the higher the minimum velocity of stable
flight becomes.

It might be thought that, by adding, as necessary to comply
with the equation, to the length and area of the tail, the velocity

phiDges, on tbo least provocation, is such as to impress vividly on the mind
the terrible nature of the risk to which Herr Lilieuthal exposed himself.
Even the fact stated in the account, that the improved type of machine
showed itself stable " in ballast," does not save the situation, for the moment
of inertia of the man himself is a serious factor in the calculation, and the
ballast, which in all probability was concentrated, would be comparatively
deficient in this respect, and the stability of the machine would be propor-
tionately improved when flying under ballast.

149

§ 82 AERODONETICS

might be reduced to any desired .extent. It may be easily shown,
however, that this cannot be carried beyond a certain point, for
the addition to the tail either as to area or length, beyond this
critical point, results unavoidably in such an increase in the
moment of inertia, as to do more harm than good, so that the
stability factor will actually be diminished instead of being
increased. Under these conditions the only method by which
an increase of stability can be brought about is by an increase in
the velocity of flight.

An examination of Herr LilienthaPs machine discloses the fact
that this critical condition must have been very near at hand,
for a large proportion of the moment of inertia is in the tail
itself ; thus an addition to the tail would have been of but little
value ; indeed, if not carefully designed it might actually have
been detrimental. A few feet a second, however, added to the
velocity of flight would have given the machine a sufficient
reserve of stability.

It is evident that, the limiting conditions once settled, the
velocities appropriate to a series of geometrically similar machines
will be determined by the law of corresponding speeds, i.e.,
the velocities will be proportional to the square roots of the
linear dimensions. Now since under the supposed conditions
the linear dimension increases as the cube root of the mass (or
weight), the minimum safe velocity will require to increase as
the sixth root of the mass. Thus, if we take it that Herr Lilien-
thal's machine (220 lbs.) would have been reasonably safe at
80 m./h., a machine of three tons total weight would require to fly
at over 50 m./h., a velocity at which the h.-p. necessary suggests
that a practical limit is being approached. We may therefore
anticipate that the flying machine will not, on the most sanguine
estimate, exceed some few tons in weight, unless some consider-
able advance is made in the prime mover, and in the near
future at least, success may be looked for in the direction of far
smaller units, certainly less than a ton, probably about half a
ton in weight, constructed to carry but one, or at the most two,

150

EXPERIMENTAL VERIFICATION

§82

Probable range of weight for flying machines
in the near future

Lilienthal, 220 lbs.

Albatros, 15 lbs.

Aerodrome, 2 J lbs. Author, 1894.
Aerodone, H3 lbs.

i > *>

Swift, 40 grams.

Mica Aerodone, 12-gram model.

Mica Aerodone, 4-gram model.

Mica Aerodone, J-gram model.

Mica Aerodone, 1 -grain model.

Fia. 74.

151

§ 82 AERODONETICS

aeronauts. This condition may cease to apply if some artificial
means be found for adding to the stability beyond that inherent
to a rigid aerodone.

The theory does not enable us to predict the conditions of
stability for phugoids of considerable amplitude. It applies,
strictly speaking, to curves of flight whose amplitude is a
vanishing quantity, but evidently from the nature of the con-
ditions and the demonstration it may be taken as applicable to
curves of small amplitude such as those numbered 2 and 3 on
the phugoid chart. This conclusion is confirmed by the
experiments.

When the flight path amplitude becomes considerable we
have at present only experience to guide us, and the author
has found that an aerodone just stable for small amplitude, is
distinctly unstable for greater amplitude; it is, however, at
present not settled whether this is universally the case. Under
all circumstances the stability factor required to ensure equi-
librium for all amplitudes of flight path is not very great, a factor
= 2 is more than sufficient under still air conditions, and an
aerodone designed to such a factor will behave quite satisfactorily
under what may be termed " ordinary working conditions/' that
is to say, in a wind to which its natural velocity is otherwise
adapted. 1

The range over which the Phugoid Theory and laws of stability
have been experimentally demonstrated does not, unfortunately,
include aerodones of the weight and velocity such as are involved
in mechanical aeronautics, although the test as applied to the
Lilienthal machine comes near to this point. There is no reason,
however, to anticipate any departure from the laws of flight as
laid down from theory and established by small scale experiment,
for the range of these experiments is in itself sufficient to show
that the equations can be considered as perfectly reliable. If we
assume for example that the equations apply merely for as great
a proportional range above the largest model experimentally

1 Compare § 40.
152

EXPERIMENTAL VERIFICATION §82

investigated, 1 as that over which the experiments extend, we have
for the upper limit an aerodone weighing 14,000 lbs., or more
than six tons. Or, if we plot the weights of the aerodones
investigated on a logarithmic scale (Pig. 74), the range from
one half ton to two tons, such as probably constitutes the range
of sizes of the flying machine of the more immediate future, is
as represented. In such a logarithmic scale equal increments
represent equal degrees of proportion, hence Fig. 74 represents
graphically the relation of the range of the experimental verifi-
cation, to the range with which the engineer is more immediately
concerned in the problem of mechanical flight.

1 The 1 lb. 7 oz. aerodone whose flight path is depicted in Fig. 25. This
flight path showed by its rate of damping that the value of <t> was con-
siderable, hence we may take the flight of this model as an experimental
verification of the equation.

153

CHAPTER VII

LATERAL AND DIRECTIONAL STABILITY

§ 83. Introductory. — Lateral and directional stability relate to
the equilibrium of the aerodone in respect of its three degrees of
freedom involved by motions other than those in the phugoid
plane. 1 Thus la teral stabilit y is concerned primarily with rota-
tion about the axis of Hight; that is, a motion resembling the

Fig. 75.

rolling of a ship ; if such a motion does not give rise to a
restoring couple an aerodone may turn over, otherwise " capsize,"
and so lose its equilibrium. Directional stability involves rota-
tion about a vertical axis and motion of translation at right
angles to the phugoid plane, these two kinds of motion being
associated in any change of course.

1 The vortical plane containing the flight path : see Chap. II.

154

LATERAL AND DIRECTIONAL STABILITY §84

In the case of an aerostat or balloon the lateral stability is
due to the relative positions of the respective centres of gravity
and displacement, the former being vertically beneath the latter,
so that if at any instant the conditions of equilibrium are dis-
turbed, a righting couple arises automatically, as shown diagram -
matically in Fig. 75 (a). It is evident that in the case of an aero-
done we cannot obtain stability this way, for the support is derived
from the reaction of the aerofoil, which is not constant as to direc-
tion, but changes with the movements of the aerodone. Thus in
Fig. 75 {b) the resultant of the weight of the aerodone W and the
supporting reaction It does not constitute a couple as in Fig. 75 (a),
but a lateral force F that gives rise to an acceleration at right
angles to the line of flight. Hence, although only one degree of
freedom is primarily concerned in the question of lateral stability,
motions in two degrees "of freedom are actually involved, and it
is by the interaction of motions of these two kinds that we must
look for the means of giving lateral stability to an aerodone in
flight.

§ 84. The Mutual Relationship of Lateral and Directional Stability.

— It is shown in the preceding section that each of the kinds of
stability that we are about to study involves two degrees of freedom,
and as there are in all but three degrees of freedom, one of these,
i.e., motion of translation at right angles to the phugoid plane, is
involved in common. One consequence of this is that the two
kinds of stability are very closely associated. To such an extent
is this found to be the case that the propriety of treating lateral
and directional stability as two separate problems is open to
question.

The mutual relationship of the two problems will be made
more clear when some consideration has been devoted to them
individually, for although the author has under consideration a
broader and more comprehensive method of dealing with the
present branch of the subject, the only treatment ready for
presentation is that which has to some extent stood the test of

155

§ 84 AERODONETICS

experience, and in which each problem is first treated as if the
other did not exist, the interaction of the two being taken
subsequently.

It is perhaps of some interest to state that the individual
problems of lateral and directional stability are comparatively
simple, and that the whole of the complication of the present
subject arises when they are taken in conjunction. The actual
discovery of the connection, simple as it appears when the case
is clearly stated, was made owing to the anomalous behaviour
of certain of the author's models, the converging spiral flight
path, mentioned in § 6, being the disorder that in the first place
drew attention to the point.

§ 85. Lateral Stability. — The problem of providing lateral
stability consists in securing that the aerodone shall not upset
by turning about the axis of flight. It is necessary to so
arrange that any departure from the plumb shall automatically
call into existence a restoring couple.

It is assumed in the present stage of the discussion that the
aerodone is only permitted the two degrees of freedom essentially
involved, in other words, it is not allowed to turn about a vertical
axis, and its motions in the phugoid plane are supposed to be
in equilibrium inter se in accordance with the principles demon-
strated in Chaps. II. to VI.

We have already examined a case of automatic stability in
the behaviour of the ballasted aeroplane. In § 5, referring
to Fig. 5, it has been shown that when the plane assumes a
laterally inclined position (owing to some initial disturbance),
the resulting motion causes the centre of pressure to move
laterally, so that the necessary restoring couple arises and
the plane returns in due course to its horizontal position.
Similarly, in the author's flight models, §§ 10, 11, etc., the
vertical fins perform the same function as is performed in the
ballasted aeroplane by the change in the position of the centre
of pressure ; when the aerodone acquires a lateral motion the

156

LATEKAL AND DIRECTIONAL STABILITY §85

pressure that arises on the upwardly projecting fins acting above
the mass centre of the aerodone supplies the righting moment.
There is another modification of design by which the same

(a)

\^

Fio. 76.

effect can be produced. If an upward inclination is given to
the two wings of the aerofoil, Fig. 76 (a), or to the extremities only,
Fig. 76 (b), a restoring couple arises as an immediate result of any
attempt on the part of the aerodone to capsize. Thus the

S

/

\

\

/

/

\

/

/f

/

Fio. 77.

\

/

/

upwardly inclined "wings" (a), or the alternative form of (b)
may be looked upon, in effect, as giving the aerofoil the form of
the arc of a circle, Fig. 77, in which case it is evident that if
oscillating its trace in space will be a part of a cylindrical surface

157

§85

AERODONETICS

whose axis is parallel to the line of flight ; this cylindrical surface
is represented in the figure by the dotted circle.

Now if we suppose an aeorodone having an aerofoil of arc
form and radius r to suffer disturbance it is evident that it will,
as the result of such disturbance, proceed to oscillate in the
cylindrical flight path trace, just as if it were suspended from the*
pointy the centre of the dotted circle (Pig. 77), and its period of
lateral oscillation will be that of a pendulum of length r. We
require to examine the problem from this point to ascertain the
conditions under which this oscillation takes place, as affecting
the growth or damping of its amplitude.

§ 86. Lateral Stability. Oscillations in the Transverse Plane. —
There is a certain similarity between the problem now before us

(d)

N.

to

i

s

Fig. 78.

and that dealt with in Chap. V., the oscillations in the phugoid
plane; there are two influences at work, one tending to an increase
and the other to a decrease in the amplitude of motion.

In the first place it may be remarked that any departure from
the arc form depicted in Fig. 77, results in a resistance to
the oscillations and consequently a diminution of amplitude.
Similarly, an artificial damping may be effected by employing

158

LATEKAL AND DIRECTIONAL STABILITY §87

a combination of the fin and inclined extremities, or the fin
method of obtaining lateral stability alone constitutes a damper
of the most effective kind. Thus, referring to Fig. 78, a, b, and
c are forms relatively deficient in damping action, whereas forms
rf, e, and /may be regarded as highly efficient in this respect.

It is one of the advantages of the finned aerodone adopted by
the author that it is a form whose behaviour lends itself readily
to calculation ; it is a disadvantage of this form that it offers a
greater surface to the air for a given degree of stability, and
hence its skin friction is greater than is necessary. A compara-
tively slight " turn up " of the wing extremities will give lateral
stability, without adding materially to the frictional resistance.

The above facts fully account for the fin form being unknown
in Nature. It is no advantage from the point of view of Nature
that her designs should be calculable by mathematical processes,
but it is of the greatest importance that the resistance should be
reduced to a minimum.

§ 87. Lateral Stability. Oscillations in the Transverse Plane
(continued). — Let us now, in the typical case represented in
Fig. 77, examine the influence of changes in the position of
the centre of pressure as due to an oscillation of small
amplitude.

We will, in the first instance, assume that the moment of
inertia of the aerodone about the axis of flight is a negligible
quantity, that it is in fact of zero magnitude. We will further
assume that the aerodone experiences no resistance to its oscilla-
tion, a condition that is most nearly approached by the arc form
of Fig. 77, it being supposed that skin-friction is absent. It is
evident that in practice an aerodone must possess moment of
inertia about the axis of the path of flight, and that there will be
damping of the oscillation due to skin-friction and other kinds
of lateral resistance, but the influence of those factors will be
considered subsequently.

In Fig. 79 it is supposed that the aerodone is in the middle of

159

§87

AEKODONETICS.

an oscillation, the direction of its motion being indicated by the
arrow. Now the lateral motion will result in the displacement
of the centre of pressure of the aerofoil (laterally) in the direc-
tion of movement, and in the ordinary way this would give rise
to a couple or torque acting on the aerodone about the axis of
flight, 1 Fig. 79 (a) ; but under the assumed conditions, owing to

/

Fig. 79.

the absence of moment of inertia, this is impossible, the result-
ing torque must- be again transmitted to the_ air. immediately,
consequently the directions of motion of the extremities of the
aerofoil will be deflected as shown in Fig. 79 (6), so that the

1 In Fig. 79 (a) the resultant of the two forces, gravity and pressure
reaction, passes through the centre of curvature and not through the mass
centre ; consequently this resultant and the equal and opposite inertia force
are not in equilibrium, but form a couple. This couple is indicated by the
horizontal forces shown in the figure.

160

LATERAL AND DIRECTIONAL STABILITY §88

change of position of the centre of pressure due to change of
aspect will be neutralised by an equal and opposite change in the
centre of pressure due to a certain screw-like motion impressed
on the aerofoil.

It would thus appear that the radius of the path of oscillation
would be less than that due to the radius r by an amount depend-
ing upon the change of effective attitude of the two wings of the
aerofoil as represented by the arrow heads in Fig. 79 (b). We
may also regard this shortening of the oscillation radius as due
to the superposition of the rotation of the screw-like motion
(consequent upon the change in the centre of pressure) on the
normal oscillatory movement, the two being always of like sense.

§ 88. Oscillations in the Transverse Plane (continued). — At and
about the limits of the oscillation, when the lateral velocity is
negligible, the radius will evi-
dently become equal to r, for the
screw-like motion vanishes. We
may, therefore, represent the path
of oscillation as generated from
an evolute whose "graph" is
given diagrammatically in Fig. 80.
Under the assumed conditions the
oscillation, once initiated, will be
permanent, neither increasing nor
decreasing in amplitude.

Since the mean radius of the
path of oscillation is less than

that of the aerofoil (assuming the arc form), it is evident that
in the case of an aeroplane, where r becomes infinite, there will
still be an oscillation about an approximate centre situated some
distance above the line of flight. This is illustrated in Fig. 81,
it being supposed, as in Fig. 79, that the aerodone is in mid
position, the displacement of the centre of pressure due to the
lateral component of motion giving rise to a screw-like motion

a. 161 m

§88

AERODONETICS

as indicated. The period of oscillation will evidently be slow in
comparison with a form such as illustrated in Fig. 79.

/

/

/

/

/

/

\

\

\

\

\

*►

Fio. 81.

\

^^»

The limiting case, the aerofoil whose motion is the equivalent
of an infinite pendulum, evidently lies in the region of inverse
curvature, that is to say, the form of the aerofoil will be laterally

Fig. 82.

arched as in Fig. 82. There must manifestly be some particular
degree of curvature that under given conditions will be neutral
in respect of lateral motion, the change in the position of the

Fig. 83.

centre of pressure due to endwise motion being compensated by
the direct pressure components acting on the inclined wing

162

LATERAL AND DIRECTIONAL STABILITY §90

surfaces. An aerodone whose aerofoil conforms to the necessary
conditions is incapable of receiving a lateral oscillation.

It is highly probable that we have here the reason of the
habit of flight of many birds whose wings have a somewhat
downward trend or " droop " ; the common swift (Fig. 88)
being a notable example.

§ 89. Oscillations in the Transverse Plane. Damping Influences.
— The damping influences at work tending to diminish the
amplitude of the lateral oscillation are of two main kinds, i.e.,
skin-frictional and aerodynamic. The former is always present
to a greater or less degree, the latter may or may not be present,
but may in any case be provided by departure from the simple
arc form as in Fig. 76, or by the deliberate fitting of fins, organs
somewhat analogous in their present usage to the " bilge keels "
of sea-going vessels, as in Fig. 78 (d), (e), (/).

It is probable that birds rely largely on the lateral resistance
of their body form, which perhaps gives rise to motion of the
discontinuous type when moving obliquely ; also the skin friction
on the wing surfaces is evidently an important factor. Beyond
this, the nicety of adjustment attainable by the living organism
doubtless permits of the oscillation period being rendered so
slow that but little damping is required.

In the construction of flying machines also it may be found
that the " body " can be relied on as a means of damping lateral
oscillation, but the author believes that a certain amount of
additional fin area will generally be found of service, as used in
his model experiments, and in several actual machines that have
been built by other investigators.

§ 90. Oscillations in the Transverse Plane. Influence of Moment
of Inertia. — The influence of moment of inertia about the axis
of flight as affecting the transverse oscillation is a matter of
great complexity, and up to the present the author has only been
able to arrive at certain elementary conclusions of a qualitative

kind.

163 m 2

§90

AERODONETICS

Let Fig. 84 represent the path of lateral oscillation of an
aerodone whose whole mass is supposed concentrated at its
centre of gravity, so that the moment of inertia about the axis
of flight is zero. It is evident that in the course of its oscillation

the aerodone in passing from A to B acquires clockwise rotation,
and in passing from B to C it loses the rotation so acquired.

Let us now investigate the change to be anticipated in the
form of the oscillation as the result of supposing the aerodone

to possess moment of inertia, as must in reality be the case.
Let us suppose that a change in the moment of inertia from
zero to some finite value take place at the point B 9 then the
rotation during the portion of the oscillation B C will tend to

Fig. 86.

persist, and the path of oscillation will be modified accordingly,
as shown in Fig. 85. W

When the point (7 is reacl^the aerodone will still possess
some residuary rotation so that its flight path will continue in

164

LATERAL AND DIRECTIONAL STABILITY §91

the manner illustrated in Fig. 86. If we endeavour to trace
the path of oscillation further, we know that on the principle of
work, assuming that there is no damping, the aerodone will rise
till it touches the datum line A C, say to the point A\ (Fig. 87),
when the conditions will be similar to those already examined

Fio. 87.

at the point C 2 , so that we may continue the diagrammatic
representation in the manner shown.

§ 91. Oscillations in the Transverse Plane. Form of the
Oscillations. — It is probable that the gliding angle is prejudicially
affected by the lateral oscillation, so that the greater ihe ampli-
tude of the latter the steeper the mean gliding angle will become.
It has been assumed in the foregoing section that the diagram of
the flight path is a projection on a plane at right angles to the
mean gliding path, and that the latter is unaffected by the oscil-
lation amplitude. When there is any damping taking place the
dissipation of the energy of oscillation will result in the aerodone
not rising to the datum line A C (Fig. 87), and the oscillation
will consequently terminate at a lower point than would other-
wise be the case. When there is no actual damping it is possible

165

§91

AERODONETICS

that the same is to some small extent true, owing to the aerofoil
not finding itself under the most favourable conditions for
economic flight, but this is a point we can afford to ignore ; we

Fig. 88.

may, for instance, suppose that any effect of this kind is in itself
a separate kind of damping ; we may alternatively presume that
the direction of projection is varied according to the changes in

Fig. 89.

the gliding angle, assuming no real damping to take place.
Either supposition for the present purpose seems legitimate, the

166

LATERAL AND DIRECTIONAL STABILITY §91

point, however, might assume greater importance if any means
were found of dealing with the problem quantitatively.

Let us now suppose that we have damping influences at work ;
then, in addition to the moment of inertia effect of the preceding
section, by which the amplitude was shown to increase (Fig. 87),
we shall have a disappearance of energy at each oscillation that,
if of sufficient magnitude, may result in equilibrium or in a
definite diminution of amplitude. Thus, in Fig. 88, the aerodone
beginning an oscillation at the point A, which in the absence of
damping would rise at the end of that oscillation to the point c,
will now, owing to the loss of energy, rise to a point C of less

Fig. 90.

altitude, and on the succeeding oscillation it will find itself at a
point A i of lower level still, and so on.

Under these circumstances it is evident that the path of
oscillation may either increase in amplitude as before, but at a
slower rate, as in Fig. 88, or it may remain constant as in Fig. 89,
or again, it may gradually damp out as illustrated in Fig. 90.

One incidental result of an oscillation in progress of damping
will evidently be that the gliding angle proper will have super-
posed on it a further angle of descent due to the dissipation of
the energy of the oscillation. In the Figs. 88, 89 and 90 the
direction of projection is supposed to be parallel to the gliding
path proper, without this added angle, so that the damping effect
makes its appearance as a separate item. \

167

*. »

§ 92 AERODONETICS

§ 92. Lateral Stability. Oscillations in the Transverse Plane in
Practice. — The conclusions of the foregoing sections are fully
borne out by actual experience. The form of the observed
oscillation path is quite in agreement with that given from
theoretical considerations in Figs. 87, 88, 89 and 90. The fact
that an aerodone in a state of oscillation has an abnormally
steep gliding angle, is also found to be true. Finally the import-
ance of avoiding excessive moment of inertia about the axis of
flight is a fact that it is quite easy to verify experimentally.

The author has frequently met with the lateral oscillation in
the course of model experiments. From an examination of a
number of models that have shown a tendency to develop oscilla-
tion it has been found that the weight of the aerofoil should in
no case exceed *5 to *6 of the total weight, although this is but
a rough and ready measure, the moment of inertia does not for
a given aerodone depend upon aerofoil weight alone; thus
the oscillation will occasionally make its appearance even when
the aerofoil weighs considerably less than *5 of the total.

As illustrating the point in question, it has been frequently
noticed that an aerodone that has been quite steady in flight
when first made, has developed a habit of " fluttering" when the
wings have been repeatedly repaired with paper patches, so that
their moment of inertia is increased.

It would thus appear that moment of inertia in any aerodone
or flying appliance is bad, whether about the transverse axis or
about the axis of flight ; the mass should be as concentrated as
possible. It is interesting to note how excellent are the forms
employed by Nature in both respects.

§ 93. Directional Stability. — In directional stability two degrees
of freedom are involved, namely, rotation about a vertical axis, and
motion of translation at right angles to the phugoid plane ; thus
direction of flight stands in the same relation to an aerodone or
flying machine, as the nautical term course to a ship at sea.

•In considering the present subject from an aerodonetic

168

LATERAL AND DIRECTIONAL STABILITY §94

standpoint we are concerned with the equilibrium of the aerodone
in respect of its direction of motion for the time being, rather than
with the continued maintenance of a given course ; the latter is
a question of navigation, given a machine that is stable; the rest
is a matter of holding the machine to the desired direction by
appropriate steering mechanism. Automatic steering to a fixed
course is a problem that has been very thoroughly worked out in
connection with the Whitehead torpedo, and without doubt, if
required, the same principle 1 could be applied to an aerodone.
This question, however, is reserved for later discussion.
It is essential to the directional stability of an aerodone that
I any change of aspect should give rise to a restoring couple. It is
; evident that if the aerodone be symmetrical about the phugoid
plane, its normal aspect will be one of equilibrium — stable or
r otherwise. It is also evident that the stability of the equilibrium
will depend upon the organs of resistance to lateral motion ; thus
if the fin area is well abaft the mass centre, as in the feathering
of an arrow, the equilibrium will be stable, whereas if the fin
area is too far forward instability will result. In the latter case
any small departure from the normal will give rise to a couple
tending to exaggerate the error.

§ 94. Directional Stability (continued). — Although it is not
necessary for our present purpose that an aerodone shall main-
tain a constant direction of flight, still it is desirable that it shall
have some power to resist any disturbance tending to change its
course. Thus in the case of an arrow shot from a bow, the
directional equilibrium is rendered stable by the feather at the
after end of the shaft, but this does not prevent an arrow from
obeying implicitly any transverse force, as, for instance, that of
gravity ; the trajectory is approximately identical with that of
any other kind of missile.

The same conditions do not apply in respect of the action of

1 For a discussion of the gyroscope in its application to problems in
stability, and direction maintenance, see Appendix VII.

169

§94

AERODONETICS

gravity in the case of an aerodone, for we are no longer con-
cerned with motion in a vertical plane ; but considerations of
lateral stability and the phugoid theory both require that the
aerodone shall preserve its direction with some degree of
obstinacy, and thus a tandem system of fins is adopted, as
already described and illustrated in § 11.

The simplest case of such a fin combination from a theoretical
point of view is one in which the one fin is situated at or about
the centre of gravity of the aerodone and acts exclusively as an
abutment, the function of the other fin, which is arranged some
distance from the first, being purely directive, that is to say, it
acts as a wind vane, or as the feather of an arrow. It is evident
that, other things being equal, the rate of displacement of the
aerodone in the face of an applied transverse force will be
inversely as the area of the abutment fin, 1 and further, the
consequent angular movement, that is the change of course,
will be inversely proportional to the distance separating the fins.

§ 95. A Study in Directional Equilibrium and Maintenance. — It is
evident from the foregoing that the question of direction involves

Fig. 91 .

two distinct problems, equilibrium of direction such as is provided
by the feather of an arrow, and direction maintenance, that is, the
capacity to resist forces tending to divert the aerodone from its
line of flight, provision for which takes the form of an abutment

1 Depending on the law of the small angle, Pp ex 0.

170

LATERAL AND DIRECTIONAL STABILITY §95

fin. The wider problem of securing an absolute sense of direc-
tion, which involves some communication between the aerodone
and the outer world (either optic or magnetic), or the employ-
ment of a gyroscope, is outside our present purview.

Let us take for the purpose of initial study a special kind of
arrow illustrated in Pig. 91. Let A be the effective 1 distance
separating the abutment fin b from the wind vane c, let r be the
radius of the path curvature, and let /3 be the angle made by the

«

Fig. 92.

abutment plane to its direction of motion through the air
(Fig. 92). Then (for small angles) r = ~ . (1)

Now /3 depends upon the lateral force, the velocity, and the
area and form of the abutment plane, and for any given value
of |3 we have r oc A.

Let us suppose that A be made infinite, then if a force act laterally

1 Strictly speaking from centre of pressure to centre of pressure ; this, under
the assumed conditions, will be from J to \ the width of the fin from its
leading edge.

171

§ 95 AERODONETICS

on the mass centre it will give rise to a sideways movement, the
lateral velocity eventually becoming sufficient to produce an
aerodynamic reaction on the abutment plane just equal to the
applied force. Under these circumstances the direction of motion
is for the time being modified, but owing to the supposition that
A is infinite, no rotational movement takes place, and so when
the external force ceases to be applied the aerodone or arrow
resumes its original direction.

But here we meet with a point of some importance. If the
" tail," instead of extending infinitely rearwards, be conceived to
extend forwards, that is, if A instead of being + <x» be taken as
— od, we find on examining the conditions that no alteration
has resulted ; the aerodone will behave in precisely the same
manner in either case. It is therefore probable that if the tail
extend forward, provided it is of sufficient length, it will give
directional stability. This result, which is somewhat unexpected,
calls for further investigation.

§ 96. A Study in Directional Equilibrium (continued). —

Let a = area of abutment fin.
„ m = mass of aerodone.
„ V = velocity.

p = density of fluid.

r, A and /3 stand as before.

It is now assumed that the guide vane precedes the abutment

fin (Fig. 98), and A will be taken as positive measured forward.

Now in absolute units the centrifugal force of the mass m when

the path becomes curvilinear is given by the expression — ; — ,

that is, for a given value of m and V centrifugal force varies
inversely as r.

But by (1) r varies inversely as /3. Hence the centrifugal
force varies directly as /3.

And we know that for small angles the aerodynamic reaction

172

a

>9

LATERAL AND DIRECTIONAL STABILITY §96

on the abutment fin varies directly as /3. Therefore the centri-
fugal force varies directly as the reaction on the abutment fin.
And if the centrifugal force is less than the reaction on the abut-
ment fin the conditions are those of stable equilibrium, and
conversely if it is in excess the equilibrium is unstable.

Hence the limiting condition of stable equilibrium is given by
equating the two, or,

m V 2

- - = c |3 a Pw = c |3 a C p V 2

but, by (1) /* =

or

/

r
A

/3

m & an

— = c j3 a C p,

m = c C p a A.

If the mass of the aerodone is in excess of that given by the
above expression, the centrifugal force that results from any

^

ABUTMENT FIN

DIRECTIVE E/N.

Fig. 93.

slight deviation from the straight line is greater than the
restoring force — the reaction on the abutment fin, and the
directional equilibrium is unstable; any deviation from the
straight, however slight, tends to increase.

If the mass is less than that given by the expression, the
restoring force will exceed the centrifugal, and the equilibrium is
stable.

On the supposition that the abutment fin is of approximately
" square " proportion, the product of the constants 1 c C p =
2 X '66 X '078 = "108 ; or, if we suppose the fin to be a plane
of n = 4, in pterygoid aspect, this value becomes 2*27 X '70 X
•078 = 124.

1 For air.

178

§ 97 AERODONETICS

§ 97. Directional Stability (continued). — The importance of the
foregoing investigation is found in the fact that the limiting
condition of directional stability lies beyond that at which the
aerodone or arrow is neutral. In the actual problem the abut-
ment fin has to deal with an accidental applied force as well as
the centrifugal force due to change of direction ; this would

Fig. 94.

naturally lead to the limiting condition being reached sooner,
but does not otherwise alter the problem.

Let us examine the two cases in which an aerodone is
supposed fitted respectively with a positive and negative tail, we
will, for the present purpose, take the rearward tail (Fig. 94) as
positive, and the forward tail (Fig. 95) as negative. 1 We will
assume that in both cases there is an applied transverse force,
as indicated, acting on the mass centre, just as would be the
case^if the aerodone had assumed a laterally inclined position.

ft j,

S» >

Fig. 95.

Now it will be seen that, in the first case (Fig. 94), the aerodone
will undergo a change of course in the direction of the applied
force. This change of course is not due to the direct sideways
motion impressed, but is measured by the change in the angular
position ; thus, the instant the applied force is withdrawn, the
direct sideways motion rapidly falls to zero, but the change of
angular position, that is, the change of course, remains. And in
Fig. 95, when the tail is negative, it will . be seen that the change

1 This seems the most natural convention ; the real tail is a positive tail.

174

LATERAL AND DIRECTIONAL STABILITY §98

of course is in the opposite direction to the applied force, in
spite of the fact that the lateral displacement directly due to
the force daring its application is the same as before.

Now it is evident that in the intermediate condition, when
the length of the tail is plus or minus infinity, the change of
course is zero, and if we can simulate this condition we shall
have an aerodone whose directional maintenance is perfect in
the sense of the present investigation. 1 It is this condition that
is above referred to as one in which the aerodone is in neutral
equilibrium, or directionatty balanced, and the result that this
condition can be reached without imperilling the directional
stability is one of considerable importance.

§ 98. Directional Stability. Practical Considerations. — It is
evidently not possible, on the lines so far investigated, to con-
struct an aerodone that shall be directionally balanced, owing

Fig. 9ti.

to the fact that whether the tail be made positive or negative,
its length requires to be infinite. It is further evident that
the required conditions cannot be directly approximated/ for
the length of tail necessary would so add to the moment of
inertia about the transverse axis as to lead to longitudinal
instability.

There is, however, an indirect method. The vane employed

1 §§ 93, 94.
175

§ 98 AERODONETICS

for directional equilibrium may be also made to do duty as part
of the abutment area ; thus, in Fig. 96, the two fins are jointly
responsible for the abutment and jointly responsible for the
directional equilibrium. Any combination of fins of this kind
will be found to possess an equivalent tail length and abutment
area, and this tail length may be made positive, neutral, or
negative at will, without the actual distance between the fins
exceeding a given desirable value. We may term the process
of determining the equivalent value of any particular combina-
tion, finjre8olutjM 9 and the object of the designer is to obtain a
combination that will render the aerodone directionally balanced,
that is, an equivalent tail length = oo, or such other value as
he may deem expedient. The resolved abutment area, also,
must be sufficient for the requirements of lateral stability, i.e.,
capable of sustaining any chance applied forces, and of rapidly
damping any side oscillation.

§ 99. Fin Resolution.

= area of front fin.

= area of rear fin.

= fin tail length (as before).

= distance from fin to fin (centres of
pressure).

= distance from mass centre to front fin.

= distance from mass centre to rear fin.

= y = ratio of pressure on rear, to pressure

on front fin,, due to force acting through
mass centre.

„ /3x = angle made by the front fin to line of

flight.

,, v\ and r 2 = the lateral components of the velocities

of the front and rear fins respectively.

176

Let

a\

>»

a 2

>>

A

>»

A

99

Ai

»»

\ 2

P2

99

LATERAL AND DIRECTIONAL STABILITY §99

r

_._J__.

i

! i

'.4_-

"IT"

ni"T

f&J, <

_L_i._ 1

§ 99 AERODONETICS

Referring to Fig. 97, we have, from geometrical considerations,

r 2 __ A — A 2
r x ~~ A + A^

whence, since A = Ax + A 2

A = A -^ A x . (1)

Now from aerodynamic considerations,

Pi = cxCi P V* fr

0r ft = „ /! _ T72

but ri = F ft, or n = *\ . (2)

The expression for r 2 is more complicated. The rear fin is
situated in the wake of the front fin, and is, consequently, influ-
enced by its " wash." We may regard the lateral velocity of the
rear fin as made up of two components; the one due to its
motion through the fluid, the value of which is computed in the
same way as for the front fin, and the velocity of the fluid itself
as represented by the motion impressed upon it by the front fin,
that is to say, the wash aforesaid.

The first of these will be given by the expression,

Pa

c 2 C 2 p r

The second, it follows from aerodynamic considerations,
will be t?i X (1 — *i)

ci C\ P V

nr r. - (1 ~ fl ) P 1 4- **

01 '2 — T7TT1; h

ci CipV • a C 2 p V

It is convenient at this point to make the assumption that the
front and rear fins are of the same aspect ratio; it is usually
easy to make them so, at least approximately. The only
quantities affected are the constants c and C, the effect of the

178

LATERAL AND DIRECTIONAL STABILITY §100

assumption being that ci and c 2t also that C\ and <7 a become
identical. In any case, we know that the variations of C are not
worth taking into account, and the changes in c for moderate
variations of n are not great. The expression for 1*2 thus
becomes,

r _ (1 — «) Vl + P* ,QV

From (2) and (8) we obtain,
'1 _ Pi

■ (4)

n — r 2 Vi — (1 — €) pi — |>2 1 — (1 — c) — y
Whence (1) becomes

A = A l Ax (5)

1 — (l — «) — y

in which y f = — J will be given by the expression,

Aa oq*

§ 100. Fin Resolution (continued). — We have thus determined
the length of the theoretical tail of which the fin combination is
the equivalent ; it remains now to show how the area of the
theoretical abutment plane, a, is related to a\ and ag, the actual
fin areas employed. We do not require to attack the problem
de novo, as the ascertained value of A may be employed in
effecting the solution.

If F be the force at any instant acting through the mass

centre, the component acting on the leading fin is^ F, and

A -J- A
the lateral velocity due to this force is T^ x times that result-
ing at the mass centre itself. Therefore the equivalent fin area
required is,

A A + A x

179 N 2

j ■• '

§ 100 AER0D0NETICS

The equivalent, or resultant;, fin is of the same aspect ratio as
those from which it is derived. This is represented in a dia-
grammatic way in Fig. 97, in which the resultant fin is shown
dotted. It is to he understood that this phantom abutment fin
together with an imaginary tail of length = A, constitutes the
equivalent of the two real fins, a\ and a*.

The position of the effective centre of pressure of the fin
combination is a matter of some interest in view of the function
of the fins in securing lateral stability. Calling the distance of
the effective centre of pressure above the centre of gravity, b, the
value of b may be ascertained by drawing a line through the
centres of pressure of the two real fins. Thus, referring again
to Fig. 97, the positions of the centres of pressure of the two
fins, ai and a 2 , are first determined, and the centre of pressure
of the resultant fin is given by the intersection of the straight
line joining them, and the perpendicular set up from the mass
centre.

§ 101. Rotative Stability. — There is a peculiar form of insta-
bility that sometimes arises as the result of the interaction
between the motions involved in the maintenance of lateral and
directional equilibrium. In separately solving theBe two prob-
lems, we have, in both cases, invoked the aid of motion of
translation at right angles to the phugoid plane, and it is neces-
sary so to harmonise the proportions of an aerodone that there
shall be nothing inconsistent in this fact. In other words, we
require to study the mutual interaction of the motions involved
in order to define the relationship of the various functional
parts of the aerodone necessary to the conditions.

The manifestation of instability of this form is that the
aerodone, apparently with little or no provocation, loses its
equilibrium and comes rapidly to earth with a kind of spiral
dive. The model, if carefully constructed, will heel and dive
Either to the right or to the left, depending upon some accidental
disturbance to initially decide which way it will go ; a want of

180

LATERAL AND DIRECTIONAL STABILITY §102

symmetrical accuracy in launching is a sufficient determinant.
It is from the nature of the motion involved, that the author
has taken the name rotative as applying to this form of
instability.

In the consideration of the question of lateral stability the
effective centre of pressure of the fins was tacitly assumed to be
situated vertically above the mass centre of the aerodone, for it
was supposed that the direction is unaffected by the fin reaction,
the function of which was taken as exclusively that of restoring
the lateral equilibrium and of damping out lateral oscillations.
In the study of directional stability and maintenance it was
found that the position of this centre of pressure is one permit-
ting of considerable latitude ; the fin-tail length A might vary
from any positive value, through infinity, to a limiting (mini-
mum) negative value. Now A = ± oo means that the aerodone
is directionally balanced, and the centre of lateral pressure
is perpendicularly over the mass centre, so that in this case
the condition assumed in the sections on lateral stability is
complied with.

Were it possible to ensure in every case that A = oo, the form
of instability that we have now under consideration would not
exist, and the present investigation would be unnecessary ; in
the case of birds, where the whole machine is under continual
surveillance, without doubt this is a point that can be relied
upon, and in actuality there is every probability that directional
balance is obtained to a high degree of approximation. In arti-
ficial flight, however, it is advisable not to approach too closely
to a limiting condition, and although it has been shown that the
limiting condition in this case lies beyond that of A = oc, still
the margin is small, as shown by the fact that the negative tail
is a phenomenon difficult or impossible of demonstration in air
from the gossamer-like construction that would be necessary.

§ 102. Rotative Stability (continued). — Given that the fin-tail
length A is positive and finite, that is, the fin combination is the

181

§ 102 AERODONETICS

equivalent of a fin-tail of finite length extending rearward,
rotative instability depends upon the fact that any change in
lateral trim has an effect on the course, and the resulting change
of course reacts on the lateral trim and so on with cumulative
effect. Let us suppose that an initial " list " be imparted to the
aerodone by any accidental cause, the immediate consequence is
that the aerodone commences to slide sideways down an
imaginary inclined plane of its own making — the approximate
plane of the aerofoil ; this lateral motion, as we have already
seen, brings pressure to bear on the fins, tending to " right " the
aerodone. At the same time this pressure, acting on the whole
behind the mass centre, results in a change of course in the
direction of the initial list (port or starboard as the case may be).
This change of course involves a curvilinear flight path, such
that the one wing travels faster than the other, and therefore
experiences a greater pressure reaction; and the wing that
travels the faster is that having the upper position by virtue of
the initial list, so that this initial list will be increased. This
augmented list results in turn in a more rapid change of course,
and so on, the ultimate result depending upon whether or not the
accumulation approaches a finite limit.

We have therefore to deal with a problem involving a rotary
motion of the aerofoil about an axis perpendicular to the flight
path, and we have to correlate such motion with the couple to
which it gives rise, both as to magnitude and direction. The
problem is one of some difficulty.

It would appear that the least compromising method of treat-
ment is to suppose that the whole lifting power of each wing of
the aerofoil is concentrated at a particular point, which we may
term the aerodynamic centre, so that we may base our calculations
of lifting effort, and therefore capsizing moment, on the relative
velocities of the two aerodynamic wing centres. It is evident
that for any given aerofoil such points must exist, although they
may require to be determined experimentally for each particular
form. We will term the distance of the aerodynamic wing

182

LATERAL AND DIRECTIONAL STABILITY §103

centre from the mass centre of the aerodone the aerodynamic
rad ius^

Similarly the resistance in the line of flight may be conceived
to be due to the whole resistance of each wing concentrated at
some stated point along its length. We will term this point the
a erodromic wing centre , 1 and its distance from the mass centre
the a erodromic radiu s.

These two quantities, the aerodynamic and aerodromic radii,
may be taken as fundamental data belonging to any particular
aerodone that must be independently ascertained. Certain
theoretical deductions relating to the same will be given at the
conclusion of the investigation.

§ 103. Rotative Stability. Basis of Investigation Defined.

Let W\ be the apparent weight, i.e., resultant of actual

weight and centrifugal force of flight path.
V = velocity of flight, as before.
p = density of air, as before.
A = fin-tail length, as before.
a = area of abutment fin, as before.
b = fin centre height, as before.
r = flight path radius.
a = aerodynamic radius.
,, s = aerodromic radius.

r = lateral velocity of mass centre.
/3 = inclination of abutment fin to its direction of
motion.
„ t = torque about axis of flight.
„ T = torque or turning moment about vertical axis.
C = addendum angle of heel, i.e., relatively to the

apparent plumb.
c and C be the aerodynamic constants of the abut-
ment fin, as before.

1 This term is employed for want of a better. Tho quantity is one that, so
far as the author is aware, is only of use in connection with the present
problem.

188

99

99
99
99

99

99

§ 103 AEE0D0NETICS

It will be supposed in the investigation that follows that the
aerodone is gliding in a flight path forming the arc of a circle,
and the investigation will have for its object to determine the
point at which the radius of curvature is constant, that is to say,
to determine the conditions under which the various torques and
reactions are balanced. The general solution to include variations
of r (the flight path radius) would be needlessly complicated ;
the particular case in which we are interested for the time being
is the straight gliding path, and while the physical supposition
will be throughout that the path is of curvilinear form, it will
be supposed that the value of r is of a higher order than the
other linear quantities concerned, and the usual consequences of
a supposition of this kind will be taken advantage of as occasion
may arise.

The torque about the vertical axis, due to the variation in the
relative wing resistances (in the line of flight), if included in the
main investigation causes some tiresome complication. This
factor has been eliminated by a preliminary process, that takes
the form of a correction applied initially to some of the data or
to the design of the aerodone.

§ 104. Rotative Stability. Preliminary Investigation. — The
velocities of the aerodromic wing centres (Fig. 98) will be given
by the expressions 1 : —

- - T and T .

r r

Now y W\ = total resistance in the line of flight at velocity V or
—^- l * 8 resistance due to each wing of the aerofoil for straight

path. And since the resistance is proportional to the velocity
squared, 2 the resistance of the wings will be respectively : —

yWxir + s)* y jTi (r-»)*
- 2 "a a™* 2 ^ '

1 This statement, as applied to the figure as drawn, is scarcely accurate ; the
actual conditions, however, are those of the small angle where cos /3 may be
taken as unity.

a Compare " Aerial Flight," Vol. I., Aerodynamic*, § 159.

184

LATERAL AND DIRECTIONAL STA BILITY § 104

But * is small in comparison to r, therefore s 2 is negligible,
and the expressions become : —

-j- X ( 1 + T ) and I_» x ( i _ _ j
and the difference of these is

. TIT ^ *

therefore the turning moment T = y n\ — (1)

and, (Fig. 98), r = £ = ±1

p v

T = y J1\ *?-? (2)

A \

Now let F (Fig. 98) represent the transverse force sustained by
the abutment fin, then F = a c /3 C p V 2

but 6 = 4> /. F = acrCpF (8)

Now in expressions (2) and (8) the only variable in respect of
the path curvature is r, the whole of the other quantities are
constants whose value is known, and thus we have T a f , or
equating the two we have,

_ r? 2 y U\ * 2

r — f

acCpPA

But where the radius of curvature is great, that is, when the
flight path is approximately straight, JFi is sensibly equal to W
the weight of the aerofoil. And WjY* is the constant of the
aerofoil denoted by A',

.-. T = F —t-r, — r- (4)

2 y K s 2
Now the quantity — — r , -v- in this equation is a constant

q. c o p a

of linear dimensions, and represents a distance I at which a force

= F must act in order to give the torque T. In Fig. 98 this

torque is represented by the couple whose constituent members

185

§104

AERODONETICS

are the forces L\ and F 2 . Of these two forces h\ is arranged
acting through the mass centre of the aerodone, 1 and is therefore
equal and opposite to F, consequently the force h\ alone remains,
and we have the result that the turning moment due to the differ-
ence of the resistances on the two iviyigs of the aerofoil is equivalent

Fig. 98.

to a disfflacement of the centre of gravity rearwards by an amount

= I where

2y Ks*

I =

ac C p A'

1 The usual method of resolution.

186

LATERAL AND DIRECTIONAL STABILITY §104

\

\

\

\

a

m

©

ft

187

§ 104 AEEODONETICS

There is one point of some subtlety that requires consideration
in relation to this correction. The quantity A is initially cal-
culated from the actual position of the centre of gravity, and the
Jin plan, as in § 99, and the displacement of the centre of gravity
in accordance with the present equation will vitiate the value of
A so obtained. Two courses are open : one is to move the fin-
plan back bodily by an amount = l 9 that is to say, actually move
it hack by this amount in the design and in the aerodone ; the
other is to repeat the calculation with the new value of A sub-
stituted, and repeat again as many times as may be necessary
to arrive at a sufficiently close approximation. In either case
the present method constitutes far the simplest solution to this
portion of the problem.

§ 105. Rotative Stability. Investigation. — Let Fig. 99 represent
diagrammatically an aerodone in flight, the symbols correspond-
ing to those given in § 108. 1

Then, as in the preceding section, the velocities of the
aerodynamic wing centres are respectively —

L+l y and L=± V,
r r

and since the lifting reaction varies as J' 2 the lifting efforts of
the wings will be respectively : —

V\ (r + <>)* , Wi (r - <r) 2
2 r 2 2 r» '

But a being small in comparison to r f <r 2 is negligible, hence
difference between lifting efforts is —

ITw, , 2(r\ Wi [ « 2

?(• + ¥)-¥(!-¥)

r

1 In Fig. 99 / is the centrifugal force due to the curvilinear flight path,
\\\ is the resultant of W and /, R x is the total reaction on the aerofoil, and
B* is the resultant of R\ and \V\.

18d

LATERAL AND DIRECTIONAL STABILITY §105

2 0" 2

and torque t = W\ -^-. (5)

This is the torque tending to capsize the aerodone, and the
condition of equilibrium is that this shall be balanced by the
righting moment due to pressure on the fin area due to the lateral
motion.

Now force F acting on abutment fin = a c /3 C p V 2 and

.-. F = ac v C p V

(this is equation (8) of § 104),

A V A V

and r = — or v =

F =

v r

acCpPA

. , . . , *bcC P V*A ia .

or righting moment = -J- , (6)

and the condition of stability is that this shall be equal to or
greater than the torque r or by (5) and (6),

a b c C P V 2 A > 2 W x o 2 ,
but for the nearly straight flight path Wi = W, .*. 7^ = K,

hence we may write the above,

aJcCpA

>1,

2 A<r a

which is the equation that must be satisfied if the aerodone
is to be rotatively stable.

It is worthy of note that the above expression is independent
of the velocity of flight ; thus if an aerodone be designed that is
just stable for any given values of weight and velocity, then it
will be just stable, if its geometrical form be the same, for all
other appropriate values of W and V. Thus if it be made four
times its original weight, so that its velocity be twice as great,
its rotative stability will undergo no change.

189

§105

AEEODONETICS

V

*

^

8 I I

LATERAL AND DIRECTIONAL STABILITY §106

§ 106. On the Aerodynamic Radius. — The value of the quantity
that we have defined as the aerodynamic radius, can, if desired,
be determined experimentally. Thus, employing a whirling
table, if determinations are made of the total lifting effort of any
aerofoil under given conditions, and if also the turning moment
about the axis of flight, due to the fact that the two wings of the
aerofoil are at different radii, be measured, the value of o- is a
matter of simple calculation. The disposition of the apparatus

+

*S I 4f j *5

I

Fig. 101.

required to carry out this method is given diagrammatically in
Fig. 100. 1 Probably the best procedure would be to find the

1 Referring to Fig. 100 the arm of the whirling table c is mounted adjust-
ably on the head of the vertical shaft a arranged to run in a fixed bearing b.
The aerofoil y is mounted on a bolster / adapted to pivot on the steel points
c and c ; the bolster / is arranged a short distance from the geometrical
centre of the aerofoil o, so that the reactions on the wings of the aerofoil
shall balance, and this distance may, if desired, be made adjustable by
suitable construction.

In the arrangement shown in the figure, the radius of the whirling table
is made adjustable by means of the screw d, instead of providing means of
varying the distance of o ; the result is the same.

In order to signal the condition of equilibrium two electrical contacts are
provided, h and h, connected by leads j and j with two commutator rings k,
mounted on the insulating sleeve n. These two contacts control two alterna-
tive circuits, and indicate the position of the aerofoil by a galvanometer, or
by electric bells in the well-known manner.

The arrangement as shown is suitable for small scale experiment ; for
large scale work modifications in the design would probably be required.

It is evident that if the radius of the whirling table be increased the
reaction on the inner wing will increase faster than that on the outer one,
and vice verm ; the adjustment should be made till the reactions balance.

The aerofoil should be initially adjusted to balance gravitationally about
its pivot support, in order to eliminate centrifugal force.

191

§ 106 AEEODONETICS

position of the point of suspension at which the reactions on the
two wings accurately balance ; the velocity of flight as a factor
being in this way eliminated. In any case the whirling table
employed should be of not too great a radius, as the magnitude
of the quantity measured will be greater as the radius of the
whirling table is less. 1

The above method may be regarded as particularly appropriate
in view of the* resemblance that exists between the conditions
of experiment and the conditions of flight as presented in the
theory.

§ 107. On the Aerodynamic Radius. Theoretically Considered. —
Let us assume as in §§ 192 and 205, Vol. I., that each element
of the length of the aerofoil sustains a reaction appropriate to its
velocity through the air; this assumption being that we may
apply to every small increment of the length of the aerofoil the
equation, W = K V* which we know applies to the aerofoil as a
whole. Thus let Fig. 101 represent the aerofoil which we will
suppose divided into a number of sections whose constants are
respectively K\ K* K% K^ etc., then we are assuming that the
lifting power or weight supported by these sections individually
is given by K\V^ K%V* K3V*, etc., where V\, V*, etc., are the
mean velocities of the different sections.

That this assumption is not strictly accurate can be easily
demonstrated ; thus, let us suppose the aerofoil to revolve about
one of its terminal elements, say K Q in the diagram ; then if
the supposed law hold good there will be no lifting effort what-
ever on Kg. But we know that K 8) although itself virtually
immobile, is preventing the immediate flow of air from the
under to the upper side of the section marked K lt consequently
it must itself experience some difference of pressure between its
under and upper surfaces.

Dealing with the matter broadly, the tendency of the " field of

1 At the time of writing the author has not actually employed the method
proposed ; an estimate on the basis of § 107 having been made to serve,

192

LATERAL AND DIRECTIONAL STABILITY §107

force M1 is to spread from the regions of greater intensity to those
of less, so that any difference from normal in the distribution of
the pressure reaction will be ameliorated to some extent by a
redistribution of the field, and the differences and therefore
the aerodynamic radius will be less than would be computed on
the assumed basis of constant K values. The extent to which
these considerations invalidate the results of our assumption we
have no means of gauging other than by direct comparison with

Fio. 102.

experiment, experiments that have not at present been made,
but it is evident that the greater the aspect ratio the more
completely will the conditions be complied with, and, as stated
in Vol. L, § 205, our assumption may be considered a close
approximation when the aspect ratio is considerable (say upwards
of four or five), and a rough approximation only when the
aspect ratio is small. At the worst the final result will be an
error on the safe side.

Referring now to Fig. 102, we have plotted with respect to

A.

1 "Aerial Flight," Vol. I., Aerodynamics, Chap. IV.

198 o

§ 107 AERODONET ICS

the origin O and co-ordinates x and y, the values of W per unit
A" for an aerofoil whose mean path radius is r; abscissae
represent radius values and ordinates = IF/A'. Then since
1V/K = I' 9 the curve is of the form y = n x 2 where n is a
constant.

Now let us suppose that the distance a a be that occupied by
the aerofoil, and let us assume, as in § 105, that we are dealing
with the nearly straight flight path, so that r is great compared
with a a, then we may take the form of the curve within the
span of the aerofoil to be a straight line. Let xi and y\ be the
absciss® and ordinates about a new origin 0\ chosen on the line
of flight : then the equation to the straight line approximation,
from the new origin, will be yi = n\ xi where v\ is a constant.
But //i is the increase in lifting force per unit A, due to the excess
of velocity consequent on the circular flight path, and is positive
on the one wing of the aerofoil and negative on the other. And
the turning moment about o\ resulting from this increase
== yi xi = v\x? which is positive whether x is positive or
negative, hence it is positive for both wings of the aerofoil. Or,
the total turning moment may be represented by the expression
2 (A iii fi 2 ) that is to say, the sum of the turning moments of
the individual small elements.

It may be recognised at once that the problem of determining
the turning moment for any given aerofoil is strictly analogous
to that of finding the moment of inertia of a thin bar in which
the mass per unit length of the bar is represented by the value
of A per unit length in the aerofoil, hence, the aerodynamic radius
of the aerofoil is equal to the radius of gyration of a thin Itar whose
mass is at every point proportional to the value of A, increment for
increment, of the aerofoil.

If the " grading " of the aerofoil be that proposed in Vol. I.,
§ 192, that is to say, parabolic, we know by Routh's rule

L
that the aerodynamic radius a- will be ----.— or, <x = *228 L,

J 2 V 5

where L is the length of the aerofoil.

194

LATERAL AND DIRECTIONAL STABILITY §108

§ 108. On the Aerodromic Radius. — The aerodromic radius could
evidently be determined experimentally on the lines proposed
for the aerodynamic radius in § 106, the axis or pivot on which
the aerofoil is mounted being arranged vertically instead of
horizontally ; the forces to be measured are, however, compara-
tively small (commonly from one-sixth to one-eighth of the
vertical reaction), and the apparatus would require to be
appropriately sensitive.

From the theoretical point of view, if the form of the aerofoil
were that discussed in § 120 of Vol. L, the problem would be
greatly simplified, for the aerodynamic and aerodromic radii
would be equal. We know, however, from §§ 190, 192, that
such a form does not comply with all the conditions, and that
the aerofoil extremities have to be formed to give rise to motion
of the discontinuous type. In nature we know that the wing
terminations are given much diversity of form to this end, and
that in general the sectional form should be flattened so as to
become virtually an aeroplane at each extremity.

Under these conditions the value of the aerodromic radius is a
quantity that, from a theoretical standpoint, we are unable to
estimate with any great degree of accuracy in the existing state
of knowledge. 1

1 Compare § 1 14.

195 o 2

CHAPTER VIII

REVIEW OF CHAPTERS I — VII AND GENERAL CONCLUSIONS

§ 109. Preliminary. — In the foregoing chapters many results
individually of importance have been demonstrated ; the main
consequence of these results — the definite proof of the automatic
stability of a properly designed aerodone — is alone a matter of
very great moment. The establishment of the laws of stability
in a form immediately available to the designer, cannot fail to
have a far-reaching influence on the future of mechanical flight,
and on the development of the flying machine for aerial
navigation.

In the present chapter we shall first carefully review the theory,
especially with regard to its aerodynamic foundations, which will
be examined minutely, and subsequently any unaccounted factors
and limitations, where such limitations exist, will be discussed.
We will then briefly consider the application of the theory to the
problem of mechanical flight, dealing with such points as the
influence of the mode of propulsion, the rate of damping of the
phugoid oscillation, the law of corresponding speed and the
conduct of model experiments, departures from the elementary
type, etc., etc. ; afterwards discussing the experimental verifica-
tion and employment of the theory of lateral stability of
Chapter VII., and in conclusion a brief note will be given of the
historical development of the present aerodonetic theory and of
its experimental confirmation.

§ 110. Aerodynamic Basis of the Phugoid Theory. — From a strictly
logical point of view the series of investigations forming the
subject matter of the present volume begin at the commencement

19G

GENERAL CONCLUSIONS §110

of Chapter II. The strictly logical order is, however, not that
in which a subject is always best presented, and the preliminary
discussion of Chapter I., with the account of some of the author's
earlier experiments, serves as a preparation for the more abstract
treatment that follows.

It is a remarkable fact that the " Phugoid Theory " of
Chapters II. and III. with which the rigid investigation opens,
rests, apart from the specific conditions laid down by hypothesis,
implicitly on one aerodynamic law and one law only ; i.e., the law
that the reaction varies as the square of the velocity. 1 Thus the
aerofoil is supposed to be rigid and to suffer no change in attitude
or aspect ; 2 and the form and amplitude of the motion set up by
it in the air through which it moves are sensibly constant ; 8 it
follows directly from aerodynamic principles that the reaction
is point for point proportional to V 2 . Apart from the fact that
this law has been established experimentally, not only for direct
resistance but also for oblique reactions, 4 we know that the limits
of its approximate application^ as due to viscosity on the one
hand, and to elasticity on the other, are far removed from the
values with which we are concerned. 5

It is a most important and striking fact that so great a
result from the aerodonetic point of view, as that of the phugoid
flight path, can be deduced from one single aerodynamic law.
One prime consequence of tins fact is that the experimental

1 In addition to the ordinary laws of motion.

8 For the specialised meaning of the terms aspect and attitude see Glossary.

3 When viscosity is excluded, and the V 2 law applies, the motion of the
fluid at different velocities is, in effect, homomorphous.

4 Proved more especially by the experiments of Dines, in which pressure
reactions are balanced by the centrifugal force of an adjustable mass; see
" Aerial Flight," Vol. I , Aerodynamics, §§ 223 et seq.

5 The influence of viscosity makes itself felt only for small bodies at low
velocities where the Newtonian Law is about to give place to that of Allen,
" Aerial Flight," Vol. I., Aerodymtmirs, §§ 50 et seq. The influence of
elasticity is probably not sensible for low flight velocities, but at high
velocities a correction is required ; compare " Aerial Flight," Vol. L,
Aerodynamics, Appendix T.

197

§ 110 AERODONETICS

confirmation of the path form and period, incidentally provides
an independent proof of the law on which the theory is founded,
as well as proving the soundness of the theory itself.

§ 111. Basis of the Equation of Stability. — The aerodynamic
basis of the extension of the Phugoid Theory dealt with in
Chapter V., culminating in the equation of stability, is constituted
by the following : —

(1) Law of pressure on the normal plane, Pgo = C p J' 2 ,
also the numerical value of the constant C, established
experimentally.

(2) Law of pressure on planes at small angles, Pp = c /3 P&
definitely established for planes of square proportion by the
experiments of Duchemin, Dines, and Langley, as obtaining over
a very considerable angle of inclination ; also found to be
generally true for planes of other proportions but over a less
angular range. It probably does not apply even approximately
for planes of extreme ratio in apteroid aspect ; planes of such
form are consequently to be avoided. 1

(8) The law of resistance to flight, R varies as F a . This law
is not the same as that which obtains for W = constant, 2 neither
is it the law of least resistance* In the problem as presented in
the Phugoid Theory the load, i.e., the mass of the aerodone, it
is true does not change, but the effect ire load does change; it is
made up of the component of the weight plvs the centrifugal
force due to the curvilinear flight path, measured plus
or minus according as it acts in the one direction or the
other (compare Chapter II. , § 20). Thus this law is merely
a direct consequence of the law on which the Phugoid Theory
itself is founded (§ 110), that the reaction on the aerofoil
is point for point proportional to I* 2 ; the total reaction can

1 Compare "Aerial Flight," Vol. I., Aerodynamics, § 151. The pheasant's
tail may be cited as an example.

2 Compare " Aerial Flight," Vol. I., Atrwlynamict, § 159.
Aerial Flight," Vol. I., Aerodynamics, § 166.

a <<

198

GENERAL CONCLUSIONS §112

manifestly be divided into its horizontal and vertical components 1
to each of which component reactions the same law must apply ;
hut the horizontal component (or rather line of flight component)
is the resistance to flight, hence the present law is corollary to
that of the preceding section.

§ 112. The Equation of Stability. Unaccounted Factors. — The
extent to which the equation of stability has received experi-
mental confirmation shows that any factors omitted in the
theoretical investigation can at present be regarded as unim-
portant. The fact, however, is not without interest that the
theory as presented does omit items whose influence may in the
future, when experimental methods have undergone some refine-
ment, be found to be of sensible magnitude.

In the first place it may be pointed out that the primary
influence of the changes in the altitude 2 of the tail as affecting
the value of j3 may not, as has been assumed, cause an exactly
proportional change in the reaction. Before the validity of this
assumption can be discussed, we require to know more about the
form of the aerofoil employed. If this be an aeroplane, and if /3
be a decidedly small angle, then we know that the assumption is
justified by direct experiment. If, on the other hand, the
aerofoil be one of pterygoid form, it would appear that the angle
a, and the curvature of the section, ought, strictly speaking,
to undergo simultaneous variation.

The above factor as bearing on the foundation of the equation
of stability probably results in an error of a lower order of
magnitude than the quantities with which the equation deals,
owing to the fact that the demonstration only applies strictly to
the phugoid of evanescent amplitude ; that is, to the stability
of the straight gliding path. The factors in question doubtless

1 It is convenient to use tbe term horizontal for the direction of flight, and
vertical for the normal thereto. The supposition here is that the flight path
is horizontal.

9 The term attitude is defined as the position of the plane or other hody
about the horizontal transverse axis in relation to the line of flight.

199

§ 112 AEKODONETICS

have some sensible influence on the conditions of stability for
flight paths of greater amplitude, but this is a wider problem,
of which, so far, no attempt has been made to give a theoretical
solution.

In the equation of stability as first presented the influence of
the " wash " from the aerofoil as affecting the motion of the tail
plane is not taken into account, but an approximate method of
dealing with this factor is suggested and embodied in the final
form of the equation (§ 68). The experimental evidence of the
validity of this correction, although at present inconclusive, is
on the whole satisfactory.

From the nature of the assumption on which the correction is
founded, i.e., that the tail is, in effect, situated outside the
peripteral region, the correction is evidently only an approxima-
tion whose accuracy depends upon the extent to which the
assumption is justified in practice. Thus if the tail length is
considerable, so that the tail plane is remote from the aerofoil,
the error should be small, whereas if the tail be short, so that the
tail plane is close to the aerofoil and so involved in its periptery,
the error will be considerable.

The method employed to deal with the "wash " of the aerofoil
is also employed in the investigation of directional stability (§ 91)).

§ 113. Basis of the Theory of Lateral and Directional Stability. —
In the portion of Chapter II. relating to lateral and directional
stability, there is very little new importation of aerodynamic
theory or data. The qualitative behaviour of the centre of
pressure of an inclined plane under changes of angle and
aspect, as known by experiment, is assumed in the explana-
tion of the behaviour of an aerodone under lateral disturbance,
and the law of pressure at small angles is again used to a
considerable extent. In § 99 the method of allowing for the
effect of the " wash " of the leading fin on the following fin is
similar to the method of making an analogous allowance in the
case of the tail plane, and involves an application of aerodynamic

200

GENERAL CONCLUSIONS §114

theory not otherwise employed, the residuary current from
the leading fin being taken as equal to the lateral velocity
component of the leading fin multiplied by (1 — €). Now €
represents the velocity of the upcurrent in terms of the velocity
of downward discharge of an aerofoil or aeroplane, the upward
direction being, of course, taken as that of the pressure reaction ;
and since the loss of momentum of the air in recess is equal to
that imparted in approach, 1 the residuary current has a velocity
(1 — c) times that possessed by the air at the moment it quits
the after edge of the plane in motion, hence the rule given.
It will be seen that in practice the " wash " will, if the present
line of reasoning is correct, have a greater velocity than that
stated, for the following plane is not an infinite distance away, so
that the air will not have finished parting with its momentum
and the residuary velocity will be greater than that calculated.
In spite of this fact, unless the one fin follows the other very
closely, the error will not be worth consideration, for by far the
greater part of the dynamic action is quite local and takes place
within a radius of one or two times the width of the plane or

aerofoil.

For the understanding of the foregoing argument, a full know-
ledge of the author's treatment of the dynamics of the periptery
is essential; reference should be made to Vol. I., Chaps. IV.
and VIII.

§ 114. Aerodynamic Basis of the Investigation on Rotative
Stability. — Beyond the free employment of the aerodynamic laws
and data already assumed, the only additional data required in
this investigation are the aerodynamic and aerodromic radii.
These quantities are fully dealt with in §§ 106, 107, and 108,
and may be considered as involving a direct appeal to experi-
ment. As independent experimentally ascertained values, no
exception can be taken to their introduction into aerodonetic
theory or into the manner of their employment. At present,

1 Comparo "Aerial Flight," Vol. I., Aerwlymnnivs, §§115 and 179.

201

§ 114 AERODONETICS

however, no experimental determinations have been made, and
the only knowledge on the subject that is available is that
deduced in § 107 from the theoretical considerations.

The author believes that in practice, in a well designed aero-
foil, the aerodynamic wing centre lies between °21 and '23 of the
length of the aerofoil from the line of flight, and that the corre-
sponding distance for the aerodromic wing centre is from # 28 to
•25. That is to say,

Aerodynamic radius, > "21 L < "28 L.
Aerodromic radius, > *23 L < "25 L.

§ 115. Limitations and Unaccounted Factors. — No account has
been taken in the present theory of the changes that take place
in the energy of the periptery, as the aerodone in its flight path
undergoes changes in its velocity. 1 It would appear possible
that, under certain circumstances, the energy of the periptery
may become a factor of sensible magnitude. Regarding the
said energy as accountable on the basis of a cojyxakd-MWM
of some definite value, then, the peripteral energy for a given
aerodone being taken as proportional to its velocity squared,
we may regard the force of gravity as in part acting on the
mass of the aerodone and in part on the concealed mass
of the periptery. According to this view only part (a con-
stant proportion) of the potential energy of the aerodone appears
visibly in the kinetic energy of its descent, the other part (also a
constant proportion) being absorbed in the peripteral system.
Thus the motions of the aerodone will be precisely as if the force
of gravity were diminished by a certain proportional amount,
that is to say, the whole of the relations given in § 86 require to
be construed with a value of g below its true value in the con-
stant proportion in which the kinetic energy of the aerodone is at
every instant less than the total kinetic energy of the aerodone
and its periptery. It seems possible that this quantity is not the

1 Compare "Aerial Flight," Vol. L, Aerodynamics, §§81, 82, 84, 85 and
123.

202

GENERAL CONCLUSIONS §116

same for different cases, but we are at present in tho dark as to
the extent of the correction involved ; the experimental verifica-
tion of § 69 would suggest that under ordinary conditions the
error from the cause suggested is negligible.

§ 116. Limitations and Unaccounted Factors (continued). — In
addition to the point raised in the preceding section, on which some
doubt must, for the time being, exist, the question of the elasticity
of the aerodone or of its parts, in particular elasticity of the
aerofoil itself, is one which may place a limit on the applicability
of the theory.

In the theoretical and experimental foundation of the law
of the reaction variation (reaction varies as square of velocity),
the rigidity of the aerofoil is a definite part of the hypothesis.
This condition is fulfilled with a sufficient degree of approxima-
tion in the case of the mica models employed by the author for
the purposes of verification, and it may be taken as complied
with by the generality of artificial flight models used by other
experimenters. In the case of birds and other living things
capable of flight, the circumstances are different ; in general, the
wings are flexible in a high degree, so that their form changes
with every change in the pressure reaction. Under these con-
ditions the phugoid equation probably only applies as a rough
approximation ; changes in the sectional form of the aerofoil
mean changes in the value of II w and although the consequences
of such changes are in every likelihood quite amenable to
theoretical treatment nothing of the sort has been at present
attempted.

The kind of deflection of the wing under load must depend
upon the wing structure. In the wing structure of birds any
addition lo the pressure takes effect principally by straighten-
ing, or bending backward, the secondary remiges, so that the
camber of the arched section is diminished ; this is illustrated by
Fig. 108, which is a reproduction of Fig. 57 of Vol. I. In the
case of an aerodone or flying machine, in which the aerofoil

203

§ 116 AERODONETICS

consists of a canvas sheet stretched between an anterior and a
posterior rail, or arranged like the sail of a ship, the reverse
takes place, the greater the pressure the more the canvas will
" belly." In the case of the pterodactyles of old, as also in the
bat, and in gliding or flying machines modelled on similar lines,
the effect of increased pressure must be considered doubtful ; it
depends upon the directions and degrees of the tensions by which
the wing membrane is restrained.

Applied to aerodones or aerodromes whose aerofoil is flexible
in any of the manners suggested, the Phugoid Theory must have
its limitations, the form of the departures from theory evidently

Ola/ a

Wing or HeRm/VQ Cull

sec r/o/v.

Fig. 103.

depending upon the form of flexibility that the aerofoil exhibits
under stress. The application of the extension of the theory
in the case of the swift and the albatros suggests, however,
that even under these doubtful conditions the results are of
value.

§ 117. Limitations and Unaccounted Factors (continued). — When
the aspect ratio n of the aerofoil is low, so that its fore and aft
dimension becomes considerable, the changes in the position of
its centre of pressure, consequent on changes of attitude, may
add materially to the longitudinal stability of the aerodone ; the
behaviour of the aerofoil becoming comparable to that of a
ballasted aeroplane, where stability is provided without any tail

204

GENEKAL CONCLUSIONS §117

organ at all. This addition to the stability is one for which
it is difficult to make proper allowance : the conditions are too
indefinite.

A further difficulty is experienced in the application of the
equation of longitudinal stability in cases where the tail plane,
instead of being of square proportion or of elongate form in
pterygoid aspect, is of elongate form in apteroid aspect. The
law of pressure on such a plane approximates to the sine square
law of Newton, 1 and we can no longer regard the small angle
law, which forms part of the basis of the theory of stability, as
holding good ; consequently, in such a case, the equation cannot
be applied. Fortunately, planes in apteroid aspect possess other
disadvantages from an aerodynamic point of view that render
their employment in artificial flight out of the question, so that
the present limitation to the theory is of but little importance.
Perhaps the best example in nature of a tail in apteroid aspect
is found in the pheasant ; this bird, if one may judge by the
form of its tail, has some special private objection to its stability
being investigated.

The most serious difficulty in the employment of the theory of
lateral and rotative stability is due to the fact that the quantities
dealt with in the equations can in practice only be inferred. In
nature the vertical fins on which this theory is based do not exist,
and we have practically no means of applying our knowledge to
this branch of the subject. In artificial flight models the employ-
ment of fins undoubtedly adds unnecessarily to the skiu-frictional
resistance, but the extent of this addition is not by any means an
intolerable burden, and the author, as will be gathered from the
experiments cited, commonly employs models so fitted.

In the case of an actual flying machine the question of power
expenditure becomes a matter of first importance, and probably
the fin area will have to be reduced to a minimum. In the
earlier or experimental stages, however, there seems to be no
reason why they should not be retained, at least, as an auxiliary.

1 " Aerial Flight," Vol. L f Aerodynamics, §§ 150, 151.

205

§ 117 AHRODONETICS

It has been shown that a large portion of the duty of the fins can
be performed by the ends of the aerofoil itself, these being
arranged to slope upward at an angle ; the question of the
equivalent of such upturned ends expressed in fin area will
be discussed in a subsequent section.

§ 118. The Influence of the Mode of Propulsion. — In the appli-
cation of the various branches of aerodonetic theory to the
solution of the problem of stability in the construction of an
aerodrome or flying machine, the mode of propulsion becomes a
matter of importance.

In the basis of the investigation of Chapter V., the pro-
pulsion was assumed as applied in the simplest possible manner,
i.e., as a component of gravity ; so that the propulsive force, for
curves of small amplitude, could be taken as constant.

In an aerodrome or flying machine the propulsion is not so
derived, it is commonly due to a thrust supplied by a screw pro-
peller, or pair of propellers, caused to rotate by some form of
prime mover. Now, in such a combination the thrust, for a
small variation of velocity, such as contemplated by the theory,
may be sensibly constant, or it may Increase when the velocity
increases, or, as is more generally the case, it may increase when
the velocity decreases. Beyond this, when the motor is fitted
with a fly-wheel the thrust is also a function of the change oj
velocity, that is to say, other things being equal, when the fly-wheel
speed is increasing the torque acting on the propeller is less than
when the fly-wheel speed is decreasing, and the thrust consequently
is also less. It is evident, therefore, that factors are introduced
into the problem foreign to the initial investigation, to an extent
that cannot fail to complicate the subject.

Before we can discuss the influence of propulsion we must
devote some consideration to the special circumstances surround-
ing the employment of motive power in connection with flight,
both as touching the motor itself and the means of propulsion.

Assuming, firstly, that the latter take the form of a screw

206

GENERAL CONCLUSIONS

§118

propeller, an assumption that the author believes the future will
fully endorse, we know that the relation between the torque and
thrust is approximately constant for variations of velocity of
small magnitude. 1 Thus instead of being concerned with the
variations of the propeller thrust, we can go direct to examine
the law of variation of the torque of the prime mover.

In Fig. 101 we have plotted two curves ; in the lower diagram
abscissae represent revolution-speed and ordinates horse-power;

if.

Yio. 104.

in the upper diagram abscissae again represent revolutions, but
ordinates represent torque. Now, referring to the upper curve, it
is evident that the law of variation of torque as a function of speed

1 It is presumed that the propeller is approximately a design of best
efficiencj' ; if this condition is infringed the fact stated may not hold good,
the essential is that the blade should be properly proportioned for least
gliding angle. Compare " Aerial Flight," Vol. I., Aerodynamics, §§ 202 et seq.

207

§118

AERODONETICS

of revolution might be approximated by an empirical equation
of suitable form, 1 but as in our equation of stability we do not
require to deal with the whole curve, it is sufficient to approxi-
mate the conditions by a tangent at the point with which
we have to deal, that is, the point on the curve at which the
abscissa represents the velocity of the motor proper to the
natural velocity of the aerodrome. This point for a given motor
and for a given value of r^will depend upon the pitch of the pro-

Fio. 105.

peller and the ratio of the gear transmission, and in the abstract
we have nothing in the conditions to determine these quantities.
The diagrams given in Fig. 104 are rough plottings from the
brake tests of a petrol motor, but this type of diagram is common
to other prime movers. The point on the curve chosen for
engines of fixed speed, or the portion of the curve utilised when

1 This id tho same law for given conditions as that of thrust expressed as
a function of velocity V,

208

GENERAL CONCLUSIONS

§119

the speed is variable, is commonly that in the neighbourhood
either of the speed of greatest torque or that of greatest horse-
power. Thus in the case of a stationary gas engine, where fuel
economy is a matter of first importance, the point on the curve
chosen is preferably that where the speed is not greatly in excess
of that of maximum torque. When, as is usual in the case of
the flying machine, the weight of the prime mover is the most
important consideration, the point on the curve chosen should be
as nearly as possible that of maximum horse-power.

§ 119. The Influence of the Mode of Propulsion (continued). —
In Fig. 105 li is the curve of resistance, and F the curve of

Fig. 106.

propulsion; the ordinates of the curves li and F represent
respectively the resistances and propulsive forces proper to
different values of V as given by abscissae. It is assumed that
the forces of propulsion and resistance are equal when the
aerodrome is travelling at its natural velocity, that is, when
V = V n the point p in the figure.

Let the ordinate of the line p a represent the constant force
of propulsion Q assumed in §§ 50, 51 and 57; then drawing

a. 209 p

§ 119 AERODONETICS

tangents to the two curves at the point p, and a line c b
perpendicular to p a, we have, for a given small change in V,
the difference between the force of propulsion and the resist-
ance represented by the line c & as compared with an amount
a b under the conditions of a constant propulsive force.

Let us denote the quantity — ~. by the symbo l ^._ ^

When the motor is working at maximum horse-power, we have
for small variations, V F = constant, or

But R a V 2 hence

or by (1) and 2),

dF
dV~

F
V

dR

dV ~

2R

V '

dF _

F

(1)

(2)

dR 2 R

When V = T 7 n , that is at point p (Figs. 105 and 106), F (the
thrust) is equal to R (the resistance), so we have 1

dF _ _ 1
dR "~ 2 '

thus referring to Pig. 106, 2

c a __ 1

<tt; ~~ "~ 2

or

* = — = ca + ah _ V5m
a b a b

§ 120. The Influence of the Mode of Propulsion (continued). —

Applying the foregoing result to the investigation of Chapter V.,

1 The minus sign merely means that a c is measured in the opposite sense
to a b.

1 In Fig. 106 a curve of the form V F — constant is indicated by a dotted
line. This curve is tangent to the torque curve at the point where horse-
power is maximum.

210

GENERAL CONCLUSIONS §121

we have, when the force of propulsion is constant (§ 61), the

expression —

dll h .

= ,, tany,

dL II

n

where h is the maximum half amplitude of the flight path.
But by the preceding section (§ 119), under actual conditions

1 is \j/ times as great as when the propulsive force is constant,
cILj

therefore under these conditions,

dll . t /«! .
dL =*W n tany '

As in § 61 this becomes

^ = * X V 2 0i tan y,

where @i represents the maximum inclination of the flight path
to the line of mean gliding.

On carrying the correction through, as in §§ 62, 63, we obtain
the final expression

*<!> > 1

as representing the conditions of stability, where <t>, as in § 63,
is given by the expression

4 / H* tan y

I \K + cC P ea $)*

§ 121. Bate of Damping of the Phugoid Oscillation. — In Chapter
V. we dealt with the damping effect of resistance quantitatively,
but only as balanced against an opposed influence ; we will now
discuss the damping of the phugoid oscillation as a progressive
phenomenon, as a measure of the degree of stability. We have
first to examine and define the conditions on which the damping
rate depends, and decide in what form we desire the solution
presented.

The quantitative verification of the theory and equation of

211 p2

§ 121 AERODONETICS

stability depends upon our being able to recognise in the flight of
an aerodone or aerodrome the particular case when for small
oscillations the amplitude does not vary. Failing an actual
photographic record, it is necessary to rely on visual observation;
and although by this means any rapid change of amplitude can
be detected with certainty, and a drop of 30 per cent, in five or six
oscillations is quite appreciable, we do not know, without making
the investigation on which we are about to embark, to what
extent any given error of observation may affect the experimental
estimate of the stability coefficient <t>. It is evident that we
require to correlate the loss of amplitude (from one oscillation to
the next) with the stability coefficient and with the other
known data.

Employing the sine curve approximation and the symbols of
Chapter V., we know that the variations of h depend both upon
the phugoid path form and upon the damping effect of resistance
where the phugoid variation is zero, as where h is maximum and

its value is represented by /<i, -w is proportional to A itself (§ 57) ;

uIj

let us go over this ground once more.

Ignoring for the present the moment of inertia effect, and
taking a point on the flight path when h is maximum, that is,

JJT

-tj as due to the phugoid equation is zero, we may write down
the following direct from the conditions : —

JJT

W -jf = energy lost by aerodone per unit length of

' J flight path. (1)

W y = energy supplied by constant force of pro-
pulsion per unit length of flight path. (2)

— jf y = energy expended against resistance per
n unit length of flight path. (8)

By (2) and (8) energy lost by aerodone per unit length of flight
path,

212

GENEKA.L CONCLUSIONS §121

But H — H n = hi and dH = dh u therefore by (1),

dL ' H

n

an = y h: (4)

which is the differential equation to the rate of damping (in
terms of flight path) at the time when h = h that is, when the
flight path is at a point of maximum amplitude.

Let us for the time being ignore the phugoid oscillation, and
suppose h to be influenced by damping alone, so that the flight
path becomes a curve of the form given by equation (4), h being
substituted for h h that is,

yh
Integrating this expression we have —

lo(J e h = -?- L. (5)

K

n

This gives the form of the resulting curve under the artificial
conditions supposed ; the amplitude damps out according to a
logarithmic law.

Let us now introduce factors representing the influence of
moment of inertia and propulsion. From an examination of the
construction of the equation of stability (comp. § § 62, 63), on the
basis ♦ = 1, it is evident that the proportion of the damping

factor neutralised by moment of inertia will be = ^ so that the

proportion remaining will be 1 — ^, which becomes when we
take account of the propulsion coefficient,

* - i, (6)

and equation (4) becomes

dh _ f ... 1 \ 7»i
dL

= (*-l)>k ">

213

§ 121 AERODONETICS

Integrating as in (5) we have,

y

(-i)'-

h(j c h = jj . (8)

So far the solution is on quite an artificial basis. The oscilla-
tion has been supposed suspended for the time being in order
that we may examine the form of the damping curve ; we now
see that this is logarithmic. A curve in accordance with the
equation is given as an illustration in Fig. 107.

If we could suppose the oscillation to result in a flight path
of castellated form, we could interpret this curve as in Fig. 108 ;

Fio. 107.

this is of course impossible, but it is not a great step to
take the integration of the sine curve as a basis, including the

2

resulting factor - in equation (7) when the integration becomes

_ 2 ?(*-cT) L

%e h = ^—-jj '- , (9)

in which values of h are only comparable for values of L differing
by multiples of one half phase length, i.e., by a multiple of the

quantity -^.

In the examples that follow, L is taken at two successive
maxima (or minima), the two values chosen being L = — L\
L = respectively. The values so obtained give a ratio that

214

GENERAL CONCLUSIONS

§122

holds good, in accordance with the well-known property of the
logarithmic law, over every subsequent phase length.

Owing to the numerical relation that obtains (for phugoids of
small amplitude) between H n and L\ 9 equation (9) is susceptible
of some simplification, thus h\ = 2 it V 2 H n9 hence, where

Fio. 108.

we are dealing with whole periods the equation becomes, for
L = ± U

log e h= ± 4V2y(*-^)

or in common logs,

log h = ± 2-46 X y ( * - * ). (10)

§ 122. Damping of the Phugoid Oscillation. Examples. — The
foregoing investigation may be illustrated by a few examples.
Firstly we will take the case of a simple aerodone presumed to
be without moment of inertia and gliding under the influence of
a constant force of propulsion.

Example I.

By the conditions ^ =

hence

y = *2.

oo, ^ = unity,
1

* - i = ! - ° = l

.\ log h = ± 2-46 X 2 = -492 or T508,

215

§ 122 AERODONETICS

giving, when L is respectively + Lor - L, values,

h = 31046
and h = '8221.

Now when L = ; log h = 0, or h = 1, so that for three
successive phases we have amplitude values,

h = 3-104
h = 1-000
h = -822

and since we know that a logarithmic series of this kind con-
stitutes a geometrical progression we may continue the series of
values by simple proportion thus : —

h = 1035

h = -0884, etc., etc.

In tabulating these series for the different examples given we
shall take L = as the starting point, hence the initial ampli-
tude will be represented as unity.

The units in which h is measured do not matter, the conditions
of the investigation are that the amplitude is small, but the law
probably applies as an approximation even when the amplitude
is considerable. It is evident that since the series is a geo-
metrical progression, any given value may be taken as the
initial amplitude and the succeeding values calculated by simple
proportion.

In the tabulated figures given in respect of each of the
examples that follow, the number of completed phases of the
curve is given against each of the tabulated values of h, beginning
at the instant of launching, as zero, thus,

Phas*.

1

2
3

Am pi it tide.

1-000
•822
•1035
•0334, etc.

210

I

^

\J

©

o

M

%>

§122

AERODONETICS

It will be noted that the amplitude falls to about one-tenth of
its initial value in two oscillations ; in this and the examples
that follow the number of oscillations required to effect this
reduction is taken as a handy criterion of the rapidity of the
damping.

The flight path as here deduced for the supposed case is given
diagrammatical ly in Fig. 109 (a), the initial amplitude being
shown as = J H n . The curves given in Fig. 109 are no more
than rough representations of the form of the flight paths drawn
by hand through the calculated maxima and minima.

§ 123. Damping of the Phugoid Oscillation. Examples (con-
tinued).

Example II

* - ^ =

y = # 2 (as before)
4> = 2

^ = 1 (constant force of propulsion)
1

l-|=-5

or

log h = — (2-46 X '2 X *5)
= — -246 = f-754,
h = -5675,

and series becomes,

Phase.

Amplitude.

1-000

1

•567

2

•3*21

3

•182

4

•103

218

GENERAL CONCLUSIONS

§124

Example III.

y and V as before ; <£ = 1*5.

1 — j^g = -838,

log h = — (2-46 X "2 X -338)
= - 164 = 1836

or h = '685,

and series becomes,

Phaiw.

Ainpliluli'.

1000

1

•685

2

•469

8

•321

4

•220

5

•150

6

•102

The flight paths in the preceding examples (II. and III.) are
depicted in Fig. 109 (b) and (c).

§ 124. Damping of the Phugoid Oscillation. Examples (con-
tinned). — It is evident that, in the examples so far considered,
the rate of damping is such as to render quite obvious the fact
that <l> > unity. We will now take two further cases, in which
<J> is taken at values 1'25 and rill respectively.

Example IV.

y and 4* as before ; <I> = 1*25

1

1 -

= 2

T25

h h = — (2*46 X "2 X -2)
= - 0981 = i'9016
// = 797,

219

§124

AEItODONETlCS

and series becomes

Phase.

Amplitude.

1-000

1

•797

2

•635

8

•506

4

•402

5

*

•320

Limit of

plotting (</) Fig. 109.

10

•102

Example V.

y and ^ as before ; <l> = 1111

1 - iiii = l

log h = — (2-46 X *2 X 1)
= — 0492 = i-9508

•
• •

h

= -8929,

and series becomes

Phase.

Amplitude.

1-000

1

•898

2

•798

3

•718

4

•686

5

•568
Limit of plotting (e) Fig. 109.

20

•108

It is evident from the foregoing examples that when as many
as five or six oscillations are under observation, the error of
determining the condition <I> = 1 is less than 10 per cent,
where the conditions for observation are favourable ; thus in the

220

GENERAL CONCLUSIONS §125

case last considered, in which <!> = I'll], the extent of damping
in six oscillations is sufficient to approximately halve the initial
amplitude, an amount that is quite obvious.

In the present state of knowledge it is evident, therefore, that
an ordinary visual estimate of the rate of damping is sufficient
for practical purposes, either for scale model experiments or for
the verification of the equation of stability.

§ 125. Damping of the Phugoid Oscillation. Examples (con-
tinued). — We will now consider a case of an aerodrome propelled
by a motor under maximum h.-p. conditions.

Example VI.

y = '2 as before ; <I> taken = unity ; ^ from the conditions

= 1-5

1-5 — 1 = 5

log h = — (2-46 X '2 X 5),

which is the same as in Example II. (Fig. 109 (/>)), hence an
aerodrome whose flight path is on the point of instability
without propulsion, enjoys a stability equivalent to 4> = 2 when
propelled by a motor working at maximum h.-p.

Example VII.

Data as before, but <I> = 2

1-5 -1=1

log h = - (246 X '2),

which is the same as in Example I. (Fig. 109 (a)). Here the
propulsion effect alone is just sufficient to neutralise the influ-
ence of moment of inertia, and the damping rate is the same as
that for an aerodone without moment of inertia gliding under
uniform propulsive force.

We will now take a few cases in which the flight efficiency is
bad, i.e., in which the gliding angle is considerably greater than
in the foregoing examples.

221

§125

/

\ERODONETICS

Example VIII.

y = -388 ; * = 1 ; 4> = oo.

00

log h = - (2-46 X -838)
= - -82 = 1-18

h = -151,

and series becomes

Phase.

1

2

Amplitude.

1-000
•151
•023

and flight path (Fig. 109 (/)), becomes nearly dead-beat.

Example IX.

As before but 4> = 5

1 - I = -8
o

lot/ h = - (2-46 X '333 X "8)
= - -656 = 1-344

k = -2208,

and series becomes

Phase.

Amplitude.

l

2

1-000
•221
•048

and again flight path (Fig. 109 (</)), is nearly dead-beat.

Finally, let us take a case of extreme inefficiency such as one

222

GENERAL CONCLUSIONS §126

of the models of M. Pline (compare § 67); we will assume
y = # 5 and moment of inertia to be absent. 1

Example X.

y = -5, ^ = 1, 4> = oo ,

i-i=i.

00

log h = — (2-46 X -5)

= - 123 = 2-77
h = -05888,
and series becomes,

Phase.

Amplitude.

l

2

1-000
•059
•008

In this flight path, depicted in Fig. 109 (h), the phugoid
oscillation appears to die away almost instantly, as actually
happens with models of very low efficiency.

§ 126. The Law of Corresponding Speed. — It is not the least of
the practical consequences of the Phugoid Theory that the law
of corresponding speeds, first enunciated by M. Beech for ships,
and independently discovered by Mr. W. Froude, holds good also
for aeronautical machines. This law, which is the basis of all
model experiments in naval architecture, and with which the name
of Froude is inseparably associated, was stated by him as follows :
— " If the ship be I) times the dimension of the model, and if at
the speeds V\ \\ \\ . . . . the measured resistances of the
model are Hi, H* H* . . . . then for speeds J\ VT>, V 2 V D,

1 When the value of y is high, the flight path is not as a rule greatly
affected by moment of inertia. Comparo Examples VIII. and IX. ; <t> is
ordinarily of considerable magnitude when y is great.

223

§ 126 AEEODONETICS

V s V D, .... of the ship, the resistances will he 7> 8 /?i, Z> 3 /i2,
lPliz .... To the speeds of the model and ship thus related
it is convenient to apply the term corresponding speech."

The ahove law may be otherwise expressed by saying that
when the speed of one model or vessel is to that of another
similar model or vessel as the square roots of their linear
measurements, their resistances are as their respective masses,
or as the cube of their linear measurements. 1

The above law of corresponding speed rests upon the negligi-
bility of viscosity as a physical quantity on which the resistance
depends. This is not the same as the neglect of resistance due
to skin friction, but rather the assumption that such resistance
follows implicitly the V 2 law ; thus, according to the law of
corresponding speed as enunciated, skin frictional resistance
must obey the law of total resistance, i.e., it will be proportional
to I) 8 ; for the exposed surface varies as D 2 , hence the resistance

per unit surface varies as D, and since the velocity varies as VT)
the skin-frictional resistance varies as V 2 .

We are acquainted with the fact that resistances due to skin-
friction and discontinuity, although definitely depending upon
the fluid possessing viscosity, are substantially independent of
the value of viscosity itself. 2 We know from the work of Allen
that this fact is not strictly true, and further we know that the
error in the case of skin friction is greater than for other forms
of resistance in the production of which viscosity is involved.
This difficulty is usually dealt with by the method proposed
by Froude, as a correction to the experimental curve ; this
method consists in the separate computation of the skin-
frictional resistances of the model and the vessel, and the
deduction from the experimental curve of resistance of the
difference so ascertained.

1 For experiments with model vessels in a fluid of stated density, the mass
must vaiy strictly as the cube of the linear measurement, otherwise the
displacement would not be proportional.

8 * 4 Aerial Flight," Vol. I., AmHlynumics, Chap. II.

224

GENERAL CONCLUSIONS §128

§ 127. Theory of Corresponding Speed. — In the case of a body
that is totally submerged, and whose density is equal to the
surrounding fluid (or where gravity is supposed inoperative), the
resistance of a body of stated geometrical form is a function of
five physical quantities, namely, linear size of the body, velocity
of motion, density of the fluid, viscosity, and elasticity.

When we have to deal with the action of gravity, as when the
body is in the region of a free boundary service (as in the case of
a ship), or where the body is of greater density than the fluid (as
in the case of an aerodrome), we have also to take account of the
gravity as an acceleration, as one of the physical quantities of
which the resistance is a function.

On the basis of ignoring elasticity (as being only a sensible
influence at high velocities), the author has shown (Vol. I., § 89)
that so long as gravity is without effect, the law of corresponding
speed is given by a certain equation due to Osborne Beynolds.
Further, in the section cited, a law is deduced (probably of no
great practical utility), which takes cognisance of both viscosity
and gravity as factors in the problem.

If now we ignore both viscosity and elasticity (a course that
when gravity is inoperative gives us the V 2 law) there remain
four physical quantities of which the resistance and the form of
the fluid motion are functions, i.e., linear size, velocity, density,
and acceleration.

§ 128. The Theory of Corresponding Speed (continued). — The
form of the fluid motion generated by a body under the present
hypothesis depends upon the geometrical form of the body, and
the quantities p I and V y and the acceleration of gravity which
we will represent by/. It is customary to employ the symbol g
for gravity, but this suggests gravity as a constant, or approxi-
mately so, whereas in the generalisation that follows we take
gravity as a variable, just as if we required to apply the resulting
law to widely different altitudes or to experiments conducted in
planets other than our own.

a. 225 q

§ 128 AERODONETICS

We will proceed to demonstrate the law of corresponding
speeds by the method oj dimensions. 1 Now since we have to
deal with models geometrically similar, the form of the fluid
motion is a function of p, Z, / and V, and of these quantities
only. And the condition of corresponding speed is that the fluid
motion is homomorphous in different cases ; that is, the geo-
metrical form of the fluid motion is constant. We therefore

write —

l p fV r P* = constant,

L'' L r M*
or, h p 7p ; r j r L3- = constant.

Hence, p + q + r — 3 * =

2 q + r =
* =
p + q — 2 <? =
or, p = q

and r = — 2 q

l« f« V-*' = constant,

I f
or =y 2 = constanc. (1)

This is the law of corresponding speed, including possible
changes in the acceleration of gravity, and would be applicable
even if we suppose model experiments to be conducted on the
planet Mars for vessels or aerodromes to be built on the earth.
We are happily under no such obligation, so we substitute for
the variable /the constant </, and the law becomes —

I
V

which is the law of corresponding speed of Beech and Fr

§ 129. The Law of Corresponding Speed in its relation to the
Phugoid Theory. — Although the abstract proof of the law of
corresponding speed as generally applicable, under the conditions

1 Compare " Aerial Flight,*' Vol. L, Acrn*hjnamics> Chap. II.

226

a = constant, or V a V^l, (2)

GENERAL CONCLUSIONS §129

of hypothesis, is in itself sufficient, it is of interest to discuss its
application to an aerodone or aerodrome in the light of the pre-
ceding flight path and stability investigations, more especially in
relation to the results of Chapters II., III., and V.

It is at once evident that, since the phugoid chart represents
the possible flight paths of an aerodone to a scale that varies as
the square of the velocity, the value of II n , the determining
factor of the scale of the chart, will always be proportional to the
linear size of the aerodone as given by the law of corresponding
speed. This is in perfect harmony with the hypothesis, for it is
evident that if the size of the aerodone did not vary directly as
the scale of the flight path, the motion of the air could not be
exactly homomorphous in different cases, and the conditions of
corresponding speed could not be complied with. Thus the law
of corresponding speed is fully consistent with the results of
Chapters II. and III.

In the theory of stability of Chapter V., the equation,

<j> _ 4 I H n * tan y

1 ( K + cC ptap)

contains quantities which vary when the size of the aerodone

varies, these are : —

H n * I <x L 3

I a L 2 X mass (assuming the proportionate distribution

of the latter).
K a I?

A a U
But for similar aerodones the mass is proportional to L 3 , for
the volumes part for part are as U, and the densities part for
part are constant ; l hence I oc L fi . Therefore for similar aero-
dones at corresponding speeds —

<I> a i — which is constant,

1 The law of similarity must in effect apply to density as well as to
geometrical form.

227 Q 2

§ 129 AERODONETICS

that is to say the law of corresponding speeds applies also in
respect of the stability equation, and if any given model be
proved stable to some stated degree, any geometrically 1 similar
model at its corresponding speed will be stable in like degree.

§ 130. The Theory of Corresponding Speed (continued). — An exami-
nation of the manner in which the law of corresponding speed is
dependent upon gravity, both in the case of the steamship or
sailing vessel, and in the case of the aerodone is instructive. In
the latter case we have seen how the law, as established from
dimensional theory, results in a constant relation between the
linear size of the aerodone or aerodrome and the linear size of
the flight path, that is the scale of the phugoid chart. If we
suppose, as in § 128 (1), that gravity be variable, then the scale
of the chart will vary inversely as gravity, for II n will vary in-
versely as gravity for a given value of V n , and H n gives the scale
value to the chart. This result is reflected in the equation in the
fact that the product of the linear size of the aerodone /, and the
variable force of gravity/, is constant for given value of V H .

In naval architecture the factor analogous to the flight path,
as controlled by the acceleration of gravity, is the wave motion
impressed on the water by the motion of the vessel. If it were
not for wave motion and wave-making resistance the law of
corresponding speed would be of no value to the marine
engineer; the law of corresponding speed ensures that the
comparison between the size and shape of the wave system to
which the vessel gives rise and the wave system developed by the
scale model holds good, so that the motion of the water is of
exactly the same geometrical form in both cases.

Thus the relationship between the size of the waves (wave
length) and their velocity is analogous to that between the
size of the phugoid curve (phase length) and the velocity of
flight.

1 Gcvmdricalhj used in its widest sense. Compare preceding footnote.

228

GENERAL CONCLUSIONS §132

§ 131. Scale Model Experiments. — In practical aerodromics, i.e.,
in the design and construction of flying machines, it is difficult
to overrate the importance of the theory of corresponding speed
and its application in experiments with scale models. In naval
architecture this method of experiment has done enormous
service in the development of the modern sea-going vessel,
enabling an exact account to be made out in advance of the
resistance and power expenditure for any particular form of hull
at stated speed. In aerial navigation we may expect the method
of the scale model to be of still greater service, for here it is not
only a matter of estimating the power required, but also of ascer-
taining definitely that proper provision has been made for the
necessary stability, both longitudinal and otherwise, and in the
investigation of forms and types of which theory may not be
able to render an exact account.

In other words, the information furnished by the law of
corresponding speed and model experiment, in the case of the
marine vessel, is of service as securing a more precise knowledge
of the performance of a vessel than would otherwise be obtain-
able, and as enabling the naval architect to secure a given result
at least cost. In the construction of flying machines we may
look to the scale model to do all that of which it has hitherto
shown itself capable in connection with naval architecture, and in
addition to render the experimental and early stages of develop-
ment free from the danger that at present exists, by placing in the
hands of the aeronaut or designer an absolutely reliable means
of testing his work without exposing himself to the smallest
danger, and otherwise without risk of disaster. A few pence
spent in small scale experiment will give as much information
as an equal number of pounds expended in a full scale machine.

§ 132. Scale Model Experiments (continued). — Owing to the fact
that density is one of the physical quantities involved in the
dimensional equation, the density of the component parts of a
model or machine is not a matter of indifference. As a factor in

229

§ 132 AERODONETICS

the equation of corresponding speed it was shown that density
vanishes from the variables, but this does not mean that it ceases
to be a factor of importance.

It must be clearly understood that the method of dimensions in
its present application, and in other cases of the same kind,
depends definitely upon the condition of similarity, and the
disappearance of density from the equation means that if a pro-
portionate change of density takes place throughout the system
such change is without effect. Thus in the case of model vessels
the difference of density between sea water and fresh water does
not matter so long as the density of the model is varied in like
ratio ; in this case the need for this variation is obvious, for
otherwise the extent of immersion of the model would be affected.
Beyond this, if the model experiments were extended to deal with
rolling and pitching, the distribution of the density in the different
parts of the model would require to simulate the actual differences
in the vessel. When the experiments are confined to resistance
this condition only needs to be complied with to the extent of
keeping the centre of gravity in its correct position.

In the case of an aerodone or aerodrome the questions of
rolling and pitching, or the equivalent, are actually a most
important part of the investigation, so that evidently the distribu-
tion of the mass becomes a matter of vital importance. In order
to adhere strictly to the law of similarity, we should require to
construct the model exactly proportional to the full scale aero-
drome in every detail, using like material for like parts through-
out. It is evident, however, that the conditions do not necessitate
any such extreme measures. We need to simulate such a model
in all essential respects, and we must appeal to our former in-
vestigations to ascertain what points must be considered essential.

Referring to § 129 in which the law of corresponding speed
was reviewed in its relation to the stability equation, we see first
and foremost that the total mass must vary as the cube of the
linear dimension ; this is analogous to the density condition in
the case of a boat ; it carries with it also the same condition as

230

GENERAL CONCLUSIONS §133

to the position of the centre of gravity being similarly situated.
In the second place we have the condition that the moment of
inertia shall be equal to that of the exact scale model ; that is to
say, proportional to the fifth power of the linear dimension.
This evidently applies not only to the moment of inertia about
the transverse axis, as involved in the equation of longitudinal
stability, but also about the other co-ordinate axes ; in all cases
the moment of inertia is reckoned about an axis passing through
the centre of gravity.

§ 133. Scale Model Experiments (continued). — It is evident from
the foregoing discussion that before a model can be constructed
of any given machine or design, not only must the weight and
position of the centre of gravity be known, but also the moment of
inertia must be calculated or measured about each of the three
co-ordinate axes. In practice, this means that in the originating
of a flying machine, the moments of inertia of the component
parts should, in the first instance, be separately computed from
the initial design, and the moments of inertia of position of
these separate components should then be calculated from the
location of their centres of gravity relatively to the centre of
gravity of the whole machine ; the total moments of inertia will
then be the sum of the moments of inertia of the component
parts as separately computed, and of their moments of inertia as
due to their position. 1

Having thus ascertained the moment of inertia of the intended
design, and having settled the relative scale of the model it
is proposed to employ, the moment of inertia of the latter, about
each of its three co-ordinate axes, should be calculated on the
basis of I oc L 5 , and when the model is made it should be care-
fully adjusted to the calculated values, 2 at the same time the total
weight being made rigorously proportional to L 8 .

1 By a well-known theorem in dynamics. The procedure given is advan-
tageous as allowing for a re-arrangement or adjustment of the design with
the least possible re -calculation.

* For the method of measuring moment of inertia, see § 1 72, also Appendix YL

231

§ 133 AERODONETICS

. The functional parts of the model must, of course, be arranged
strictly in accordance with the design, that is to say, there must
be complete similarity of position in the arrangement of the
aerofoil, tail-plane, fins, or directive organs, etc., between the
model and the full-sized machine that it represents.

It is probable, in fact it is almost certain, that the conditions
as to skin friction that apply in the case of model experiments
with ships apply also with regard to aerial experiment, that is to
say, the coefficient of skin friction diminishes to some extent as
the size increases ; in consequence of this it may be found neces-
sary to make a correction on the score of skin friction, deducting
on the basis of the coefficient proper to the model and adding on
the basis of the coefficient proper to the full-sized machine, after
the manner proposed by Froude and employed in experiments
with model vessels. The two cases are not, however, altogether
parallel ; in the aerodynamical problem we have many considera-
tions foreign to the conditions that obtain in the model ship
experiments.

§ 134. Scale Model Experiments. Allowances. — The influence of
the skin-frictional error is more far-reaching in the case of
an aerodrome than in the case of a ship, and consequently
the corrections required will be more extensive.

The direct correction for the relative reduction of resistance on
account of size is the first, one may say, the primaiy, correction.
This depends upon the difference in the coefficient of skin friction
in the two cases, i.e., the full-sized aerodrome and the small scale
model. It, however, also depends upon other factors of which our
knowledge, in the absence of an actual design, is limited ; thus in
the case of a model we know that we can make an aerofoil that
is structurally sufficient without stays, struts, or guys of any
kind, but in a full scale machine it is credibly established that
this is not possible. Now all these guys, struts, etc., offer resist-
ance which to all intents and purposes may be regarded as an
augmentation of skin friction since this added resistance varies

282

GENERAL CONCLUSIONS §135

as the square of the velocity, and farther it increases in some
ratio with the size of the aerofoil. Consequently, although the
larger model or machine is less resisted owing to its lower
coefficient, it experiences a greater resistance owing to these
structural necessities ; it will he only possible to ascertain by
careful experiments and calculations whether and to what extent
any allowance will be required.

As a consequence of any primary correction that may have to
be made, others of a secondary character are required ; thus if
the resistance of the model be proportionately greater than that
of the aerodrome, its y value will be greater, and consequently
it will possess a greater degree of stability, and allowance will
require to be made for this, otherwise the small scale model may
have a sufficient margin of stability, but the full scale machine
may yet be deficient.

Given the extent of the primary correction, i.e., the relative
values of the gliding angle y, the secondary correction could be
provided by deliberately making the moment of inertia an appro-
priate amount greater (or less as the case may be) than calculated
on the L 5 basis, in accordance with the equation of stability, so
that the value of 4> shall remain constant. This correction may
either be made in the model as compared with the design, or the
correction may be made in the design to comply with the
performance of the model.

§ 135. Scale Model Experiments. Summary. — The fundamental
reason that corrections are required in accordance with the fore-
going section, is that the influence of viscosity was ignored in the
dimensional equation. This is definitely a weak point in the
theory of corresponding speed, at least in the form enunciated by
Beech and Froude. The author has shown (Vol. I., § 39) that
in order that Froude's law should hold good without correction
the kinematic viscosity of the fluid must be made to vary as
the § power of the linear dimension. The inconvenience of
conducting flight experiments in specially prepared fluids is,

288

§ 135 AERODONETICS

however, too great to allow this result to possess much practical
interest ; in spite of this it is highly probable that, when once the
physical relations that exist between one fluid and another are
properly recognised, many of the aerodynamic data required
for the theory of flight will be more accurately determined
from experiments in water than those conducted in air. It
is worthy of remark in this connection that unless some great
discovery as to the physical properties of fluids remains to be
made, determinations of resistances and pressure reactions made
in water, at velocities small in comparison to the velocity of
sound, should apply to air in the ratio of the densities of the two
fluids, provided the product of velocity and linear dimension is, in
the aquarium experiment, one-fourteenth of that represented in the
actual conditions ; fourteen being the ratio of the kinematic
viscosities of the two fluids. 1 On the question of the resistance
of the structural parts of the aerofoil or wing member, the
struts, ties, etc., the magnitude of this can be determined
approximately from independent data, the various elements of
construction being separately assessed from tabulated experi-
mental determinations.

It would seem desirable, at any rate during the earlier
stages of practical aviation, to precede the construction of a full
scale flying machine by two sets of small scale models, the first
being on a quite small scale, perhaps ^ full size, followed by
a second trial in which the model could be conveniently made
about J full size, and furnished with all the constructional
detail of importance in miniature. On this basis for a machine
of 1,200 lbs. total weight, the first model would weigh about
4| oz., and could be constructed principally of mica, and could
be flown indoors, the larger model would weigh some 5J lbs., it
could be fitted with twisted rubber propulsion, and could be
tested out of doors in any reasonably calm weather.

§ 136. Departures from Elementary Type. — The type of aero-
done or aerodrome chosen for the purpose of investigation and

1 Compare "Aerial Flight," Vol. I., Aerodynamics, §§ 25, 3G et seq., and 137.

234

GENERAL CONCLUSIONS §136

exclusively dealt with in Chapters II., III., and IV., etc., is one
of the most elementary simplicity. There are certain forms of
departure from type that are more apparent than real, and which
evidently in no way affect the results of our investigation ; for
example, the aerofoil may consist of a number of superposed
elements like the double deck machine of Lilienthal or the
" Venetian blind " construction of Phillips, without invalidating
the theory in an : important respect. There are other modifica-
tions that probably require to be taken into account, as involving
some departure from the conditions, thus if the aerodone is
acentric, that is to say, if the centres of mass and resistance do
not approximately coincide we have to deal with new conditions ;
for example, if the weight be hung some distance below the aero-
foil, it is evident that since the velocities of the mass and the
aerofoil on a curvilinear flight path are not equal, the Phugoid
Theory will cease to apply with exactitude. Again, in the case of
a machine that employs more than one aerofoil in tandem for the
purpose of support, instead of a single aerofoil and a directive
organ, it is doubtful to what extent the theory is applicable
without some modification.

In the tandem arrangement it is evident that if the aerodone
be constrained to follow a phugoid flight path, other than that of
uniform gliding, it will require a variable applied torque, about a
transverse axis, always acting in the direction of its rotation, thus
when passing from left to right and troughing the applied torque
must be counter-clock. If the distance from the leading aerofoil
to the trailing aerofoil is small in comparison with the radius of
curvature of the flight path this effect becomes negligible, just as
the similar effect which exists also in the elementary type also
becomes negligible when the tail length is small compared to the
radius of curvature, one of the conditions of the initial hypothesis.
It is evident therefore that this effect may be legitimately
dismissed.

In a type of flying machine at present in some vogue, 1 the

1 The WrigLt brothers are reported as employing a machine of this kind.

285

§ 136 AERODONETICS

longitudinal stability is maintained by a horizontal rudder in
advance of the main aerofoil. As a hand-operated device for
regulating the flight of the machine the theory of such an
arrangement is simple enough, but it would appear that it may
also be used to secure automatic stability. 1 The theory under
these circumstances is obscure. If the load per foot super on
such a " leading plane " be greater than that on the aerofoil, the
present theory probably covers the arrangement in question.
We may regard the machine as one with a hypertrophied tail, on
which the greater part of the weight is carried. If this explana-
tion is found inadmissible, as must be the case if the pressure on
the " leading plane" is less than that on the aerofoil, then the
behaviour of this type of machine is a matter that awaits
investigation.

The author believes that in general the tandem form is amen-
able to the same laws as the elementary type, and it would
probably not be a difficult step to generalise the theory to include
both types. It would appear possible, on the principle dealt with
under the heading "Fin Resolution/' § 99, that any such tandem
combination might perhaps be resolved into an equivalent aerofoil
and tail plane, and it would plausibly appear that we may have
here the solution to the problem of obtaining an effective tail of
great length without being penalised by excessive moment of
inertia. This is one of the points left for future investigation.

§ 137. The Theory of Lateral and Rotative Stability.— The theory
of lateral and rotative stability has not up to the present been
subjected to so rigid an experimental examination and verifica-
tion as the Phugoid Theory and equation of longitudinal stability.
The general qualitative results unquestionably hold good; for
example, the lateral oscillation as depending to some extent on
the moment of inertia about the axis of flight, the damping
effect and lateral stability obtainable by the employment of

1 The flying models made by Clarke, of Kingston-on-Thames, are con-
structed with a " leading plane."

236

GENERAL CONCLUSIONS §138

vertical fins, the importance of obtaining approximately neutral
equilibrium (in respect of lateral motion) about a vertical axis,
etc., these are all facts that may be demonstrated by simple
experiment with mica models such as employed by the author
and described in the present work.

The theory of rotative stability was actually developed as the
result of an investigation to account for the anomalous behaviour
of a model, which exhibited a form of instability that at the time
was new to experience, at least so far as the author was concerned.
It is of some interest to state that the aerofoil of this model had
been designed with a view to testing the value of a tapered form
whose ordinates were proportional to the curve of grading —
approximately parabolic (comp. Vol. L, § 120). Thus the
centre of pressure of the aerofoil must have been nearly neutral
in respect of aspect and the aerodone had to rely almost entirely
on its fins for its restoring couple. Now at the time of this
experiment it was the author's practice to arrange the centre of
pressure of the fin area well abaft the centre of gravity with a
view to making sure that the aerodone should be directionally
stable. The result of these two facts in combination was that
the whole design of the aerodone seriously infringed the conditions
of rotative stability, both as exhibited by experiment, and — as
was subsequently ascertained — by computation from the equation.

Similar confirmation was obtained from another model in
which the theoretical conditions were inadvertently infringed,
but here the aerofoil was not of " neutral " form and the
necessary data were not obtainable, the relation of the righting
moment due to the aerofoil itself being an unknown quantity. 1

§ 138. Application of the Theory of Lateral and Rotative
Stability. — It has already been stated that the vertical fin is not
without disadvantages ; it lends itself to exact calculation better

1 The plan-form of the aerofoil in this case was an ellipse with approxi-
mately segmental grading. Stability was restored by increasing the area
of the leading fin.

237

§ 138 AERODONETICS

than other arrangements by which lateral stability can be secured,
but it offers an unnecessary extent of surface to skin friction.
The importance of minimising resistance losses of every kind in
the construction of a flying machine is bo great that the very
idea of resistance being incurred for the sake of calculation is
inadmissible, except at most for initial experiments. It is con-
sequently evident that the method of calculation or computa-
tion must be rendered more comprehensive to deal with such
forms as from economic considerations may be found best.

Now in the case of the gliding bird there exists nothing in the
way of a fin of any kind, although the streamline form of the
bird's body must be regarded as possessing some considerable
directive power, and thereby contributing to the rotative stability.
The lateral stability proper is in all probability due, so far as
it is independent of nervous control, to the considerations dis-
cussed in §§ 85 — 88 and to the upturned extremities of the
primary feathers, a feature common to nearly all the larger
birds when gliding or soaring. 1 This turning upward of the
extremities is due to the natural flexure of the tips of the
primary feathers owing to the pressure reaction, but it is none
the less effective on this account.

In a flying machine it is evident that the provision for lateral
stability will require to be proportionally more extensive than
in the case of a bird, owing to the absence of any delicate
mechanism of adaptation. It is, moreover, obviously better to
give direct stability by the provision of adequate areas to the
functional parts than to attempt to minimise these by any system
of complex automatic adjustment.

The problem then resolves itself into determining the equiva-
lent value in vertical fin area of wing or aerofoil terminations
of known area and angle. The problem as thus presented is

1 This feature may be observed in most gliding or soaring birds , but not
always. In the swift and other members of the swallow family the wings
have a very marked downward trend; even here the extreme tips of the
wings are somotimes visibly flexed iu the manner stated.

238

GENEttAL CONCLUSIONS §139

one for which it would be impossible, from theoretical considera-
tions alone, to give more than an approximate solution.

As a practical solution, the first step is to investigate the
critical point of rotative instability by means of a model designed
to comply strictly with the requirements of theory, i.e., with an
aerofoil as nearly as possible neutral in aspect, and with fins of
a form for which the aerodynamic data are accurately known.
Then having verified the equation (or determined its defect),
the proportions should be ascertained experimentally of the up-
turned extremities 1 that are equivalent to some definite fin plan.

It is not anticipated that the fin system can be entirely dis-
pensed with by the employment of inclined terminals, but rather
that it shall be reduced to the minimum necessary to ensure that
the aerodone shall be directionally neutral (comp. §§ 96 et seq.,
and § 101).

§ 139. Conclusions. — Before passing on to the subject of the
following chapter, which constitutes to some extent an indepen-
dent section of the work, the conclusions that we have been able
to reach may be enumerated, and a few words added as to the
history of the present theory and of its experimental confirmation.

In the foregoing chapters it has been shown : —

(a) That the longitudinal stability of an aerodone is deter-
mined by purely dynamical considerations, without any adjustable
organs or mechanism whatever.

(b) That the normal flight path may be regarded as a straight
line trajectory in which the velocity of flight is just sufficient to
produce the necessary lifting reaction of the aerofoil, and the
gradient is at such an angle as may be required to supply by
gravity the power required for propulsion, or in the case of an
aerodrome propelled by motive power to supply (or absorb) the
difference between the power supplied by propulsion and that
expended in overcoming resistance.

(c) That the flight path otherwise consists of a series of

1 Or such other feature as may be provided to give lateral stability.

289

§ 139 AER0D0NET1CS

undulations, whose amplitude is either constant or increasing or
diminishing in accordance with certain equations which may be
regarded as established beyond question as a first approximation.

(d) That the stability of the flight path depends upon the
amplitude being either constant or diminishing, and the condition
of such stability may be determined from the present theory in
combination with known aerodynamic data.

(e) That in face of an isolated gust of wind the stability
depends upon the relation of the velocity of flight or natural
velocity to the velocity of the gust of wind ; the latter in terms of
the former must not exceed a certain critical value which may
be deduced approximately from theoretical considerations alone.

(f) That in face of a synchronous periodic aerial disturbance
the continued stability of an aerodone or aerodrome depends,
under given conditions, upon the damping factor, a quantity
that is calculable from the known data and which follows a
logarithmic law.

(g) That lateral stability is intimately associated with the
maintenance of direction and with a kind of stability termed by
the author " rotative stability," and that the necessary conditions
to ensure lateral, directional, and rotative stability can be satis-
factorily met by suitably proportioning the various functional
parts, without adjustable organs or equilibrium mechanism of
any kind.

(h) That the law of corresponding speed enunciated indepen-
dently by Beech and Froude in connection with model ships
holds good mutatis mutandis for aerodones and flying machines.

Beyond the above the detail results reached are almost too
numerous for recapitulation.

§ 140. Historical Note. — The series of investigations by which
the foregoing results have been reached, published for the first
time in the present volume, has been carried out by the author at
odd times during the last fifteen years. In the first instance,
about the years 1892-3, the general qualitative theory of automatic

240

GENERAL CONCLUSIONS §140

equilibrium, both longitudinal and lateral, was developed.
At that date the author had no knowledge of the work of
Mouillard or Penaud ; the account of these experiments has
been added subsequently. The experiments of 1894 were
devised with the unique object of putting to the test the theory
in its then rudimentary form, and constituted in every respect a
complete demonstration.

The next step of importance was made in 1896-7, when the
quantitative solution to the flight path was found in the investi-
gation of Chapter II. and the plotting of the phugoid equation,
which was finally effected in October, 1897, in the form pre-
sented in Fig. 42, and which appears as a frontispiece to the
present volume. The special cases had been solved about a year
previously, and the result is to some extent embodied in a portion
•of the descriptive matter of Patent 3,608 of 1897, given in
Appendix II.

In 1905 * the author succeeded, by the employment of mica,
in constructing very light aerodones of low velocity, and thereby
effected the verification of the flight path curves of the Phugoid
Theory, both as to the general form of the curves and as to the
phase length and time relationship. It was in connection with
these experiments that the influences of resistance and of moment
of inertia were discovered. The damping effect of the former
was first recognised, and a preliminary investigation as to the
rate of damping was made ; this and other observations resulted
in the further discovery of the moment of inertia effect. 2 Both

1 During the period intervening between 1898 and 1905 the author waa
too closely occupied in other ways to give much time to the present subject.

2 The manner in which the importance of moment of inertia was eventually
brought home to the author is perhaps worth recording. In endeavouring
to construct a small low velocity indoor model with indiarubber propulsion it
became evident that a further investigation was necessary. The author was
already aware that a serious discrepancy existed in the damping rate calcu-
lation, but for the time being was unable to locate the fault. The miniature
aerodromes, instead of damping, deliberately increased the amplitude of their
flight path, and in watching their behaviour in flight it became evident that
the cause was a kind of fly-wheel action ; in other words, it was vitibly a
moment of inertia effect.

A. 241 B

§ 140 AER0D0NETICS

these influences were then subjected to rigorous theoretical
investigation, with the results recorded. The investigations
relating to lateral and directional stability followed shortly
after, and that on rotative stability was undertaken owing to
this peculiar form of instability having been accidentally en-
countered when making experiments with a quite different end
in view. With the exception of a few embellishments and
confirmatory experiments that have been added subsequently,
the whole of the investigations were completed before the end
of 1905.

242

CHAPTER IX

SOARING

§ 141. Introductory. — The phenomenon of " soaring " is not
witnessed with sufficient regularity or frequency in the British
Isles for it to impress itself on the ordinary observer as a
distinct mode of flight. It is not at all unusual in fact to find
the term entirely misapplied, and in view of the dictionary
definition it is, perhaps, necessary to admit two distinct usages
of the word, the one meaning being simply to fly to a height,
quite regardless of the mode of flight employed, and the other
being that commonly accepted and used in scientific works,
the equivalent of the French term vol a voile; it is in the
latter signification that the word is now employed.

In the active flight of birds, that is in flight such as that
of the pigeon, where the wings are energetically employed (the
vol mine of the French), we are rightly accustomed to regard the
work performed in the flapping of the wings as the source of
the power expended in flight, and it is a matter of ordinary
observation that under normal conditions, when the wing
flapping ceases, the flight path takes on the whole a downward
trend ; under these passive conditions we know that the energy
that is still being expended in the continued flight is derived
from the descent of the body under the force of gravity, the
angle of free descent being indicated by the symbol y throughout
the present work. The conditions under these circumstances
are analogous to those of a cycle or motor vehicle when " free-
wheeling" or "coasting"; the flight path must on the whole
be at a sufficient downward inclination to supply the enerpy
necessary for propulsion; the analogy also comprehends the

243 r 2

§ 141 AERODONETICS

fact that in either case undulations may exist in the flight path,
the latter even at times rising so that work is done against
gravity, but this work is supplied from the kinetic energy of
the body in motion which constitutes an energy reservoir, and
any depletion has to be made good by a compensating descent
at a greater angle than the natural gliding angle (y), if the
flight is to continue. 1

The above facts are, as has been stated, a matter of common
observation, and in still air there is no exception possible; it has,
however, long been noticed that in windy weather the pheno-
menon of flight appears in an entirely different aspect, birds
may be seen to glide for an indefinite period without any descent
whatever, and without so much as a flap of the wings, or; motion
of any kind that could be of any material value for the purpose
of propulsion. This is the type or mode of flight known as
soaring, and it manifests itself under a considerable diversity of
circumstances ; the one important fact is that in all its varieties
the presence of wind is essential.

The idea of sustained flight without any visible source of
power is so foreign to the instincts engendered by modern
training, that it has been the common experience of nearly
all writers on the subject to have had the facts of observation
received with doubt and incredulity; this scepticism, as con-
stituting a barrier to the serious study of the subject, has frequently
been made the subject of comment, and it is thus, perhaps,
that at one time the problem was allowed to remain in the hands
of persons ignorant even of the principles of relative motion, and
those who, having no mechanical instinct to enable them to grasp
the difficulties of the problem, could glibly explain the whole
subject by a supposed analogy to the ordinary string-kite 2
(cerf volknt) !

1 The essential is that the velocity at the beginning and end of the
observed flight should be the same.

8 The term •' string-kits " is employed in the present work in the same
sense that "jmper kite" is frequently used, in order to avoid possible
confusion with the bird commonly so called.

244

SOARING § 142

In order that there should be no misapprehension as to the
facts, the author has prefaced the present account of the theory of
soaring flight by a brief description of the phenomenon as observed
by him in these latitudes, including certain elementary deductions,
and by quotations from the writings of authorities of unquestion-
able standing whose observations cover a wider field.

§ 142. Author's Observations on Soaring Flight. — Soaring flight,
as defined by the preceding section, is distinguished from other
modes of flight by two facts that are easy to recognise by atten-
tive observation ; firstly, that the bird when soaring does not
sustain or propel itself by the performance of muscular work,
as it does in active flight ; secondly, that it does not continuously,
or on the whole, lose altitude, as is the case in the gliding mode.
Thus in appearance the soaring bird is a gliding bird that in
some mysterious way glides continuously but does not come to
earth.

The common swift (Hirundo apus) may, in a comparatively
light wind, be seen to soar for considerable periods. The spectacle
is not nearly so impressive as is the case with the larger birds, but
as the swift is by far the most common example of a soaring bird
to be witnessed inland in Great Britain, it is an example that we
cannot afford to ignore. The author has on several occasions
observed swifts to soar definitely without a wing flap for periods
of 15 to 30 seconds, and ignoring a kind of spasmodic flutter
that seems to take place at intervals of some 20 or 30 seconds,
they may be seen to soar for periods of many minutes' duration.
The " flutter " referred to is a very peculiar feature in the flight
of the swift, it seems to be a kind of spasm that does not occupy
more than a small fraction of a second ; it leaves, however, a
loophole for the sceptic, and prevents us regarding the evidence
in this case as absolutely conclusive ; there is a possibility that
these spasmodic efforts contribute more considerably to the pro-
pulsion than in the author's opinion is the case.

More frequently than not the site chosen by the swift for

245

§ 142 AERODONETICS

soaring is the immediate vicinity of some terrestrial obstacle
by which the motion of the wind is disturbed ; in this case it
probable derives its support from the upcurrents developed, or
otherwise from the variations in the wind velocity, in a manner
subsequently to be discussed. On many occasions, on the other
hand, the author has observed swifts soaring at greater altitudes,
and apparently in no way associated with any obstacle or surface
irregularity, although this is a fact of which it is difficult to make
certain. Observations on the swift alone are not entirely conclusive
as evidence of the fact of soaring flight, but so long as the fact
is not questioned the demonstration is all that could be wished.

In January, 1898, the author witnessed the spectacle of some
twelve or fifteen herring gulls following in the wake of the
ss. Germanic the whole voyage from Liverpool to New York.
The weather was stormy, and the passage occupied some ten
days; in spite of this the gulls followed without difficulty,
soaring continuously day and night. The author watched these
gulls sometimes for upwards of an hour, and scarcely a
single flap of the wing amongst the whole flight could be
observed. 1

1 In an article on "Flight" in Newton's "Dictionary of Birds" the
following passage occurs: "Gulls may be seen to soar for a prolonged
period of time in front of or above cliffs on the shore when the wind comes
from the sea. This, however, cannot be taken as an example of soaring
proper, seeing that gulls can only remain suspended in the air without loss
of relative horizontal motion under special conditions which do not apply in
the case of the typical soaring birds. Gulls are not seen to soar except
under conditions which point towards upward currents being a veiy obvious
explanation of the phenomenon. In the case, on the other hand, of the
typical soaring birds, such as those named above, soaring is observable
under conditions and at heights where there is no sufficient reason to assume
that upward currents exist. For example, eagles and adjutants are seen to
rise continuously by soaring for miles above the ground or sea."

The above passage serves to illustrate how difficult it is to reconcile the
statements of different observers, also the danger of asserting a neyativt
unless the experience on which the assertion is made is universal.

Parenthetically it may be remarked that the latter portion of this passage
is misleading. If upward currents exist in the atmosphere, other than as
due to a terrestrial obstacle to the flow of the wind, then they will evidently

240

SOARING § 142

It is said that, generally speaking, the gulls change from an
out-bound to a homeward-bound vessel when one or two days
out, but on this occasion nothing of the kind took place. 1 The
flight path consisted in what appeared to be gentle undulations,
the birds describing orbits (relatively to the vessel) of oval form,
the major axis being inclined backwards from the vertical, and
* probably of some 6 feet or 8 feet length. 2

The herring gull may also be observed to remain in the air
soaring for considerable periods of time at almost any point along
the south coast where there are cliffs on the sea shore. The
soaring here is evidently on a different footing and depends
in most cases on the upcurrent set up the cliff itself. We shall
find later that it is necessary to discriminate between certain
different kinds of soaring, the simple case where the bird utilises
an upcurrent does not differ, from the aerodynamic standpoint,
from the gliding mode of flight, there is a superposed upward
motion of the aerial system, nothing more.

In the autumn of 1906, on the west coast of Ireland, in
particular Achill Island, the author observed the following
gulls in soaring flight: —

Great Black-back (Larus mannas). Numerous.

Glaucous 5 (Larus glaucus) . . Two only.

Herring Gull (Larus argentatus) . Very numerous.
Kittiwake (Larus rissa) . . „ ,,

be greater at considerable altitudes, for the surface of the earth is a boundary
surface to a fluid region to which the motion in the immediate vicinity is of
necessity parallel, but it is not so constrained elsewhere (comp. § 145).

The reason that gulls do not soar to a height like eagles and adjutants
is possibly connected with the manner in which they get their living.
To assume that the mode of soaring is different seems to the author
unwarrantable.

1 The weather was thick and stormy, and no homeward-bound vessel was
sighted.

9 It is impossible to vouch for the accuracy of this estimate; the orbit
may have been considerably larger than stated ; distances at sea are
deceptive.

8 Some doubt attaches to identification.

247

§ 142 AERODONETICS

In general, it was observed that in weather in which the gulls
are able to soar in the open, the velocity of the wind is consider-
able, not very much less than the velocity of flight of the birds
themselves. Observations were made for the most part on the
nearly flat foreshore of Keel Bay, in the neighbourhood of Keel
Lough.

With the wind in the south-west a pair of ravens were seen on
several successive days to make use of the upcurrent of a cliff
when returning from scavenging the foreshore at low water.
Having reached the point where the cliff commences, these birds
made no further wing effort, but merely sailed lazily up and up,
till where the cliff merges into the steep of the mountain side, high
above the " Cathedral Rocks," they became lost to view. On three
separate occasions the author watched this simple, but effective,
labour-saving device in operation ; it was evidently part of a daily
routine carried out with a familiarity born of long experience.
What might happen when the wind is in some other quarter is
an open question ; the occasion did not arise, in all probability it
is some other part of the foreshore that receives attention and
the wind is still made humbly to do its duty, evidently whatever
part of the island presents for the time being a " lee shore" 1 is
that where the food supply will be most abundant, and it is there
also that the wind provides the upcurrent for the journey back
into the mountains.

On the slopes of Mweelin mountain, in September, a golden
eagle 2 was seen soaring high on the mountain side; unfortunately,
at too great a distance for minute observation. In contrast to the
slow deliberate sailing of the eagle, one moment in a direct line,
then in a mighty circle, could be seen the livelier but equally
astonishing evolutions of a kestrel hawk; it was the hawk's
hunting ground, the eagle was the usurper. First above, then

1 A sailor's term, meaning that the shore is to leeward of the ivater.

2 It would appear that this bird is now extremely rare in the British
Isles ; in 1906 only one pair were known to exist in the neighbourhood of
AcLill, reported locally as nesting in Clare Island.

248

SOARING § 143

far below, then completely lost to sight, and again high above,
the hawk eventually drove the larger bird from the scene ; the
eagle took its departure still soaring, but in a straight course
without " circling " or orbital motion of any kind.

§ 143. Soaring Flight Described by other Observers. — The

albatros furnishes one of the most remarkable examples of
soaring flight; its evolutions have been vividly described by
many writers. To J. A. Froude 1 we are indebted for the
following graphic passage: — "The albatros whe*els in circles
round and round, and for ever round the ship — now far
behind, now sweeping past in a long rapid curve, like a per-
fect skater on an untouched field of ice. There is no effort ;
watch as closely as you will you rarely or never see a stroke
of the mighty pinion. The flight is generally near the water,
often close to it. You lose sight of the bird as he disappears
in the hollow between the waves, and catch him again as he
rises over the crest; but how he rises and whence comes the
propelling force is to the eye inexplicable ; he alters merely
the angle at which his wings are inclined ; usually they are
parallel to the water and horizontal, but when he turns
to ascend or makes a change in his direction the wings
then point at an angle, one to the sky, the other to the
water.' '

No less vivid is the description of the flight of the condor,
given by Darwin; 2 the following extract will suffice. "When
the condors in a flock are wheeling round and round any spot,
their flight is beautiful. Except when rising from the ground I
do not recollect ever having seen one of these birds flap its wings.
Near Lima, I watched several for nearly half an hour without
once taking off my eyes. They moved in large curves, sweeping
in circles, descending and ascending without once flapping. As
they glided close over my head, I intently watched from an

1 " Oceana" (1886), p. 76.

2 " Voyage of II. M.S. Ucayte," Chapter IX.

249

§ 143 AERODONETICS

oblique position the outlines of the separate and terminal feathers
of the wings ; and if there had been the least vibratory movement
these would have blended together, but they were seen dis-
tinct against the blue sky. The head and neck were moved
frequently and with force, and it appeared as if the extended
wings formed the fulcrum on which the movements of the neck,
body, and tail acted. If the bird wished to descend, the wings
for a moment collapsed ; and then when again expanded with an
altered inclination, the momentum gained by the rapid descent
seemed to urge the bird upwards, with the even and steady
movement of a paper kite. In the case of any bird soaring its
motion must be sufficiently rapid so that the action of the
inclined surface of its body on the atmosphere may counter-
balance its gravity. The force to keep up the momentum of a
body moving in a horizontal plane in that fluid (in which there
is so little friction) cannot be great, and this force is all that is
wanted.''

Some information of importance may be gleaned from the
description of the evolutions of the soaring bird given by
Mouillard in his " L'Empire de l'Air " ; the following paragraphs
may be quoted as to the point : —

" Sans vent le voilier tombe, son vol n'est plus possible, il est
oblige de devenir rameur ; c'est ce qui fait qu'il est rarement
matinal, parce que la matinee est ordinairement calme, surtout
dans les pays chauds. ^

" Admettons maintenant l'existence d'un cour^nt d'air, ce qui
arrive presque toujours a une certaine hautej* dans Tatmosphere.

" La scene change, le voilier decrit des cercles, s'eleve en Fair
\ une grande hauteur, puis de la se laisse glisser dans la direction
ou il veut aller, meme contre le vent."

And later he says : —

" Un voilier qui s'eleve par un temps calme rame ordinaire-
ment jusqu'a une centaine des metres, et arrive a cette hauteur
commence a decrire ses ronds, moitie ramant moite planant,
diminue les battements a mesure que l'elevation augmente et

250

4

SOARING § 144

finit par les cesser tout a fait ; ce qui demontre que Pair n'est
immobile qu'a la surface du sol. 1 '

Langley 1 gives the following account of the soaring of the
turkey buzzard : — " On the only occasion when the motion of one
near at hand could be studied in a very high wind, the author
(Langley) was crossing the long 'Aqueduct Bridge' a over the
Potomac, in an unusually violent November gale, the velocity
of the wind being probably over 85 miles an hour. About
one-third of the distance from the right bank of the river, and
immediately over the right parapet of the bridge, at a height not
over 20 yards, was one of these buzzards, which for some object
which was not evident, chose to keep over this spot, where the gale,
undisturbed by any surface irregularities, swept directly up the
river with unchecked violence. In this aerial torrent, and appa-
rently indifferent to it, the bird hung, gliding, in the usual manner
of its species, round and round in a small oval curve, whose
major axis (which seemed toward the wind) was not longer than
twice its height from the water. The bird was therefore at all
times in close view. It swung round repeatedly, rising and
falling slightly in its course, while keeping on the whole, on one
level, and over the same place, moving with a slight swaying,
both in front and lateral direction, but in such an effortless way
as to suggest a lazy yielding of itself to the rocking of some
invisible wave."

The foregoing observations must be considered sufficient not
only to place any question of fact beyond doubt, but also
to define the characteristics of soaring flight as known to
observation.

§ 144. The Different Modes of Soaring Flight.— In a letter to
Nature* in 1883, Rayleigh laid it down that in order that a bird

1 "The Internal Work of the Wind" (S. P. Langley); "Smithsonian
Contributions to Knowledge."

2 Washington City.

8 Nature, XXVI [., r . 534.

251

§ 144 AERODONETICS

should consistently maintain its flight, without working its wings,
either —

(1) The course is not horizontal ; or

(2) The tvind is not horizontal ; or

(3) The wind is not uniform.

These alternative conditions, first clearly enunciated as above,
must be looked upon as the essential basis on which all
explanations of soaring must be founded.

It is evident that if a wind were uniform and horizontal the
condition of the bird in continued flight, so far as the dynamics
of flight is concerned, would be precisely the same as for still air ;
the difference is merely one of relative motion ; therefore soaring
is impossible under these conditions. If the horizontal velocity
of the wind be greater than the maximum velocity at which the
bird is able to travel, that is if it exceed the value of V n by more
than a certain amount, it is evident that the bird will be unable
to stem the current; this, however, does not affect its power of
flight as such one way or the other, it merely affects the utility or
otherwise of the exercise of that power. 1

The first of Rayleigh's conditions is framed to cover the case
of a downward flight path. It is evident that if the flight path
is not on the whole horizontal and that the maintenance of flight
is due to this cause, the bird (or aerodone) will eventually come to
earth. The case is one of ordinary gliding, hence we have already
excluded this by our definition of soaring, and only conditions (2)
and (8) remain, thus Rayleigh's dictum may be expressed in the
following modified form : — In order that a bird should remain con-
tinuously in flight without performing work and without loss of
altitude, that is to say, in order that a bird should soar, either —

(a) The wind being uniform possesses an upward velocity
component; or,

(b) The wind is not uniform.

1 Thus a horizontal wind exceeding the maximum velocity of flight,
renders flight impossible to a bird that requires to return to its point of
departure, but it may bo of inestimable value to a bird in migration.

252

SOARING §145

So far we have not demonstrated in what way these conditions
render it possible for soaring to take place, the problem has been
discussed as a negative proposition. We shall in the following
sections show that these conditions form the fundamental basis
of two distinct modes of soaiing, both of which are practised
extensively by birds in flight. In some cases it is probable
that the soaring bird takes advantage of favourable meteoro-
logical conditions of both kinds, at one and the same time ;
such mixed conditions without doubt frequently exist. This,
however, in nowise prevents the subject being dealt with in
theory as if the two conditions are entirely separate and
distinct.

There are two kinds of want of uniformity that will be dis-
cussed as coming under condition (b); firstly, one in which
there is a fluctuation due to a general turbulent condition of
the atmosphere, 1 as evidenced by the "gusts" that commonly
form part of the phenomenon of wind; and, secondly, that
due to the proximity of an obstacle to the flow of the aerial
stream, such as a building or other obstruction by which a
" dead-water " region is caused and maintained. The first is thus
a want of uniformity of velocity expressed as a function of
time for any particular point in space ; the second is a want
of uniformity of velocity as a function of position in space for
any particular instant of time. These two kinds of want of
uniformity give rise to two recognisably different modes of
soaring; a subdivision of the mode due to condition (b).

§ 145. The Vertical Component of the Wind. Meteorological
Considerations. — The explanation of soaring on the assumption
that the bird is sustained by an upcurrent, or by an upward
trend of the wind, is one that has frequently been advanced, and
also on which doubts have been repeatedly cast. There would
appear to be considerable prejudice existing against the idea of
vertical motions of the air, and before discussing the probable

1 " Aerial Flight," Vol. I., Aerodynamics, §§ 37, 131.

253

§ 145 AERODONETICS

influence of upcurrents on the phenomenon of soaring, it is
desirable to examine the question of such currents from a
meteorological standpoint.

At the outset it may be pointed out that since the whole of our
everyday experience of wind motion relates to movements of the
air in the immediate vicinity of a boundary surface of the
atmosphere, the whole of our ordinary wind data relate to
motions parallel to that surface, that is on the whole to horizontal
motion ; this fact alone is sufficient to account for an innate
dislike to any other kind of wind. Probably if we had spent our
lives at the top of the tower a few thousand feet in height we
should have become accustomed to specifying the direction of the
wind both in altitude and azimuth, and the motion of a vertical
wind would not have seemed strange or unusual.

In an article by C. S. Roy in A. Newton's "Dictionary of
Birds/' the following passage occurs :— " That the direction of the
wind even at great heights, and above a comparatively smooth
sea or plain, is by no means always parallel to the surface of the
globe is more than probable; but we know of no reason for
assuming that the upward currents are sufficiently predominant
over the downward currents to justify us in looking on the former
as capable of explaining observed facts as to soaring." This
passage is delightfully ingenuous ; firstly, it is evident that the
upward and downward currents, measuring them by their flux,
must on the whole be exactly equal, otherwise there will be an
accumulation or an attenuation of the lower strata of the
atmosphere which we know is not the case, so that we cer-
tainly have no reason to suppose that either predominates.
Secondly, it is evident that it is entirely unnecessary to the
proposition that the upward currents should predominate,
unless it be first shown that more than half the area of
the earth is simultaneously available to the soaring bird.
Thirdly, it is unnecessary that the upcurrent theory should
explain every case of soaring as implied, it is sufficient that
it should explain those that are not otherwise explicable. The

254

SOARING

§145

above is quoted as showing, in connection with wind motion,
the bias under which the human mind labours.

Now as to positive evidence; of this there is no lack. The
phenomenon of land and sea breezes is one of which the rationale
has been understood for a very long time. 1 When the influence of
the solar rays begins to be felt in the early part of the day, the
surface of the land quickly becomes warmed and gives up some
of its heat to the air in its vicinity, which thus becoming speci-
fically lighter than the surrounding air tends to rise. 2 The
regions covered by water (rivers, seas, and oceans) do not so
readily become heated, partly because the sun's rays penetrate to

^T^Z^T^^^

Fig. 110.

a greater depth, and therefore have to heat a considerably greater
quantity of matter, and partly because the specific heat of water
is greater than that of dry soil, and also probably because more of
the heat vanishes in supplying the latent heat of evaporation of
the water that is taken up in the state of vapour. The well-
known consequence of this is that in the middle part and early
middle part of the day, when the power of the sun has made itself
felt, there is a strong upcurrent over the regions occupied by land

1 Reference may be made to any encyclopaedia or text-book of Meteoro-
logy.

2 There is, perhaps, an academic objection to the use of the phrase " tends
to rise " ; this objection, however, is without effect on the discussion.

255

§145

AERODONETICS

the displacement of which has to be made good by air flowing in
laterally across the coast line. This is illustrated in the case of
an island in Fig. 110. Later in the day, when the solar heat has
declined, the land loses heat rapidly by radiation, and the
conditions are, to some extent, reversed.

The above is the universally accepted explanation of the
sea breeze that springs up in the course of the forenoon at the
" sea-side " in the summer in our own latitude and in a more
pronounced manner still in lower latitudes where the mid-day
beat is more intense. It is scarcely necessary to remark that
in rough weather the sea breeze is on too small a scale to

Pig. 111.

overcome the greater aerial disturbances of which the atmo-
sphere is the prey; it is, however, worthy of note that if
there be a slight breeze in the early morning in a direction
opposed to the " sea breeze," that is to say, a breeze oflf shore, it
is said by yachtsmen that the sun eats up the wind, for such a
breeze in fine weather nearly always drops later in the forenoon,
when we may regard it as having been neutralised by the sea
breeze that would otherwise have arisen.

Thus the sea-breeze and the vp-current are two parts of one
and the same stream. The position of an observer on the coast
line, Fig. 110, is comparable to that of an observer situated at
the bend of a river, Fig. 111. If he measures the velocity of the
stream in his immediate environment (a), he will have some idea

256

SOABING § 146

of its magnitude at (b). Now a sea-breeze of ten or fifteen miles
per hour is not uncommon, so the up-current by which such
breeze is generated may be expected to have a velocity of this
order of magnitude. Again, from the purely theoretical point
of view, let us suppose that the column of heated air is but
one degree 0. hotter than the surrounding air, then its density
will be approximately ^&Tjth part less than normal, and if the
height of the heated column be 300 feet the difference of pressure
by which it is propelled will be equivalent to a "head " of one foot ;
this, by the principle of Torricelli, corresponds to a velocity of
8 ft/sec., which is more than sufficient to sustain a gliding bird
without loss of altitude. 1 It is of interest to note that in the
passage already quoted from Mouillard's work he speaks of the
rmlier (soaring bird) commencing to soar at about 100 metres
altitude, up to which point active flight requires to be employed
if the weather is calm. The passage also, " G'est ce qui fait qu'il
est rarement matinal, parce que la matinee est ordinairement
calme, surtout dans les pays chauds," is of considerable interest.
The word matinee must evidently, from the context, be translated
as " early morning."

§ 146. The Vertical Component of the Wind (continued). — The
question of the vertical component of the wind is actually far
broader than might be supposed from the foregoing discussion.
The considerations that apply to the generation of a sea-breeze
in reality apply universally to the origin of almost every kind
of aerial disturbance. Thus the changes of density due directly
and indirectly to the solar heat constitute the mainspring from
which nearly all wind energy is derived. There are exceptions of
some slight importance, as, for instance, electrical action, the tidal
action of the moon and sun on the atmosphere, also the internal
heat of the earth ; but the heat energy, whether solar or terrestrial,

1 It is only necessary that the up-current should exceed the velocity of
least descent in order that a gliding bird should be indefinitely sustained.
Compare " Aerial Flight," Vol. I., Aerodynamics, § 176.

A. 257 S

§ 146 AERODONETICS

acts on the density of the atmosphere from point to point either
by change of temperature or by change of water content, and so
gives rise to vertical motions, to which the horizontal currents
are counterpart. Thus in considering the horizontal component
of the wind first and the vertical as a secondary and doubtful
quantity we are reversing the order of cause and effect.

In spite of the above, it is probable that the vastly greater
proportion of the energy stored in the atmosphere is in the form
of horizontal motion, so that the customary view is to some
extent justified. Using an analogy, we might express the position
by comparing the vertical motions of the air to the piston move-
ment of an engine, and the horizontal motion that results, and
persists, to that of the fly-wheel.

In the summer of 1905 the author observed a particularly
clear example of the type of upward flow that takes place under
the conditions discussed in the preceding section when becalmed
off Yarmouth, in the Solent. About mid-day, approximately at
the turn of the tide (low water), the whole eastern portion of the
Isle of Wight and mainland in its vicinity could be seen mapped
in the sky. Over the land the sky was closely studded with a
number of bolster-like cumulus clouds, apparently situated at a
height of some few thousand feet. Each cloud forms the head
of an ascending column, with a network of interspaces through
which return currents descend from higher altitudes. The broad
avenues and masses of blue sky, faithfully representing the water-
ways beneath, also are occupied by the descending currents.

Perhaps one of the most striking and convincing cases of an
upward current on record is that most graphically described by
Santos Dumont 1 in connection with a ballooning adventure. He
says : " Shortly after this I let the balloon go down again, hoping
to find a safe air current, but when within 800 yards of the ground,
near the Yar, I noticed that I had ceased descending. As I had
determined to land soon in any case, I pulled the valve rope and
let out more gas.

1 " My Airships " (A. Santos Dumont), p. 54.

258

SOARING § 146

" I could not go down. I glanced at the barometer, and found
indeed that I was going up. Yet I ought to be descending, and
I felt, by the wind and everything, that I must be descending.
Had I not let out the gas?

" To my great uneasiness, I discovered only too soon what was
wrong. In spite of my continuous apparent descent, I was,
nevertheless, being lifted by an enormous column of air, rushing
upward. While I fell in it I rose rapidly higher with it.

" I opened the valve again ; it was useless. The barometer
showed me that I had reached a still greater altitude, and I

could now take account of the fact by the way the land waB dis-
appearing under me. I now closed the valve to save my gas.
There was nothing but to wait and see what would happen.

" The npward rushing column of air continued to take me to a
height of 3,000 metres (almost two miles). I could do nothing
but watch the barometer. Then, after what seemed a long time,
it showed that I had begun descending."

In general it may be stated that cyclones are regions of
ascending currents; waterspouts, sand " pipes," and tornadoes,
are also evidence of upward motion. It is in fact the potential
259 s 2

§146

AERODONETICS

energy released by the ascent of the lighter air (and descent of
the heavier) that passes into the vortical system, originally of
sluggish motion, and renders it dangerous by reason of the
enormous velocity that it acquires. 1

§ 147. The Vertical Component of the Wind as dependent upon
Geographical Conditions. — The vertical component of the wind as
due to the physical features of the country, i.e., mountains, cliffs,
etc., is of a kind that is too self-evident to require any argumenta-
tive support. When a wind passes over a ridge, such, for
instance, as a range of hills, the various layers flowing like strata
one above the other, are diverted at first to the same extent as

Fig. 113.

the ground, and in the upper layers to a lesser degree. If, instead
of a ridge, the hill or mountain is isolated, or, as it is sometimes
termed, of "sugarloaf" form, the disturbance close to the
ground is parallel to its surface, as before ; but the layers above
are less deeply affected owing to the fact that the air can pass
round the hill sideways as well as over the top.

The behaviour of a wind in passing over a ridge of an isolated

1 When a mass of matter in rotation or circulation is drawn towards the
centre of motion, its velocity is increased to an extent equivalent to the
work done in overcoming centrifugal force. It is perhaps of some interest
to note that the regions of the earth where the soaring bird performs its
evolutions on a graid scale are also the regions where the phenomena
mentioned may be most frequently observed.

260

SOARING

§147

hill may be diagrammatically illustrated by the hydrodynamic
plottings given in Figs. 112 and 113, representing the flow of a
" perfect fluid " round a cylinder and a sphere respectively ; the
horizontal axis of symmetry may be regarded as representing the
ground surface, and the upper part of the diagram a ridge of
semicircular section or a hemispherical hill, as the case may be.
In practice, in cases such as those illustrated, the motion is
not of the form shown. Owing to the fact that the air is not a
perfect fluid, the flow will be of the discontinuous type, with a
dead-water region in the lee of the obstacle ; this is unquestion-
ably the case with forms far less abrupt than those figured,

'/////»'

'////////////////A

Fig. 114.

hence the shelter afforded by a range of hills or mountains; if • o ^\
the flow were of the Eulerian form such shelter would not exist, (t**** 4 **!
The general character of the stream-lines, on the windward
slopes of the ridge or hill, is not, however, greatly different in
the two types of flow, so that we may look upon Figs. 112 and
113 as roughly representing the conditions, so far as the
ascending currents are concerned.

For gentle undulations, where the gradient is not too great, we
may evidently take the upward velocity of the air current as
approximately proportional to the gradient ; the assumption here
is that the air stream follows the surface of the ground, and that
its velocity is not seriously affected.

261

§147

AERODONETICS

In the case of a vertical cliff the form of flow will closely
resemble that calculated on the Rayleigh-Kirchhoff basis, 1 this
being represesented in Fig. 114. The surface of discontinuity
of course becomes the seat of turbulent motion, as indicated dia-
grammatically in Fig. 115, in which an attempt has been made
to indicate the Btream flow aB it must actually take place. The
region of greatest up-current is evidently immediately in advance
of the upper edge of the cliff, and the "useful" region maybe
roughly defined as a circle (shown dotted in Figs. 114 and 115),
whose centre is slightly above the level of the plateau, and whose
diameter is about equal to the height of the cliff.

It is evident that similar up-currents will be generated by other

' "//////////////ft

Fig. 115.

kinds of obstruction, svch as houses, walls, etc., and, further, in
some cases the up-current may arise from the motion of a body
through the air in lieu of the air moving past the body. In this
way ships, especially sailing vessels, may conceivably become the
cause of an aerial disturbance that the Boaring bird can turn to
its advantage.

§ 148. The Up-Current in its Relation to Soaring Flight. — It is
not by any means every case of soaring flight that is to be
attributed to an upward motion of the air. There are, however,
many cases in which this explanation is clearly adequate, and is
indicated by the conditions. There are also many cases in which

1 " Aerial Flight," Vol. I., Aerodynamics t §§ 94 et aeq.

262

SOARING § 148

the upward motion of the air, though perhaps not in itself
sufficient, unquestionably plays a prominent part.

It is in the author's opinion comparatively rarely that in these
latitudes a sufficiently powerful up-current arises from the simple
heating of the air to support a bird without loss of altitude. We
may take it that the velocity of the soaring bird is usually about
85 to 50 feet per second, and that its natural gliding angle is
approximately 1 in 5 or 1 in 6 ; it may be somewhat less, but
that it is greatly leBS is scarcely probable. Consequently the
downward velocity of a bird in gliding flight relatively to the air
is commonly about 7 to 8 feet a second, and it will therefore
require an up-current of a velocity equal or superior to this in
order that soaring should become possible.

In connection with this subject it must be remembered that it
is not only necessary that the conditions should be occasionally
favourable, but that they should be favourable with sufficient
frequency or regularity for the bird to take advantage of them in
connection with its means of livelihood. If the necessary up-
current exists over some particular region half a dozen times in
the course of a year, and no more, it is hardly likely that we shall
find that on these occasions the air will be infested with eagles
and adjutants, or other of the larger exponents of this mode of
flight. Neither will an essentially marine Bpecies leave its
fishing ground in order to give a demonstration of soaring flight
for no useful purpose. The most we can expect to witness on
these occasions is that many of the birds that usually fly in
whole or in part by wing exercise have taken to soaring, and that
active flight is only resorted to occasionally and at infrequent
intervals.

The connection between the frequency and regularity of con-
ditions that admit of soaring and the presence — one may almost
say the very existence— of the soaring bird that makes use of
these conditions, is probably more intimate than has been
hitherto imagined. A most suggestive passage occurs in Darwin's
" Voyage of the Beagle" as follows : — " This day I shot a condor.

263

§ 148 AERODONETICS

It measured from tip to tip of the wings eight and a half feet,
and from beak to tail four feet. This bird is known to have a
wide geographical range, being found on the west coast of South
America, from the Strait of Magellan along the Cordillera, as
far as eight degrees north of the Equator. The steep cliff near
the mouth of the Bio Negro is its northern limit on the
Fatagonian coast; and they have there wandered about 400
miles from the great central line of their habitation in the
Andes. Further south, among the bold precipices at the head
of Fort Desire, the condor is not uncommon; yet only a few
stragglers occasionally visit the sea coast. A line of cliff near
the mouth of the Santa Cruz is frequented by these birds ; and
about eighty miles up the river, where the sides of the valley are
formed by steep basaltic precipices, the condor reappears. From
these facts it seems that the condors require perpendicular cliffs."
It is curious that Darwin should not have associated the fact to
which he here testifies with the explanation of the soaring flight,
which obviously suggests itself. The idea can hardly have
escaped him, but presumably as a matter of observation the
direct evidence was insufficient to justify the suggestion being
presented. 1 In spite of this, the author feels that the circum-
stances are such as to render it almost certain that the flight of
the condor depends very much upon the existence of these cliffs
and the up-current to which they give rise. 2

Up-current soaring may be seen at almost any point along the
coast in this country where cliffs abound, especially in the south,
where the prevalent wind (south-west) lends itself appropriately.
In suitable weather the gulls can be Been gliding, and continually
gliding, rising or falling at will, in front of the almost perpen-
dicular chalk cliffs, where, with a breeze of some 15 or 20 miles

1 It is not usually supposed that the up-current region in front of a cliff is
so extensive as evidently it is (Figs. 114 and 115). Modern hydrodynamic
theory did not exist at the date of Darwin's observations.

9 It might be added, the existence of the condor may depend upon these
geographical features, both in the sense of its past evolution and as to its
future continuance.

264

SOARING § 148

an hour over a considerable region, the up-current would appear
to be at least twice that actually necessary for mere sustentation.
In a letter to Nature 1 Mr. S. E. Peal gave a sketch diagram
and description of the soaring of some of the larger birds, those
especially mentioned being pelican, adjutant, vulture, and cyrus, 3
the point of observation being situated near Sibsagar, Assam,
that is, in the valley of the Brahmaputra, at the foot and on the
slopes of the Naga Hills. The sketch is reproduced in Fig. 116,
and the particulars given and to be inferred from the description
are as follows : — The birds rise by flapping the wings vigorously,
and begin to soar when they reach an altitude of from 100 to
200 feet from the ground. The flight path then consists of a
series of large sweeping circular movements of about 150 feet

Fig. 116.

diameter, rising from 10 to 20 feet at each "lap." When soaring
in this way the wings are rigidly extended and quite motionless ;
the birds frequently soar thus to a height estimated at 8,000 feet.
The velocity of soaring flight is computed as from 15 to 85 miles
per hour. The general trend of the spiral that constitutes the
path of flight is to leeward, the loss of position, or drift, being
about 20 to 50 feet each lap.

If we take the velocity of flight to be the mean of the limits
estimated, i.e. = 25 miles an hour, the time taken to complete
one lap of the spiral is approximately 13 seconds, which, on the
rate of ascension stated, is about one foot a second. The probable
rate of descent through the air of one of these birds when soaring

1 Nature, XXIII, p. 10.

* The name hero given does not appear to be in general usage. The
author has been unable to identify this bird.

265

§ 148 AEEODONETICS

is 7 ft./sec., bo that the upward velocity required to account for

the phenomenon is 8 ft./sec. total.

If we take the difference between the maximum and minimum

estimates of the velocity of flight as due to the superposed motion

of the air, that is as due to the horizontal component of the wind,

gg jg

we obtain for the latter ^ — = 10 miles per hour, or say

15 ft./sec. Thus the motion of the wind as estimated from the
data supplied has an upward trend of about 1 in 2, and a
velocity of between 11 and 12 miles per hour. It is of some
interest to note that the mean upward course of the spiral flight,
taken from Mr. Peal's sketch, corresponds with the upward angle
of the wind as given by the above estimate. This coincidence
suggests that if the foregoing reasoning is correct, the stratum
or column of ascending air may be one of quite limited dimen-
sions, a kind of stream in fact which the birds in their flight
are careful not to quit.

Mr. Peal offers an explanation different from that here given,
but as it appears to involve the usual fallacy of supposing that the
bird can draw upon the translation energy of a horizontal air
current, after the manner of a string kite, 1 it has not been
thought suitable for discussion in the present work.

Lord Bayleigh in discussing Mr. Peal's observations points out
the impossibility of the explanation offered, and advances the
alternative one that the layers of the air at different altitudes
are moving at different velocities. On this basis he shows on
dynamic principles that a bird could extract energy from the
wind. He says, however : " A priori, I should not have supposed
that the variation of velocity with height to be adequate for the
purpose ; but if the facts are correct some explanation is badly
wanted." The author is quite in agreement with this passage ;
in the light of present day knowledge it is evidently impossible
to account for soaring in the manner suggested.

About the same date (1883) Mr. Hubert Airy offered an alterna-

1 See footnote, § 141.
266

SOARING § 149

tive explanation, founded on the supposition of horizontal vortices
in the atmosphere, due to, and constituting part of the pheno-
menon of wind. The author does not believe that in the caBe in
point this explanation is indicated by the facts as presented, but
the subject may be of great importance in its relation to soaring
of another kind, and further reference will be made to this
suggestion.

§ 149. The Up-current in its Relation to Soaring Flight (continued).
— The late Mr. W. Froude was one of the most staunch adherents

Fig. 117.

to the up-current theory of soaring, and a passage l in one of his
letters to the late Lord Kelvin, shows that he fully appreciated
the conditions essential to this view of the problem. Froude,
however, endeavoured to apply the up-current explanation
universally to all cases of soaring, and though many of his
arguments are of very great interest, they are at times scarcely
convincing, and eventually Froude himself voluntarily admitted 2
that the explanation in some cases was quite inadequate.

1 Proa Roy. Soc Edin., XV., pp. 256-8 (1888).
8 Proc. Roy. Soc. Edin., XVTIL, pp. 65-72 (1891).

267

§149 AERODONETICS

The form of the flight path of a soaring bird has been given
by many observers, and in the kind of soaring we have now
under consideration this consists of a number of circles, often
with a superposed motion of translation, Fig. 117. When the
gain in altitude is taken into account this form becomes an
oblique spiral. Many theories have been built on this peculiarity,
it being sometimes explained that the bird in going with the wind
acquires an enormous velocity, and when it turns to go against
the wind it can utilise this velocity in order to gain altitude. All
such explanations, founded on the assumption of a wind of
uniform velocity, involve a fundamental fallacy, and are inad-
missible ; on the principles of relative motion the position of a
bird in still air or in air moving horizontally and with uniform
motion is identical. 1 In spite of the absurdity of an argument of
this kind, unless supplemented by some additional hypothesis
touching the irregularity of the wind, it has appeared again and
again ; Mouillard 2 has presented it ; Esterno 3 has advocated it ;
Marey 4 has taken it seriously and devoted nearly half a chapter
to it ; S. E. Peal 6 has offered it, and many others have fallen
into the same pit. Bayleigh drew attention to this fallacy in
1883, when he enunciated the necessary alternatives already
cited.

It has frequently been stated by observers that the circular
orbit described by the soaring bird is not horizontal, but at a
slight inclination ; this supposed inclination is usually made to
fit in with the requirements of the fallacious theories offered, and
the orbit is not always described as leaning in the same direction.

1 In the sense of the article. Compare § 114. Assuming such an expla-
nation as valid, a bird could soar in still air, for, by gliding in the direction
of the earth' 8 rotation, it could acquire an enormous velocity, and on turning
to face the current the altitudo to be gained would be measurable in miles !

2 " L'Einpire de l'Air," p. 43.

8 D'Esterno, " Da Vol des Oiseaux," p. 40.
4 Marey, " Le Vol des Oiseaux," Chapter XX.
* Nature, XXIII., p. 10.
8 See Marey, '• Vol des Oiseaux," § 17.

268

SOARING § 149

In view of the influence of perspective as affecting the apparent
attitude of these flight path curves, and the disagreement between
different observers, it would appear that this feature, in the
absence of more conclusive evidence, may be safely ignored. The
author believes that in general any attempt to found a theory of
soaring on the horizontal orbital form of flight path is doomed to
failure ; the soaring bird moves in circles for the same reason
that a man sits in a chair, — because he wants to stay where he is. 1
There are doubtless exceptions, but these are discussed in a later
section.

When the up-current by which a bird soars is due to the wind
passing up a slope or over a range of hills, the ability of the
bird to utilise the up-current does not depend solely upon its
upward Velocity, as is the case when the air is rising vertically ;
it depends also upon the steepness of the gradient; there is a
certain minimum inclination that must be exceeded. Thus if the
gradient be exactly equal to the gliding angle, there is one critical
wind velocity — the natural velocity V n — at which the bird can
soar ; if the wind is too slow the bird will sooner or late): come to
earth ; if the wind velocity is too high, the bird must either come
to earth or be swept over the top of the ridge. Soaring is there-
fore only just possible on such a gradient, and in practice the
angle needs to be considerably steeper than the gliding angle, so
that there shall be a reasonable range of wind velocity available.
It is possible that careful observations on the soaring of birds on
slopes of known gradient may furnish data that will enable the
gliding angles of different species to be determined with consider-
able exactitude.

In estimating the natural gliding angle y and natural velocity
V w , the considerations dealt with in the author's u Aerodynamics "
require to be taken into account. Thus the velocity of slowest fall
is l/l a 315 of the speed of least resistance, and the gliding angle of
slowest fall is 1*155 times the gliding angle of least resistance. 2

1 This was clearly the case in the example given in § 142.
a Vol. I., §§ 164, 17'

269

§ 150 AERODONETICS

§ 150. Dynamic Soaring. — We now pass to the consideration of
the second of the alternative conditions under which soaring may
become possible (§ 144), i.e., the wind is not uniform. This is the
third of the three conditions as laid down by Rayleigh. It will
be shown that we now have to deal with an entirely different kind
of soaring from that so far discussed, one of which the theory is of
far greater difficulty, involving complex dynamical considerations.
A distinction between the two kinds of soaring being desirable,
the author has termed that kind we are about to discuss, i.e., that
which depends upon want of uniformity of the wind, dynamic
soaring.

There is more than one kind of aerial disturbance that results
in a want of uniformity (comp. § 144), and there are corre-
spondingly different kinds of dynamic soaring ; all, however,
depend upon one fundamental principle, the utilisation of the
internal enei'gy or, as it has been termed by Langley, 1 the internal
work of the wind. This term does not carry the Bame meaning
as in thermodynamics, but rather an analogous meaning; it
signifies in the present usage the kinetic energy additional to that
of mean velocity, but excluding that of the thermodynamic system*

It is established that wind comprises in addition to its main
motion of translation a superposed motion of turbulence ; 9 and
just as we may regard the energy of these two component motions
as separately put into the wind, so we may recognise in the
resultant motion two separate energy accounts which are virtually
independent one of the other.

We are accustomed, in considering the aerodynamics of flight, 8
to regard the aerofoil or wing spread of a bird as dealing
exclusively and uniformly with a stratum of air of some definite
limited thickness, instead of, as is actually the case, dealing with
an indefinite stratum in a variable degree, and in the hypothesis

1 "The Internal Work of the Wind," "Smithsonian Contributions to
Knowledge,"

* Vol. L, §§ 37, 131.

s Vol. L, §§ 160, 172 et seq.

270

SOAEING

§150

of constant " sweep " this method has been shown to give results
in very close agreement with experience. Employing, again, the
same idea, it is evident that the turbulence energy of the wind
with which we are immediately concerned is that contained in
the stratum of air with which the bird is supposed to deal, not of
necessity confined to that coming within the sweep area, but
perhaps that further included by the area of the periptery. 1

It is evidently fundamental that the bird or aerodone cannot
change the total momentum of horizontal motion of the air dealt

D

Fig. 118.

with, without itself definitely undergoing a change of velocity, for
any such action involves a horizontal force applied from without,
consequently the mean horizontal velocity remains unchanged
and the energy of translation of the wind cannot be touched.
Thus if a bird act on some portion of the wind to impart to it
change of velocity in the direction of flight, itself undergoing
retardation, then it must sooner or later act on some other part of
the wind, imparting to it a change of velocity opposite to that of
flight, so that it may itself undergo acceleration and so preserve
its mean velocity through the air unchanged.

1 " Aerial Flight," Vol. I., Aerodynamics, Appendix V.

271

§ 150 AER0D0NETICS

Now if the motion of the wind were originally uniform, any
such procedure as above implied would mean imparting energy
to the wind, otherwise superposing turbulence on an existing motion
of translation. If, however, the wind is initially in a state of
turbulence, and the bird by the exercise of intelligence or other ■*
wise act in such a manner as to equalise the velocity of the air
with which it is able to deal, it will have at its disposal the energy
of turbulence so removed from the wind.

Taking an example, let us suppose that the bird, flying in
air having a mean velocity represented by the line A B (Fig.
118), deals with two equal masses of air, one having a velocity
represented by A C, and the other a velocity A D, and if the
bird by means of temporary variations of its own velocity and
altitude (its initial and final states being the Bame) increase
the velocity of the first mass of air to equal A B and decrease
that of the second to like degree, then it will have to supply
energy equal to twice the area AB f B C minus the square on B C 9
and it will have at its disposal energy equal to twice the area
A B,B D plus the square on B D. But B C=BD, hence there
is an available balance of energy in the bird's favour equal to
twice the square on B C.

It may be noted that the magnitude of the mean velocity A B
is a matter of no importance; it cuts out, thus for given conditions
as to turbulence the motion of translation is unimportant,
it has nothing to do with the problem except so far as in
practice the turbulence cannot exist without the motion of
translation. In the limit we might suppose that the turbulence
alone exists, then the bird's duty will be to bring all the air
with which it deals to rest, and the energy at its disposal is
obviously that shown, i.e., due to the integration of the velocity
squared.

Before discussing the actual nature and theory of dynamic
soaiing it is very important to fully comprehend the underlying
principles, of which the above may be taken as a preliminary
exposition. The matter has been dealt with from a quantitative

272

*

SOARING § 151

standpoint in Vol. I., Appendix V., from which the following
demonstration is taken substantially unaltered.

§ 151. Dynamic Soaring. Preliminary Investigation. — Without
discussing the means whereby a soaring bird operates to play off
one portion of the wind against another, we may, from the fore-
going considerations, form an outside estimate of the available
energy. Thus, if we prescribe some conventional form as repre-
senting the motion of turbulence, such as a uniform reciprocating
motion in the line of flight, or a simple harmonic motion, or a
compound harmonic or circular motion, of known velocity, we
can calculate the turbulence energy per unit volume, and we may
convert this into an equivalent thrust force per unit area of the
stratum " handled " ; if, then, we know the extent of this area
in the case of any particular bird, we can determine the gliding
angle y, the minimum value of which is a quantity otherwise
known. Conversely we may, starting from the gliding angle
and other data, determine the minimum velocity of turbulence
on the convention chosen that will render soaring flight
possible.

A question that presents some difficulty is the estimation of
the area of the stratum liandkd. At first sight this might be
supposed to be the " sweep " of the aerofoil, i.e. = k A (Vol. I.,
§§ 109, 160, "Aerodynamics"), but the energy estimated on
this basis from known fluctuation data appears to be insufficient.
The conception of the peripteral area (Vol. I., § 210) suggests
that, as in the case of the propeller blade, the cyclic or peripteral
system may distribute the momentum over a much greater
mass of fluid than that coming within the sweep of the
aerofoil, and so a larger mass of the air than that coming
within the sweep area will be " handled " in the sense of the
present discussion. On this basis the sectional area of the
stratum from which the energy can be drawn is given by the

expression, 1 _— k A (comp. Vol. L, § 210).
a. 278 t

§ 151 AERODONETICS

In the following example the turbulence velocity is computed
necessary to provide the requisite energy to a hypothetical
albatros, whose data are : —

Weight . = 14 lbs.

Area . • =5 square feet.

n . . =12, hence k = 1*195 and * = # 75.

y . taken = }.

The computation will be made on the basis of the peripteral
area, and figures will in the first place be obtained on the basis
of uniform motion, either a uniform reciprocation, or a circular
motion may be presumed, the assumption being in either case
that the whole of the available energy is utilised. It will be
shown subsequently that there are other conditions limiting the
proportion of the total energy that can be usefully employed.

Now resistance to flight = W y == 14 -=- 7 pounds =

= = 64*4 poundals.

1 4- c
And peripteral area = .. __ k A, or, mass of air handled

per foot traversed,

= J-i-^ p k A = -?J X '078 X 1-195 X 5 = 38 lbs.
1 — € r "25

Let v = velocity of turbulence under supposed conditions,
we have

8-8 X t; a

— 2" " = 64 4

.-. v 2 = 89 ; or, v = 6'25.

If, instead of supposing uniform motion, we take fluctuation
to be of simple harmonic form, and let vi be the maximum
velocity of the fluctuation ; then we know that the energy for a
given value of t\ is half as great as for the same value of v on the
previous supposition, hence in order that the requisite energy

should be present, t?i = V 2 X v,

or, tfi = 8*8.

274

SOARING § 152

§ 152. Dynamic Soaring. Historical Development of Modern
Theory. — The present-day theory of soaring of the kind under
discussion may be regarded as the outcome of many suggestions
and theories that have been offered from time to time during the
last thirty years. Thus the earliest clear statement of the con-
nection between the wind fluctuation and the soaring of birds
with which the author is acquainted is due to Mouillard. 1 The
following passage is of considerable interest, not only on account
of the explicit manner in which the matter is stated, but also
as containing the germ of an idea that has been developed
independently by Mons. A. Bazin and by the author into a
means of demonstrating the principles of dynamic soaring to
an audience.

Mouillard says : —

" Le coup de vent est une puissance qui est Tame de 1' ascen-
sion : c'est la baguette qui frappe le cerceau de l'enfant, qui lui
donne la force de rester debout, 2 et meme de franchir des eleva-
tions. Supposons que nous abandonnions ce jouet a une descente
rapide ; l'attraction lui communiquera un mouvement qui le fera
rouler jusqu'au bas. Si en bas il se trouve une montee, le
cerceau pouss6 par sa vitesse acquise, par son inertie, remontera
a un hauteur egale a sa descente, moins les frottements sur le sol
et la resistance de l'air."

* * * * *

" Si maintenant nous supposons autre chose, qu'on puisse,
lorsque le cerceau est on train de remonter, d^placer le sol, de
mani&re a ce qu'il aille en sens contraire du jouet, c'est a dire lui
venir dessus, nous activerons encore l'ascension en lui communi-
quant une force supplemental, independents de son individu,
dont la resultante sera encore une elevation."

Unfortunately, although here Mouillard gives the gist of the
matter, the whole of the passage cited is given as a kind of
supplement following on an explanation that is quite inadequate

1 » L'Empiro de PAir," p. 47.

2 Mouillard' 8 analogies are not always easy to understand !

275 t2

§152

AERODONETICS

0*

6

M

a

=

=

&

O

CD

#)-

*

V

276

SOARING § 153

and which rests on the " string-kite fallacy," to which allusion
has already been made.

In 1885, Rayleigh, in a letter to Nature (§ 144), suggested
the variation of the wind at different altitudes as a possible solu-
tion, although the conditions enunciated in his letter will bear a
much wider interpretation. Hubert Airy, about the same time, 1
propounded a similar theory, but suggested further that the
irregular motion of the air due to horizontal vortices may
constitute a factor of importance. Next we have Bazin, 2 who,
in 1890, was the first to demonstrate the principles of dynamic
soaring by means of a " switch-back " model (Fig. 119) ; this
device was " re-invented " independently by the author a few
years later, and exhibited to the Birmingham Natural History
and Philosophical Society, in 1894 ; the author's model is repre-
sented in Fig. 120. The coincidence is most remarkable, the
author's model has one less "wave" than that of M. Bazin,
but the two are otherwise almost identical. 8 In 1898, Langley
published his investigations on " The Internal Work of the
Wind." 4 Although (apart from the meteorological portion of
this work) Langley had no new fact or theory to present, the
rationale of dynamic soaring is described by him in a particularly
lucid and simple way ; the publication is one well worth con-
sulting, as presenting the subject in an elementary form divested
of all mathematical complication.

§ 153. Dynamic Soaring. Theory of the " Switch-back " Model. —
The switch-back model is intended to represent the type of
dynamic soaring in which the flight path lies in a vertical plane,
and in which the bird in flight encounters masses of air moving
with different velocities. From the Phugoid Theory we may

1 Nature, Vol. XXVII., p. 690.

a Marey, "Le Vol dee Oiseaux," § 192.

* Bazin 's "montagnes russes" model was not known to the author till
many years after his paper to the Birmingham Natural History and
Philosophical Society.

4 " Smithsonian Contributions to Knowledgo," I.e. ante.

277

§ 153 AERODONETICS

evidently regard the soaring bird or aerodone as if it were
travelling on a " track " in space whose form may be any one of
the series of curves that comply with the equation, or in practice
the metamorphosis of such curves as due to the added conditions
of resistance and rotational inertia. Owing to the fact that a
bird in flight may alter its " constants " at will it is further
evident that we are not confined to curves of the phugoid series ;
in fact the bird can, within certain limits, carve out any curve it
likes, without departing from the analogy. We may regard the
resistance to flight as closely analogous to a frictional resistance
in which the gliding angle y represents the angle of friction, so
that the conditions would be approximately represented if we
suppose the ball, instead of rolling, to slide along the track. 1 If
the analogy is to hold it is evident that the condition must be
imposed that the velocity of motion must at no time fall below
a reasonable limit. 2

Now when a bird is gliding through the air in soaring flight,
it has the power of adapting the form of its flight path to the
changes in the wind velocity ; but we have no means of directly
representing this power of adjustment in the model. Instead of
adjusting the flight path to the pulsations it is evident that for
the purpose of demonstration we may do the converse and adjust
the pulses to a prearranged flight path, so in the " switch-back "
model the ball is arranged to run on a rigid track to which an
oscillatory motion is imparted in a longitudinal direction. Thus
in Fig. 120, if the ball be placed at the lower end of the " switch-
back " C and set in motion, it may be made by a properly applied
series of pushes and pulls to run up hill to the point D. The
correct way to apply the pulses to the apparatus is to impart a
motion counter to the "direction of flight" when the ball is

1 In this case, of course, there need bo no actual lifting of the ball (or
sliding block); the mountains of the "switchback" may be made of
uniform height.

s If the velocity of a bird or aerodrome becomes too low in relation to that
of leant resistance, a disproportionate increase of y takes place (compare
Vol. I., §§ 164, 176).

278

SOARING

§153

ascending, and in the same direction when it is descending.
This means that the maximum applied push takes place when
the ball is " troughing," and the maximum pull when it is
" cresting." 1

In reality, of course, the wind does not move en masse as is
the case with the " switch-back " track in the model ; but as the
bird is not concerned with the motions of the wind at distant
points this fact has no influence. In the model the motion of
the ball is practically frictionless, the energy imparted to it being
exhibited by its increase of altitude, or, if a sliding block be
substituted for the ball, whose coefficient of friction represents

Fig. 121.

the gliding angle, the energy is expended in overcoming the
frictional resistance.

Now the form of the " switch-back " track gives the form of
the flight path through the air ; that is to say from point to point
relatively to the air in which it is supposed the bird is flying.
Let this be represented in Fig. 121 by the curve a a a a.. It is
evident that the motion relatively to a point fixed in space will
be somewhat of the form given by the line bbbb, for, during the
passage of the ball (representing the bird in flight), from crest
to crest the track makes the reciprocation c to d and back again.
If, now, a piece of paper be cut to the shape of the track (the

1 The push or pull represents acceleration, positive or negative, of the
apparatus.

279

§153

AERODONETICS

curve aaaa) and the motion of the ball be followed out by sliding
the supposed track to and fro through the range c d y represen-
tative positions being given in Figs. 122 and 128, it will be
seen that the normal to the track, which in the case of the

/

Fig. 122.

ball represents the track reaction, has a propulsive component
on the real flight path b b 9 so that were the air to offer no
resistance to flight the velocity of the bird would receive a
substantial increase during each descending and each ascending

period. We consequently have a means of propulsion that may

be in actuality employed to overcome the flight resistance, and,

if sufficient, to enable the bird to increase its altitude in addition.

If we substitute for the normal reaction of Fig. 122-8 a

280

SOARING

§153

reaction inclined to the direction of the normal, Fig. 124, as repre-
senting the resultant of the lifting and retarding components of

I

Fig. 124.

the flight reaction, the condition of no energy loss, that is, the
condition of soaring at any point, is that this resultant shall be

Fig 125.

normal to the curve h b. If it is in advance of the normal
the bird is receiving energy faster than it is being expended ; if

281

§ 153 AEEODONETICS

it lies behind the normal the supply of energy is insufficient.
These two conditions are indicated in the figure.

It is evident that since the propulsive component falls to zero
twice during every phase, while the resistance remains much as
before, in order that soaring should be possible for the whole
flight path, the propulsive component must be considerably in
excess at the intermediate points, as illustrated in Fig. 125.

The foregoing theory is the basis of the following quantitative
investigation.

§ 154. Theory of Dynamic Soaring. Quantitative Treatment. —
Let A B y Fig. 126, represent a small element of the flight path

relatively to the air in which the bird is travelling, the direction
of flight is down-hill (following the lettering). Let A C represent
the same flight path element relatively to the " mean wind."
Thus in the model the path A B represents a portion of the
switch-back track, and A C the actual path of the ball relatively
to a point fixed in space.

Let A D be normal to A B, and let A E be the direction of the

282

SOARING § 154

aerial reaction on the bird or aerodone ; that is the resultant of
the normal reaction and the resistance. Then the condition of
"just soaring" for the element of the flight path under considera-
tion is that A E is normal to A C, that is to say, the angle B AG

= y.

Now draw A F perpendicular to the horizontal line C F, then
F C is the horizontal distance covered by the element of the
flight path under consideration, 1 and B C is the distance moved
by the wind pulse in the same time. We require to determine
the conditions under which the motion of wind fluctuation is
least for a given horizontal component of flight path. This will

Fig. 127.

be defined by the relative magnitudes of B C and F C ; given y

B C
we have to solve ir ^ = minimum.

i< C

This may immediately be recognised as the problem of the

efficiency of an element of the blade of a screw propeller, the

solution of which we already know, 2 if as in Vol. I, § 204, we

represent the angle F A B by the symbol the solution is

90° - y°
$ = 2~

or the mean of the angles and + y is 45°.

1 Relatively to the mean wind.

2 " Aerial Flight," Vol. I., Aerodynamics, §§ 202 et 8eq.

283

§154

AERODONETICS

Again, let Fig. 127 represent an ascending element of the curve
in which A B as before is the flight path relatively to the air in
which the bird is soaring at the time being, and A C is the same
path plotted relatively to the " mean wind " (relatively to a fixed
point in the switch-back model) ; the condition of " just soaring "
is as before when B A C = y. Here F C is the horizontal com-
ponent of the true flight path, and the required condition is that

B C

-zrji * s minimum. This condition may be otherwise expressed,

B C F C

^-^ + 1 = minimum, or -rr-j = maximum, which again is

Fig. 128.

identical with the problem of maximum efficiency of the screw
propeller.

Let us now imagine a flight path composed of alternate
elements, as in Figs. 126 and 127, to form a zigzag, Fig. 128.
There are difficulties in such a supposition, but these can for the
moment be ignored. We must suppose that the change of
direction at each angle of the zigzag path is effected without loss
of energy by some external agency. Let us further assume that
the individual elements are so small that the velocity of flight
does not undergo changes of sensible magnitude. Then, whilst
the bird (or aerodone) travels from A to Ai (Fig. 128), the wind
pulse motion is from C to B and back again ; hence the velocity
of the wind fluctuation in terms of the mean velocity of flight is

B C

2 -7— p. But the efficiency E in the sense used in the analogous

A A\

284

SOARING § 155

propeller theory is , ,— ^, so that 2 -j— j- may be expressed in the

form 2 X J-^J

the value of E being calculated from y as in Vol. I., § 204.
Alternatively the diagram may be drawn to scale for any stated
value of y and the ratio thus obtained graphically.

The results are given in Table I. for stated values of y,
both as to efficiencies and corresponding values of wind fluctua-
tion in terms of flight velocity, as given by the above ex-
pression.

It is evident that the present case corresponds to the uniform
motion conditions of § 151, conditions that are there dealt with

J

Fig. 129.

on a purely dynamical basis. We shall institute a comparison
between the two aspects of the subject when we have considered
one or two further examples of a less artificial kind.

§ 155. Theory of Dynamic Soaring. Quantitative Investigation
(continued).— The zigzag form of flight path studied in the pre-
ceding section enables the theory to be initially presented in a
very simple form, but the conception is essentially artificial. In
practice such a form of flight path is not possible ; the zigzag
must be replaced by a path of undulating form, and the theory
must be extended to deal with the modified conditions.

We will now suppose that the flight path is a curve of sines ;
that is to say, it is composed of a vertical harmonic motion super-
posed on a horizontal uniform translation. We will for the time

285

§ 155 AERODONETICS

being adhere to the supposition that the wind pulses consist of
simple alternations in the line of mean flight of uniform motion,
and that the vertical amplitude of the flight path is of sufficiently
small magnitude to permit us to regard the flight velocity as
sensibly uniform. 1

We will now, as in the preceding section, draw the flight path
relatively to the air, and relatively to the mean wind. Thus in
Fig. 129, the curve a a may be regarded as representing the
contour of the switch-back railway, and the curve b b the
path of flight relatively to a fixed point. Now these two curves
(under the supposed conditions) differ only in respect of their

\

Fig. 130.

horizontal scale ; that is to say, they are both sine curves. Let
us suppose that the one curve is made to slide horizontally over
the other, as in § 153, the intersection representing the passage
of the bird in flight. Then if we examine the two relative paths
at their intersection, owing to both curves being sine curves,
their angles will be always related so that cot Bjcot (0 + V)
will be everywhere constant, the sign T being employed to
denote the difference between the angles, now no longer equal
to y. That is to say, that if Fig. 130 represents a small element

F B

of the flight path, the relation ^—^ is constant, or the efficiency

is the same at all points.

But the propulsive force of which r is the measure now varies

1 Bolatively to the mean wind.
286

SOARING

§155

from point to point, being zero at the top and bottom points of
the curves and maximum at the points of greatest inclination.
And the condition essential to soaring is that the mean value of
T over the whole curve is equal to y, hence from the known
properties of the sine curve, if T x is the maximum value of r,

ri = 2 y.

The highest possible efficiency, therefore, that is to say the leas t
velocity of fluctuation, is given by employing -^ times the
actual y value, proper to the bird, as the y of Table I.

TABLE L 1

■f

E

V

V

2°

•982

•070

8°

•900

•105

4°

•869

•140

5°

•889

•175

6°

•811

•210

7°

•782

•245

8°

•756

•280

9°

•780

•815

10°

•704

•850

11°

•680

•885

12°

•658

•420

18°

•683

•455

14°

•611

•490

15°

•589

•520

16°

•567

•550

1 The value of y given in the table is calculated as stated in the text. It

v

is, however, worthy of remark that the single expression, y = 'Q$o y° gives

the same result for all ordinary values of y°, or if y be expressed in circular
measure (radians), y = 2y.

287

§ 156 AERODONETICS

§ 156. Theory of Dynamic Soaring. Harmonic Wind Pulsation.—
When we introduce the further condition that the motion of the
wind pulse shall be harmonic the problem becomes still more
complex. This condition is justified as a theoretical basis for
investigation on the general ground that every periodic motion
may be regarded as a series of superposed harmonic com-
ponents, and in practice the fundamental component is com-
monly the most important.

The supposition that the horizontal component of the flight
path is uniform will be adhered to, and the form of the flight
path will, as in the preceding section, be taken as a sine curve.

It will also be taken as part of the hypothesis of the present
investigation that the amplitude of the flight path is small.

Fig. 131.

Let V be the constant velocity of flight.
„ vi = vertical velocity component of the flight path.
„ v 2 = velocity of the wind pulse.

From the conditions i-i and i- 2 are both variables whose change
is harmonic, and whose phase is the same.

Let the curve b b t Fig. 131, represent the sine curve flight
path ; then the values of the variables n and r 2 may be repre-
sented by two other sine curves one quarter phase out of step
with the flight path curve, in the manner shown.

Let Fig. 132 represent a diagram of velocities at any instant.
The aerodone situated at A is travelling relatively to the mean
wind in the direction A C with velocity V; B G = r a represents
the velocity and direction of the wind pulse ; consequently A B

288

SOARING § 156

is the flight path and velocity relatively to the air in which the
aerodone is travelling.

The angle r (= C A B) is that by which the propulsion is
effected, and the condition of soaring is, as before, that the mean
value of T is equal to or greater than y.

We require to find an expression for V in terms of n and r 2
whose manner of variation is known.

It would evidently be possible to lay off a series of values of T
as a curve, from a number of measurements made from a
geometrical construction, and to integrate such curve by means
of a planimeter, the procedure here suggested being to construct
a series of diagrams such as Fig. 182, assigning values to i\ and

F B ^ % C

Fig. 132.

r 2 in accordance with the curves of Fig. 181, and then obtain

the r values by measurement.

If in the present investigation we confine ourselves to the

special case where the amplitude and gliding angle (y) are both

small, we have v% and v* both small compared to J', and the

simple relation T a ri r 2 obtains. But r\ and r 2 vary in like

ratio, hence,

T a n 2 .

Now, since the curve of i\ is a sine curve, the curve of r* is a

sine curve of half the phase length, 1 Fig. 188, and the curve of r

is likewise of this form ; consequently the mean value of r is

equal to half its maximum value. Thus, if we write T\ for the

maximum value of T we may express the conditions of just

soaring,

Ti = 2 y.

1 A woll-known result.
A. 289 it

§ 156 AERODONETICS

The solutions that we have effected in this and the preceding
sections can only be considered as approximations, the assump-
tions that have been made are not, strictly speaking, justifiable.
The assumption of uniform horizontal velocity holds good only
where the amplitude is small, both in relation to the phase length
of the flight path and absolutely compared to the value of i/ n .
If the relative amplitude is great so that the flight path curve is
steep, the assumed condition breaks down, owing to the fact that
a uniform horizontal velocity betokens an increased velocity on the
steep portions of the curve ; this is an error that can be condoned
if we suppose that the form of the wind pulse motion is modified
to suit, that is to say, it is not strictly uniform or accurately
harmonic as the case may be. If the amplitude is considerable
in relation to H n , then the velocity will be perceptibly greater

Fig. 133.

when " troughing " than when " cresting," just as it is in the
phugoid curve.

In spite of defects of hypothesis, the author believes that the
magnitude of the errors introduced is not serious, and that the
results stated may be looked upon as satisfactory approximations
to the truth.

§ 157. The Form of the Orbit in Dynamic Soaring. — It not
infrequently happens that a bird may be seen soaring against
the wind, making but little or no headway relatively to the earth.
Under these circumstances the form of the flight path appears as
an orbit, commonly of oval form, approximately elliptical.

The same flight path, if observed under different conditions,
will appear as an undulating curve of a form closely resembling
the phugoid curves of Figs. 37, 40 and 42 ; the difference between

290

SOARING § 157

the undulating path and the orbit is merely one of relative
motion.

In considering the flight path as an orbit it is only necessary
to suppose that a plotting be made on a co-ordinate chart
travelling in the mean direction, and with the mean velocity of
the aerodone in flight. Thus, in the rudimentary case considered
in § 154, if we suppose such a plotting to be effected, it is evident
that the " orbit " will be constituted by an inclined straight line
drawn to bisect the angle made by the component elements of the
saw-tooth flight path, that is, the angle A C A\ of Fig. 184 : for,
since the velocity is taken as constant the time taken for the
aerodone to travel from A to C, and from C to A will be equal

to the time taken for the co-ordinate chart to travel from A to D,
and from D to A, part for part, this relationship being ensured
by the construction given. It may be observed that by this con-
struction the " set-back " angle of the straight line orbit path is

equal to ~.

The condition V = constant, assumed in §§ 154 et seq. 9 can
only apply when the size of the individual path undulations is
vanishingly small. In practice, in considering the form of the
orbit we cannot afford to ignore the variations of horizontal
velocity which are due to the interchange of kinetic and potential
energy.

If the motions of the aerodone or bird in its flight path were
the natural oscillations of the Phugoid Theory, we know that the

291 u 2

§ 157 AERODONETICS

variations in the horizontal velocity would result in an orbit

whose horizontal amplitude is - . - of its vertical amplitude. If

V 2

we suppose that the oscillations of the soaring path are phugoid
oscillations, then we must evidently consider the horizontal com-
ponent of the phugoid oscillation superposed on the inclined line
of Fig. 184. This is shown in Fig. 136, although owing to the
artificial character of the initial assumption, i.e., the saw tooth
flight path, the figure can only be regarded as diagrammatic.

It is evident, at least in the case of a bird, that the flight path
oscillations may be other than those of phugoid theory; in fact,
by variations in the adjustment of the organs of flight, a bird

Fig. 135.

may for a given velocity (of flight) cause the undulations to be of
greater or less length than that resulting from the phugoid
equation.

Now the proportions of the orbit, on the approximate basis of
§§29 and 80, are related to the values of H n and the phase
length. It may be readily demonstrated on the lines of § 30
that, on the assumption of an elliptical orbit, the ratio of the
vertical to the horizontal axis is given by the simple expression,

vertical axis 4 it H n

horizontal axis L

where L is the phase length. This is represented in Fig. 185
by a number of examples, giving the proportions of the
elliptical orbits in several different cases. In setting out this

292

SOAKING

§157

diagram a fixed length is taken as representing the value of L as
indicated by the undulating line, and different values of //„ are
defined by the numbered datum line3 ; the vertical axis of the
ellipse has been drawn in each case as equal to //„ the horizontal

i being a constant quantity equal to

- in accordance with

the equation. Obviously no attempt has been made to show the
orbits of a proportionate or suitable size. The construction given
enables one to readily visualise the type of orbit proper to any
given LjH n ratio, without actually plotting the ellipse.

Referring to Fig. 135, it will be seen that datum line No. 1
gives the orbit proper to the phugoid night path, of ratio V 2 : 1.

Fio. 137.
Datum No. 2 gives an orbit whose ratio is 2 : 1 ; in No. 3 the
ratio is 8 : 1 ; No. 4 gives an ordinary trochoid; here the axes
are equal and the orbit is consequently of circular form ; similarly
in the other examples.

It has already been remarked that u bird or machine with
adjustable parts may have the power of varying the proportions

2yjt

§ 157 AERODONETICS

of its orbit at will, by carving out a curve of longer or shorter
phase length for a given value of If n9 and the soaring orbit may
not be by any means of constant form. Thus in Fig. 186, three
examples are given in which (1) represents the orbit of the phugoid
path, and Nos. (2) and (8) represent cases where the phase length
is artificially shortened so that the ratio of the axes becomes 2 : 1
and 8 : 1 respectively.

The present method of delineating the orbit becomes more
appropriate and exact when we deal with the sinuous form of

Fio. 138.

flight path discussed in §§ 155 and 156. If under the conditions
of § 156 we examine the problem on the assumption of uniform
velocity of flight, that is, on the supposition that the phase length
is very small in relation to H„, we find that the form of orbit
will be a line having a general backward inclination, the maximum
set-back angle being tw 7 ice as great as in the case of the saw-tooth
path, that is to say, it will be equal to the gliding angle y, but
the angle will diminish towards the upper and lower pointp,
where it will terminate vertically. As a rough approximation

291

SOARING § 157

we may represent such a path by a straight line having the same
mean set-back, and then add the superposed horizontal com-
ponent as in the previous example ; we may then draw the
resulting orbit forms given in Fig. 187, which otherwise
correspond to the similarly numbered diagrams in Fig. 136.

It is of interest to compare the orbit thus theoretically deduced
with the form as known to observation. Fig. 188 represents tho
orbit as given by Baste ; l this diagram is in close agreement with
the theoretical result, the only difference appears to be that the
set-back or slope of the major axis appears to be greater than
we have deduced. This may be due to the gliding angle of the
bird on which Baste made his observations being greater than

Fig. 139.

has been supposed in setting out Fig. 137, or the difference is one
that might easily be due to the observer (Baste) having recorded
an exaggerated impression of the feature in question. It is also
possible that on the occasion or occasions when Bast6 made his
observations the wind velocity was greater than the ordinary
flight velocity of the bird in question, but that associated with
this wind was an ample supply of fluctuation energy. Under
these circumstances the bird would evidently be induced to employ
a velocity of flight in excess of the ordinary which would account
for the extravagant gliding angle betokened by the set-back angle
recorded.

The major and minor axes of the ellipse or oval as observed
by Baste, appear roughly to be ratio of 2 : 1, which corresponds

1 Le Vol ties Oiseaux ; Marey, § 15.

295

§ 157 AERODONETICS

to a phase length less than that of the phugoid path the pro-
portion of a/2 : 1 ; this is given in diagram No. 2 of Figs. 136
and 137.

It is possible that the difference in the proportions of the
observed ellipse and that of the phugoid equation is due to wing
flexure, and not upon any conscious action of the bird in flight. 1

A photographic record has been made by Marey 2 of the motion
of the ball in Bazin's " switch-back " model; a diagram of this
is given in Fig. 139. It would appear to denote that the velocity
of the ball in relation to the length of the path undulations was
too slow when this record was made to properly represent the
actual conditions of soaring. The orbital path as recorded
corresponds approximately to orbit No. 6 in Fig. 135.

§ 158. Dynamic Soaring in its Relation to the Phugoid Flight
Path. — Let us examine the elementary case of §§ 154 — 5, in the
light of the Phugoid Theory ; that is to say, let us suppose that
the wind pulse consists of an alternation of gusts of uniform
velocity.

We will assume that the period of the wind alternation corre-
sponds to the period of the phugoid curve, and that the two are
arranged in respect of phase in accordance with the requirements
of soaring theory.

The present point of view is of interest as bearing on the
possibility of automatic soaring. The author has on several
occasions, when experimenting with mica models in suitable
weather, witnessed true soaring evolutions, models having stayed
in the air and risen to a considerable altitude, the motions being
un distinguishable from those of a living bird.

Let us suppose, in Fig. 140, that an aerodone when gliding
uniformly, encounters at the point B an adverse gust, so that
the value of // is increased, and let the increase be represented
by a transference of the datum line from x\ to x*, then the flight

1 Compare § 116.

2 Le Vol des Oiseaux, § 192.

296

SOAKING

§158

path passes from a straight line jh to a phugoid curve j> 2 . We
now arrive at the point C one half phase further along, and the
wind reverts to its former velocity, so that the datum line
descends. Now if movements of the datum were proportional
to changes in the wind velocity, the datum would return to its
original position, and nothing would have been gained, hut such
is not the case. The movements of the datum are due to changes
in the value of // consequent upon the changes of velocity, that

JC,

ft

Fia. 140.

is to say, consequent upon the change of the velocity of the
aerodone relatively to the air.

Now 2 tj II = F 2 . /. ^ = -

d \ if

and approximately for small finite increments,

AH V

V

A V

we may take A V to represent the change in velocity that takes
place periodically, and A // the corresponding variation in //,
and we see that the value of the latter is proportional to the
velocity for the time being, so that in the changes in the datum

•

of Fig. 140 the rise will always be greater than the fall. The
same fact may be rendered self-evident from the graphic con-
struction given in Fig. 141, in which distances along the line

297

§158

AERODONETICS

x represent velocities and the squares on these distances II
values. Let the velocity V undergo an increase from xi to # 2
= A r, then the change in H and rise in the datum is represented
by the shaded area. Now when the aerodone is " cresting " let
V undergo a decrease of the same value, from x 9 to x^ then
the decrease of // and therefore fall of the datum is represented
by the corresponding cross hatched area, the net gain in altitude
is the difference of the two.

Thus, referring again to Fig. 140, the fall from x* to x 3 is less
than the rise from x\ to ar 2 , and so at every oscillation there is a

W///////////////////////// A

OC

Fig. 141.

gain of altitude. If we take account of the fact that the aerodone
experiences resistance to flight, we can exchange this gain of
altitude for the necessary propulsive force ; in other words, the
energy gained from the wind instead of being stored as potential
is expended in flight.

Ro far the result is precisely that which might be anticipated
from the previous investigation, but we have now to encounter a
difficulty of a serious kind. It will be noticed on reference to
Fig. 140, that not only does the datum line (which is a measure
of altitude) rise as the net result of the wind pulses acting on

2U8

SOARING § 158

the aerodone in flight, but also the amplitude of the curve
increases at every step, consequently if nothing is done to damp
down the phugoid oscillation, the aerodone will sooner or later
find itself in the danger zone and instability will result.

Now we know that the resistance to flight is a potent factor in
the damping of the phugoid oscillation, and since the increase of
amplitude is caused by the same agency as that by which the
resistance is overcome, there would seem at first sight to be some
chance of the damping rate automatically varying in the required
degree ; it is questionable, however, if any such conclusion would
be justified either from theoretical considerations or from observed
fact. The author has endeavoured to formulate a regime under
which the phugoid oscillation may be used as the basis of soaring
flight tvithout an increase oj amplitude, but the difficulties have so
far proved too great. In fact if the author had not himself
witnessed the phenomenon of a simple aerodone soaring
dynamically under conditions that appear to leave no room for
doubt, he would certainly have been incredulous. 1

Let us examine the experimental conditions. The model
employed was one of about five grams weight (Figs. 56 and 57),
with a comparatively low damping capacity (about c or d, Fig. 109).
The natural velocity was 14 to 15 ft./sec. and the gliding angle
approximately one in five, i.e., y° = 11° ; the velocity of descent

1 In the rammer of 1905, when exhibiting model A 2 (Figs. 56, 57) to
Mr. C. W. Dixon, at his residence (" Westbourne," Edgbaston), the author
observed on several occasions that the flights were abnormally prolonged.
On one occasion the model, launched from about 7 ft. above the level of the
lawn, soared for some considerable period, rising at times to a height of
about 20 ft., eventually alighting in a tree 13 ft. above the ground level.
The model had already been in the air a long while before the time was
taken ; the actual period recorded was 11 seconds. Since the gliding path,
in the ordinary way, occupied about three seconds, tho probability is that
the total time of flight exceeded one quarter of a minute.

There was a light wind with fitful gusts at the time of the observation in
question, and the path of the aerodone was of a switch-back type, giving it
every appearance of being a living bird.

There were present, besides the author, Miss Dixon and Mr. C. W. Dixon.

299

§ 158 AERODONETICS

was therefore about 8 ft./sec. In order to soar this model seemed
to require a light gusty breeze, but under the best of conditions
anything in the way of a prolonged flight such as that recorded
was a matter of so rare occurrence as to suggest that the
fitting in of the wind pulses and the phugoid oscillations may
have been a matter of pure chance. Even if such be admitted,
and if subsequent experiment show that the occurrence of
abnormal flights is a matter that can be explained by the
doctrine of chances, the whole difficulty is not removed, for
although the flight path consisted visibly of a series of undula-
tions of considerable amplitude, much as in curve 5 of the
phugoid chart, the aerodone never once lost its equilibrium or
performed " tumbler " evolutions.

Only two possible explanations of this difficulty suggest them-
selves. Either the case was not one of dynamic soaring ; or, the
damping action increases very greatly with the amplitude of the
path, so that although the oscillation damps very slowly for
small amplitude, when the amplitude is great the damping is
sufficiently rapid to prevent the flight path from entering the
danger zone.

The first explanation, although possible, seems in every detail
contrary to the impression of those present, the first few cycles
of the motion were visibly dynamic soaring; the alternative
explanation is one which requires proof, either experimental or
theoretical, and is somewhat opposed to other of the author's
observations. The difficulty is one that must be left for future
investigation.

§ 159. The Relation between the Energy of Turbulence and the
Velocity of Flight. — In § 151 it was shown that in order that there
should be a sufficient store of energy in the wind for the flight
of a given soaring bird, having a known gliding angle, the mean
velocity of turbulence must reach a certain minimum value, this
value depending upon the nature of the assumption as to the
form of the disturbance to which the fluctuation is due. There

300

SOARING § 159

is no condition as to the flight velocity or the relation of the
flight velocity to that of the aerial disturbance.

In the method of the preceding sections (§§ 154 et seq.), in
which the manner of the employment of the turbulence energy
is taken into account, we iind as a result that the velocity
of flight and that of turbulence are related ; in other words, if
soaring is to be possible the velocity of the wind fluctuation must
exceed a certain minimum value in proportion to the velocity of
flight.

Now these two results do not on the face of it appear to be in
harmony, and it might readily be supposed that they represent
two separate conditions that require to be fulfilled. Such is not
the case : a careful examination of the hypothesis in both cases
shows that in reality there is no want of agreement ; the two
conditions represent two aspects of the same problem and they
are entirely in accord.

In the investigation of § 151 the basis is that whatever the
velocity of the bird or aerodone may be, the value of y is known.
This is a convenient assumption, for y is a quantity whose value
is always known within fairly close limits, and which varies but
little with the velocity of flight, and which is not very different
in birds of different species. The investigation proceeds on the
basis of the known data of the aerofoil, and if changes of velocity
are contemplated the above assumptions involve, firstly, that the
aerofoil undergoes no change either as to its sweep or its
periptery ; and secondly, that in the face of changes of velocity
the conditions of least gliding angle are complied with ; that is
to say, the form of the aerofoil remains unchanged, but the
weight supported increases as the square of the velocity. 1

Now if we modify the above conditions to comply with the
hypothesis of the switch-back method of investigation, we have
firstly to suppose that the area be made to vary with variations
of velocity, instead of the weight being variable, and in order
that the pressure-velocity relationship of Vol. I., Table IX.,

1 " Aerial Flight," Vol. I., Aerodynamics, § 169 ; also Chapter VIII.

301

§ 159 AKRODONETICS

shall be complied with the area must vary inversely as the, velocity
squared. But the sweep and peripteral area vary with the
aerofoil area (the proportions remaining the same), so that we
shall have the sweep and the peripteral area varying inversely
as the square of the velocity of flight. And if, in accordance
with the conditions, the gliding angle remains unchanged, in
order that soaring should be possible, the turbulence energy
of the wind per unit distance must also remain unchanged.
Hence the square of the velocity of the wind fluctuation must
increase in the proportion that the peripteral area and sweep
are diminished. Therefore the velocity of the wind fluctuation
must vary directly as the velocity of flight. Thus the total
turbulence energy of the air handled, and the energy available
according to the switch-back theory, for an aerodone or bird of given
weight both depend upon the existence of a definite relationship
between the velocity of wind fluctuation and the velocity of flight.

§ 160. The Efficiency of Dynamic Soaring. Energy Available in
Terms of Total Energy of Turbulence. — We have now investigated
the question of dynamic soaring from two different points of
view; firstly in § 151 we have dealt with the quantity of energy
existing in the wind in a possibly available form ; and, secondly,
we have in §§ 154 et seq., on certain simplified hypotheses, found
a means of expressing the quantity of energy that is usefully
employed. We now require to find an expression for the latter
in terms of the former.

Owing to the widely different lines of argument adopted in the
two investigations, the results are not immediately comparable,
and further treatment is necessary. On the zigzag basis, § 154 : —

Let E = internal energy of wind per unit distance, i.e. 9 per
foot of flight path.
,, A = area of aerofoil.
,, A i = area of stratum dealt with by aerofoil.
v = a uniform velocity of wind fluctuation.
V = velocity of flight along the zigzag flight path.

302

SOARING § 160

Let W\ = normal component of weight sustained (poundals).
,, y x = theoretical least gliding angle, as in Vol. L, Table

VI.
„ y = actual (experimental) gliding angle.
„ p = density of air.
„ k and c be aerodynamic constants as in Vol. I., § 181,

Table III.

Now

E = ^, (1)

and on the basis that the whole energy of the peripternl area is
available,

A\ = K

1 — e

. . by (1) £ = — a (1 — e) • (2)

Now by Vol. I., § 185

A V
where by § 181 y! = (1 — e) /3,

or /3 = yi

1 — e

or

H'i 1 + t

-AT* = pKyi 1- f

_ U\ (1 - €)

p « yi (1 + F

a

••• ^ (2) E= ^ lP Ml+e )(l-,^

2 /) k yi (1 + c) (1 — €) M

r _ *.a

Wit

~~ 2 yi r 2 '

But the real y is greater than the theoretical y (= yO. Let
us write y = n y t where n is a constant, or yi = -.

808

§ 160 AEE0D0NETICS

And we will arrange the expression as representing energy per
foot per poundal sustained, thus,

E _ n r a

But of this the energy usefully employed is that which would
propel one poundal through one foot in the line of flight, that
is = y. Hence the efficiency, i.e., the energy usefully employed
in terras of that in the air handled is,

n v 2

n \ v )

And for the conditions of best efficiency we know that for all

v V
useful values of y, the quantity - — is approximately constant

= -5 (§ 155, footnote to Table L), or ( y -^- \ = "25.

Now the best value of y obtainable in practice is commonly
from 1*5 yi to 2yi ; that is to say, n = 1*5 to 2, so that in general
the soaring efficiency l will be from about "83 to '25.

1 If the theoretical value of y were actually realisable the soaring efficiency
on the basis of the preceding section would become *5, the bird thus employ-
ing usefully ono half of the turbulence energy contained in the wind coming
within its grasp. The reason of this limitation is not altogether clear.

It is of interest to note that, neglecting body resistance, the energy utilised
in soaring flight in covering a given horizontal distance is, under the conditions
of least resistance, equal to that expended in gliding uniformly over the same
distance. This result is unexpected, but it arises from the fact that the
reaction normal to the soaring flight path W\ is less than the weight W.

Let E\ and E — energy required to cover one horizontal unit distance

soaring and gliding respectively. Let be the angle of the flight path to

the horizontal, as in Chaps. II. and III. ; and let the aerofoil area be supposed

correctly proportioned for least resistance, so that the angle y will have the

same value in each case ; then

E = 7 W (I)

E\ = y — r (^J

1 ' cos v '

But by resolution of forces ~j~ = cob 0, or Wi = W cos 0, substituting in
(2) E\= y W ™- e = y]V=E.

' Ci»S

804

SOARING § 161

§ 161. Efficiency of Dynamic Soaring (continued). — The theo-
retical computation of least y values in Vol. I., Chapter VIII., is
founded on a hypothesis of an incomplete and imperfect kind. 1
It is for this reason that there is so large a discrepancy between
the theoretical and actual values, apart from the influence of
aided surface (comp. Vol. I., § 161).

It cannot be pretended that the foregoing investigations
on dynamic soaring are of an altogether exact kind. In view
of the difficulty of attacking problems in aerodonetics by direct
analytical assault, and the comparative impotence of the most
advanced mathematical methods in the face of the simplest of
problems in real fluid dynamics, some kind of stop-gap theory
giving results of the right order of magnitude is the most that
can at present be expected, and as such the theory here presented
appears to be satisfactory.

As a numerical example we may take the hypothetical albatros
of § 151. Let us take the velocity of flight V = 50 ft./sec.
(Vol. I., § 187).

By § 155, since y = ^ 7 = 7

2 100

or v = = V = -=- = 14 ft./sec. (approximately).

By § 151, on the peripteral area basis, the velocity of wind
fluctuation required to give the necessary energy is 6*25 ft./sec.
And efficiency will consequently be .

( -— j = -2 (approximately).

But if we take the value of y\ given in Table VI., " Aero-
dynamics," on basis f = "02, n = 12 (to harmonise with V = 50
ft./sec. as in § 187), we have theoretical gliding angle = 1 in 16'8

(Table VI.), or n = -^ = 24 ; .*. efficiency = - = -208, which

is in approximate agreement with the previous determination
based on the two separately computed values.

1 "Aerial Flight," Vol. I., Aerodynamics, § 172.
A, 305 X

§ 161 AEKODONETICS

The above is in no sense an independent check on the theory,
it is merely a numerical illustration based on a hypothetical
example.

The assumed value of the " actual " gliding an^le of the
albatros is possibly in excess of its true value, the difference
between the values 1 : 7 and 1 : 16*8 suggests that this may be
the case; it is, however, equally possible that the "plausible
values" given in the tables of constants in "Aerodynamics''
give too high a theoretical value for an aspect ratio n = 12; the
question is only one that can be settled when more accurate
experimental data are to hand. In any case it is most unlikely
that the true gliding angle of the albatros is less than 1:10, it is
in fact improbable that so small an angle is reached by any bird
or other flying creature. 1

§ 162. Dynamic Soaring as Determined by Different Kinds of
Aerial Disturbance. — Dynamic soaring is a phenomenon that
presents itself in much diversity of form, and the different
types of soaring of this class depend upon certain quite distinct
kinds of aerial disturbance.

In one variety of dynamic soaring, such as witnessed by the
author from the ss. Germanic (§ 142), it would appear that there
is a periodical wind fluctuation only remotely connected, if con-
nected at all, with any terrestrial obstacle. In this variety it
may be at least provisionally supposed that the fluctuation is that
of turbulence pure and simple, and the wind consists of a u pack "
of vortices consisting of cyclic motion containing rotation, and
that the flight path of the bird in traversing such vortices meets
with masses of air moving with different velocities of which it is
able to take advantage in the manner already set forth. It is
true that in the example cited there is considerable evidence that

1 The only estimate extant of the flight angle of a bird that can be
considered roliable is that of Bretouniere, who, from observations made on a
number of storks, concluded that the gliding angle of this bird is 10°, or 1
in 5*7.

306

SOARING

§162

the wind pulsations possess some definite periodicity, and this
may betoken that their existence is in some way associated with
the ocean waves beneath ; we have, however, at present no proof
that such is the case.

The frequency and magnitude of the wind pulsation has been the
subject of an investigation by the late Professor Langley, 1 who by
means of anemometers of special construction showed that the
fluctuations are of very considerable magnitude, more so than had

o

z

K
U

I*
hi

-I

Fig. 142.

been currently supposed. In Fig. 142, which is a plotting given
by Langley. it will be observed that in a wind having a maxi-
mum velocity of about 40 miles per hour there is a fairly
constant fluctuation having a range averaging some 17 to 18
miles per hour, that is, about 25 f t./sec. Although the proximity
of the observations is not sufficient for the periods recorded to
represent those of most useful frequency from the point of view
of the soaring bird, it is evident thai; the fluctuations that exist

1 " The Internal Work of the Wind," Smithsonian Institute.

807 x 2

§ 162 AEROPONETICS

are of ample magnitude to provide the energy necessary to the
soaring bird in flight ; in the example cited the velocity fluctua-
tion is approximately twice and the energy 4 times that
necessary, on the -hypothesis of uniform motion, to permit of the
soaring flight of an albatros. If we take the harmonic basis
the energy is still considerably greater than that required, after
allowing for the loss of efficiency on the lines laid down. It
is evident that a wind of far less internal energy than that given
by the plotting will suffice for an albatros in flight.

It may be remarked that the records relate to the changes in
the wind velocity in respect of time at one fixed point, and they
do not, therefore, represent precisely the changes that a bird
would have experienced in the course of its flight; evidently,
however, although the effective fluctuation would be different, its
general character and magnitude will be unaffected.

The relation of the fluctuation velocity to the mean velocity or
" velocity of translation " is a matter of some interest. It would
appear that the greater the mean velocity the greater in proportion
is the velocity of fluctuation ; this point is one on which Langley
makes comment as follows : — " A prominent feature presented by
these diagrams is that the higher the absolute velocity of the
wind, the greater the relative fluctuations which occur in it. In
a high wind the air moves as a tumultuous mass, the velocity
being at one moment, perhaps, 40 miles an hour, then diminishing
to an almost instantaneous calm. . . ."

It is very questionable whether the relation between the
translational and fluctuation velocities of the wind depends alone
upon the absolute magnitude of the former, even at altitudes that I

uould preclude the possibility of local disturbance due to obstruc-
tions ; nothing is known with certainty on this point. Mouillard,
who was a very keen observer, noticed that in the soaring of
birds, " Les vents n'ont probablement pas tous la meme puis-
sance de sustention. II semble, en regardant attentivement les
oiseaux, qu'il y a des jours ou l'air porte mieux que d'autres." It
must certainly be supposed here that the proviso for a given

1*08

SOARING § 163

velocity is understood, bo obvious a fact would not otherwise have
been recorded ; on the other hand it may have been a difference
of vertical velocity component that gave rise to the observed
variation.

9 163. Dynamic Soaring; as dependent on Dead-water Region. —
In all probability many of the cases of dynamic soaring that have
l>een observed depend upon the direct utilisation of the aerial
disturbance set up by an obstacle, the flight path of the bird
lying in part in the wake region or dead water and part in the
live stream. In such cases the want of uniformity of the wind

exists as a variation of velocity as a function of ylace instead of
as a function of lime.

In Fig. 143, let A li represent in plan an obstacle of some kind
causing a system of flow of the discontinuous type, and let
}' Pi Pi Pu be the flight path of a bird, part of which, pp, is situated
in the live stream and part,»i pi, in the dead-water region. Then,
the centrifugal component of the reaction of the bird in flight
will tend, during the portion of the flight path p p, to diminish
the velocity of the live stream, and during the portion p t p. to
impart velocity to the dead water. Thus the motion of the bird
in its flight tends to equalise the velocity of the air coming within
the peripteral system, and so the conditions of dynamic soaring
!)0D

§ 163 AERODONETICS

are fulfilled. The minimum wind velocity required under the
present conditions can be readily computed from § 151.

§ 164. Dynamic Soaring. Mixed Conditions. — In cases such as
discussed in the preceding section with reference to Fig. 148, it
frequently happens that the obstacle by which the dead-water
region is maintained gives rise to an up-current ; in fact it may
be said that this always occurs to a greater or lesser degree.
A consequence is that soaring in the neighbourhood of an
obstacle is usually of a nondescript kind ; sometimes the bird
can be seen to rise evidently borne aloft by an uprush of air, at
other times it becomes equally evident that the soaring is of the
dynamic kind, and that the dead-water and eddy currents are

SS^^.^OQ

Fig. 144.

being utilised. It seems evident that many of the soaring
birds are able to trap the available energy of the wind in
whatever shape it presents itself, and it is almost certain that
many of the larger soaring birds that habitually make use of
natural up-currents are not above taking any opportunity of
soaring dynamically that may present itself. Where a wind
exists with an upward component, having at the same time
horizontal fluctuations, it would seem from the remarks of many
observers that the soaring is a mixture of the up-current and
dynamic varieties. It is possible that when the conditions are
not sufficiently favourable for either kind of soaring separately,
by employing both the bird may be enabled to maintain itself in
the air.

The form of flow of the air current or wind in the vicinity of

310

SOARING § 164

deep sea waves is, in all probability, of the discontinuous type,
especially when the amplitude of the wave motion is considerable.
If this were otherwise, i.e., if the flow were of the Eulerian form,
it would appear that a wind in the contrary direction to the
waves would be of the same effect in maintaining the wave
motion as a wind in the same direction, except for the imme-
diate effect of surface friction. We know that this is by no means
the case ; a wind to be effective in creating or maintaining wave
motion requires to blow in the same direction as the wave motion
itself; hence the motion of the air involves discontinuity.

We may take Fig. 144 as representing the type of flow as we
may credibly suppose it to exist, each wave crest giving rise to a
surface of discontinuity and a dead-water region, the vicinity of
the surface of discontinuity becoming a region of turbulence as
indicated in the figure.

It is evident that if the system of flow is as thus depicted, a
soaring bird will have an ample supply of energy at command,
available by playing off the live stream against the " dead-water "
as in examples already discussed, a possible flight path being
indicated by the dotted line p\ p 2 p 3 , etc. Thus let us suppose
that the bird descending in the region p\ and gathering velocity
from the wind, enters the " dead-water " region from which it
emerges at p*, it can evidently during this period impart much
of its momentum to the "dead-water," and, as it were, leap
once more into the live stream p* p a , where it again gathers
momentum to repeat the operation at the next wave crest. The
flight under these conditions will become a series of leaps, in
which most of the sustentation will take place in the. portion of
the flight path, p\ p* and p& p i9 etc., and comparatively little in
the portions p* p 9 , etc.

It is probable that there are other ways in which a bird could
make such a system as that depicted serve its purpose, but the
one illustration will suffice.

The above exposition, although perhaps not the whole story,
would appear to be at least a possible explanation of the bewilder-

5)11

§ 164 AERODONETICS

ing mystery that attaches to the flight of the albatros and other
marine species that skim and sail perpetually over the surface of
the ocean, and whose apparent independence of anything in the
nature of a source of motive power, has been for centuries the
subject of comment and wonderment.

312

CHAPTER X

EXPERIMENTAL AERODONETICS

§ 165. Introductory. — The present subject is one of quite
recent growth, in fact, it would appear that no comprehensive
attempt has previously been made to deal with the question of
stability in flight either by theory or experiment. Although
other investigators have worked in the field, notably Penaud,
Hargrayes, L ilienthal , and Pitcher (apart from the builders
of flying machines proper, such as Farman and the brothers
Wright ), the author has no knowledge of any systematic experi-
mental work having been done or published, and the present
chapter consists principally of an account of his own work.

The author's experiments comprise observations on the flight
of aerodones or aerodromes of various sizes and proportions.
The more important of these experiments, so far as data and
results are concerned, have already been given (Chap. VI.), so
that the present account is in the main a description of method
and of the detail construction of flight models.

The construction of the flight models or aerodones, though
originally a matter of comparative simplicity, is a subject into
which, bit by bit, a considerable amount of technicality has
crept, until the building up of a large low-velocity mica model,
such as that illustrated in Fig. 55, is a matter that requires a
certain amount of detailed instruction. Beyond this there are,
in the handling of mica, and in the designing and building of
models generally, many points in respect of which the author's
experience will doubtless prove of value to others who may
contemplate entering this fascinating field of research.

818

§ 166 AERODONETICS

§ 166. Method of Experiment. — The author's method of experi-
ment and apparatus are of the most primitive simplicity ; a tape
measure, a stop watch, and a launching staff (Fig. 2), together
with the flight model itself, constitute the complete outfit.

A necessary adjunct to the above apparatus is a large room or
other enclosed space. Failing this, it is necessary to experiment
out of doors, a procedure that involves many tedious delays,
waiting for suitably calm moments when the weather is favour-
able. At the best, outdoor experiments, unless on a rather large
scale, are unsatisfactory.

Much of the author's work has, in spite of difficulties, been
done in the open, but all the low velocity experiments (under
10 ft./sec.) have been conducted indoors, in most part in a room
approximately 28 ft. by 20 ft., and in a few cases (the models of
§ 72), in a room about 60 ft. by 80 ft. The larger the room
available the better, it is useful to have plenty of width as well
as length, a room 50 ft. wide by 60 or 70 ft. long would, for
most purposes, be found of ample size.

The aerodones may be launched either by hand or by means
of the launching staff. In general, except for ballasted aero-
planes, 1 and for model 3 of very rapid descent, 2 the author has
found hand launching quite satisfactory, but a certain degree of
skill, soon acquired by practice, is necessary, both in order to
avoid giving the model initial rotation and to ensure imparting
to it the correct velocity (= V n ), and gliding angle.

It is evident that any initial want of accuracy in the launching
of the aerodone will affect its flight path, the amplitude of the
resulting phugoid will be greater the greater the inaccuracy ; it
is thus necessary to ignore flights in which the amplitude is

1 The ballasted aeroplane is difficult to launch with certainty by hand,
owing to its having two alternative flight paths, cos ® = + 1. There
is but little difficulty when the launching staff is used.

2 For models of rapid descent the launching staff is convenient, as, by the
greater initial altitude, a longer flight is obtained. In the case of a model
whose y value is small tho flight is limited usually by the length of the
room, in which case the launching staff is of no advantage.

814

EXPERIMENTAL AERODONETICS §167

considerable, and in general the flights recorded by the author in
the measurement of the quantities V n and y have been nice
uniform glides in which the phugoid oscillation if existent at all
is barely perceptible.

From the above considerations it is manifestly desirable to
have a range of flight of at least two or three phase lengths, and
where the amplitude is fairly constant it is evident that if the
length of flight be made an exact multiple of the phase length
(Li), any error due to the phugoid oscillation will vanish, even
though the flight path be of sensible amplitude. With due pre-
cautions the error in the determination of V n due to the method
of launching is far less than the possible error due to the method
of timing by a stop w T atch, and the inaccuracy in the determina-
tion of tan y certainly need not exceed one or two inches out of a
total drop of 80 inches or thereabouts, or a probable error not
greater than 2J per cent.

§ 167. Construction. Materials Employed. — It being necessary
to the designing of any appliance or machine to understand
something of the materials utilised, even if only in order to be
able to calculate approximately the weights of the various com-
ponent parts, a few words are required as to the materials
employed in the making of flight models.

Mica. Sp. gr. 2*8.

The mineral mica is obtained either in more or less regular
lumps or slabs as when first taken from the earth, or as trans-
parent or semi-transparent plates or laminae, the form in which
it as a rule finds its way to the market. There are several kinds
of mica, but the only kind with which we are concerned is the mica
of commerce, i.e., Muscovite mica, frequently but erroneously called
talc. The slabs in which the mineral is found in Russia consist in
reality of rough crystals belonging to the monoclinic system, pos-
sessed of a very pronounced plane of cleavage ; so markedly is this
the case that it is possible to split mica up into laminae in a manner
and to an extent not possible with any other known mineral.

815

§ 167 AERODONETICS

Mica is most conveniently obtained from a mica merchant
ready cleft to some few thousandths of an inch in thickness
and cut to sizes of rectangular form. The lamina employed for
any particular purpose or job is preferably cut from a piece of
about the required size. The cost of large sheets or laminae
increases (as might be expected) in a manner quite disproportionate
to the area.

For the further splitting up of laminae when thicker than
required, a thin blade made of clock-spring, and sharpened to a
point like a lancet, is found to be most serviceable. If the blade
of a knife be used, it is, owing to its thickness, prone to damage
the mica when the laminae are of but a few thousands of an inch
in thickness. A clock-spring of about 1/100 inch gauge is quite
as stout as is necessary for the purpose.

Down to a thickness of about two thousandths of an inch the
behaviour of mica in cleavage is very consistent, and is such as
to suggest that the process of splitting and splitting again
might be carried on ad infinitum. A limit is, however, soon
approached : it is difficult to obtain with any degree of certainty
laminae of much less than 1/1000 inch thick. For a small model
it is very often that a lamina of but half this thickness is desired,
and advantage should be taken of the accidental production of
very thin laminae of this degree of tenuity, such as are sometimes
split off accidentally, these being put carefully on one side until
required.

For many purposes in the making of flight models the mica
is required of varjing thickness in its different parts. Occa-
sionally laminae will split off of markedly different thickness
at one end to the other, and sometimes these may be employed
with advantage. In general, however, it is easier to scrape
the mica down from point to point to the varying thickness
required; examples will be cited where this may be done to
advantage. When mica is less than one-thousandth of an inch
in thickness the scraping requires to be done with great
care, but in such cases resort may be had to etching with

31o

EXPERIMENTAL AER0D0NET1CS § 167

hydrofluoric acid; the process still requires the exercise of
considerable judgment.

The proneness of mica to cleave at the slightest provoca-
tion, useful as it is for the preparation of thin laminae, is a
great source of weakness when the mica is considered as a
structural material. Thus when used for large models it is
necessary to bind the edges in some way to prevent the develop-
ment of cleavages; for this purpose paper or silk may be
employed, made to adhere by ordinary gum or mucilage; in
other cases, especially for outdoor models, where damp is to be
feared, gutta-percha tissue may be effectively used, or the edges
may be treated with a solution of celluloid. 1

A further difficulty that is experienced, owing to cleavage, is
that of the attachment of mica laminae effectively to the shaft

dH^s — s

Fig. 145.

or backbone of the aerodone. However securely a mica plate
be cemented in place, it is always able to escape by cleavage, a
mere film of mica being left adhering to the cemented surface.
This feature is convenient enough when preparing a model for a
trial flight, as the tail plane, fins, etc., can be readily removed
for adjustment, but it is equally inconvenient for the finished
model. The method of overcoming this difficulty is by rein-
forcing the mica plate on the opposite face to that by which the
attachment is to be effected, and then scraping a furrow at the
point of attachment so as to expose the full thickness of the mica
plate to the action of the cement, thus the latter obtains a grip
on the whole on the constituent laminae instead of a merely
superficial hold on the outer surface. The above is illustrated
diagrammatically in Fig. 145.

1 Ordinary transparent celluloid dissolved in a mixture of 10 per cent,
ainyl acetate and 90 per cent, acetone, to the consistency of treacle.

317

§ 168 AERODONETICS

§ 168. Materials (continued). — In addition to the mica used for
the aerofoil, fins, and tail plane, the principal materials employed
by the author are : —

Wood (for backbone), " White- wood " (or " Canary ") planed
to various thicknesses, from about 1 m.m. upwards; also veneer
for use in very small models. The specific gravity may be taken
as roughly = *5.

Paper of various thicknesses, cigarette paper for binding
edges (and covering mica surfaces in some cases), weight about
•0014 gram per sq. cm. or "009 gram per sq. in. Allow when
gummed in place '0018 gram per sq. cm. or "012 gram
per sq. in.

Ballast. — The most convenient form of ballast is made by

Fig. 146.

facing thick lead-foil with gutta-percha tissue, the two being
warmed and pressed together. Stouter and heavier ballast of
the same kind may be similarly prepared of several sheets
arranged alternately. It is convenient to prepare composite
ballast of this kind of some definite standard weight per square
cm., so that any desired addition to the weight of a model may
be made approximately, without resort to a balance. Composite
ballast of this kind has the merit of being very readily applied ;
it is rendered adhesive by warming and sets almost at once.

Split lead shot (fishing shot) may also be used, and are
occasionally found convenient.

§ 169. The Aerofoil. — The aerofoil is constructed of a plate of
mica, which may either be in one piece, or built up of two or

318

EXPERIMENTAL AERODONETICS §169

more laminae joined together, if of any considerable size. For
small models a single lamina of mica of appropriate thickness
cut to the desired shape is all that is necessary; this is tied
and cemented with fish glue or other adhesive to a "bolster"
(Fig. 146), cut to the designed sectional form. 1 When required,
as when working to some specified grading, the curvature can be
maintained and regulated by means of cotton or silk ties arranged
at intervals along the length of the aerofoil, as in Figs. 60, 62,
and 68.

When the author's standard combination of parabolic grading
and elliptical plan form is used, it is a property of the resulting
aerofoil that the radius of curvature is constant at all points
along its length ; thus the aerofoil forms approximately part of

Fig. 147.

the surface of a cylinder. Strictly speaking, it is not a cylinder
but rather part of the surface of a parabolic prism (Fig. 147).

The above fact renders the correct grading of a laminar aero-
foil a particularly simple matter. A supporting pillow should
first be made of somewhat the form of the parabolic prism of
Fig. 147, but divided at intervals where it is intended to arrange
the ties, Fig. 148 ; on this the aerofoil, cut to shape and other-
wise prepared, is placed and secured by winding tape temporarily
round the whole ; the ties are then made fast and the bolster
affixed in position, and the glue or cement allowed to set ; the
temporary winding is then removed and the aerofoil is complete.

The preparation of the aerofoil for the above mounting process
depends upon circumstances ; for very small models it is only

1 Fig. 146 shows a convenient way of forming the bolster. Tho edges of
the wood are first bevelled as shown and the bolster is cut from the bevelled
edge on a fret saw machine.

81 ( J

§169

AERODONETICS

necessary to reinforce the mica lamina at intervals with small
squares of paper diagonally folded and gummed in position, to
prevent the ties cutting into the edge of the mica, Fig. 149 (see
also Fig. 68). In larger models it is necessary to reinforce also
the whole front edge of the aerofoil, either with paper, gutta-percha,
or celluloid, to render it capable of withstanding the impacts
to which it is liable. In cases where it is required to reduce the

Fig. 148.

weight to a minimum, as for low velocity models, the preparation
of the aerofoil is far more elaborate, the whole surface being
scraped to graduate the thickness according to a prepared scheme ;
in this case the scraped surface may require to be in whole or in

<

Fig. 149.

part covered with cigarette paper in order to prevent it from
flaking (Figs. 55 and 60).

Where the aerofoil is built up of a number of smaller laminae,
these are given a coat of cigarette paper in the first instance
where they are to be joined, and are then united with fish glue
or gelatine. The surfaces before being served with the prepara-
tory coat of paper are preferably scraped in an irregular way so
as to expose several layers of the mica to the adhesive, and not to
rely entirely on the surface lamina.

8-20

EXPERIMENTAL AERODONETICS

§170

§ 170. The Aerofoil (continued). — In the preparation of the mica
lamina for the aerofoil, consideration has to be devoted to the
form required along that part where the curvature is given by
the ties. It is evident, if the thickness of the mica is uniform,
that since the only restraint is due to the cotton or silk cord
in tension, the section will be an "elastic curve" (Fig. 150),
instead of the desired arc form. In order to approximate more

Fig. 150.

Fio. 151.

nearly to the latter the thickness is preferably graded, being
left greater in the middle and reduced towards the edges ; the
edges themselves, however, are left full thickness (Fig. 151) to

taVTWT Lt#£

TtC LtH£

Fio, 152.

belter withstand the rough treatment to which flight models
are subject.

The grading of the thickness as above described is carried
out by scraping, the disposition of the thickness in the finished
aerofoil being indicated roughly in Fig. 152, by contour
lines, appropriate thicknesses being given for an aerofoil 14
inches by 2 inches, for a model having a velocity of, say, 10 or
12 ft./sec.

An alternative plan has been tried for maintaining the curvature

a. 321 y

§170 AERODONETICS

of the aerofoil. Instead of tying, as described in the preceding
section, rihs are arranged at appropriately spaced intervals,
shaped to give the desired form, and tied and glued in place in
a similar manner to the central holster. This arrangement,
although quite effective, suffers from certain disadvantages,
not only is the moment of inertia about the axis of flight

Fio. 133.

unnecessarily increased, but the wings are too rigid to escape
injury for long. When the simpler method is used the wings
are very supple ; a model will sometimes fall, apparently
hopelessly injured, Fig. 153, but the instant it is picked up
the wings open out in a most surprising way and are quite

EXPERIMENTAL AEKODONETICS

§171

§ 171. The Pin-plan and Tail-plane.— The fin-plan is constituted
by the two fins, the backbone, and the ballast, thus including the
whole aerodone with the exception of the aerofoil and tail-plane.
These parts are taken as a collective unit on account of the
measurement of moment of inertia.

In the method employed by the author for the determination
of the moment of inertia, the fin-plan is suspended as a pendulum
and its oscillation period is counted; the moments of inertia
of the aerofoil and tail-plane being
separately computed and added sub-
sequently.

The reason that the aerodone is
not swung bodily and the moment
of inertia measured as a whole, is that
owing to the exposed surface, normal
to the direction of oscillation, and the
consequent resistance, the damping is
too rapid to permit of the oscillations
being counted with sufficient precision.
This difficulty would be removed if the
determination were made in vacuo.

The fins are mica laminae cut to the Q \\ \

form required and glued to the back- j, 154

bone ; for small models nothing more

elaborate is required, but for large models, especially where it is
intended that the job shall be permanent, thefront edge is protected
by a paper or other binding, and the attachment is strengthened
by means of a stouter paper reinforcement (Fig. 154). If it is
desired to attain the minimum velocity possible, the fins are
scraped down as thin as consistent with proper strength, the
leading edge is left full thickness, but the mica is thinned off
towards the " trail " and towards the upper extremities.

The tad-plane is a lamina of mica cut to shape and similarly
prepared, i.e., except for very small models the edges should be
bound or otherwise treated to avoid injury. The tail-plane is

823 y 2

§171

AERODONETICS

attached to the backbone by glue or cement of some kind, the
point of attachment being prepared as described in § 167.

The backbone is made either from a single slip of wood planed
down to an appropriate scantling, Fig. 155 (a), or it may be built
of two slips glued side by side with the fins inserted between,
Fig. 155 (b). In the latter case the reinforcement is not
necessary.

The ballast is attached to the front end of the backbone and is

adjusted with the tail-plane temporarily
affixed to bring the centre of gravity into
its correct position, namely, slightly in front
of the geometric centre of the aerofoil.

§ 172. The Measurement of Moment of
Inertia. — The method of measuring moment
of inertia employed by the author is one
that has the advantage of requiring but
little or no apparatus. The " fin-plan "
of which the mass centre has been first
determined, is suspended on a knife edge
(Fig. 156) and set swinging, the time period
being taken by counting fifty complete
oscillations.

The calculation is then made as follows : —
Let I = length of pendulum, i.e., tbe distance from the centre
of suspension to the centre of gravity of the piece.
„ t = time of single oscillation.

„ A = radius of gyration of the piece about its own centre
of gravity.
Then A is given by the expression : —

* = i (*£-')■

Iii the case of the " 12 gram " model (§ 71), two trials were
made : —
(1) I = 2§ inches = '2185 ft.

32i

(a)

(I)

Fig. 155.

EXPERIMENTAL AERODONETICS § 172

Fifty complete (double) oscillations occupied 81*6 seconds, or

t = 316.

A 2 = -2185 ( 322 ' 8 Jf - -2185 )

= -0235.
(2) I = 1-2 inches = '1 ft.

t = *83 (50 double oscillations = 83 seconds)

A 2 = -1 ( 82-2 ^? - 'I )

= '0255.

In the above example a repetition of the experiment showed
the second of the two observations to be the most accurate,
hence A 2 = 025.

The square of the radius of gyration as above ascertained,
multiplied by the mass of the piece, gives the moment of inertia
about its own mass centre. Owing to the fact that the moment
of inertia requires to be measured about the centre of gravity of
the whole aerodone, there must be added to this the moment
of inertia of the mass of the " fin-plan," supposed concentrated
at its centre of gravity, about the mass centre of the aerodone.
This may be effected in the usual manner by adding the square
of the radius of gyration, as determined, to the fequare of the
distance between the centres of gravity, and then multiplying
the sum by the mass of the " fin-plan " to obtain the total value.

As an alternative a piece of ballast may be substituted for the
tail, this brings the centre of gravity of the " fin-plan " almost
into its final position, when the measurement, made in the
manner already indicated, gives the final result including the
tail. The author has employed both methods, the latter is
perhaps a trifle the more convenient.

There is some tendency for the " fin-plan " to turn round into
an undesirable position when suspended in the manner shown in
Fig. 156. The author has tried as a substitute suspending by

325

§172

AERODONETICS

fine silk fibres, but the results are not consistent. It is safest in
any case to make two independent determinations from different
points of suspension, the results of which should agree ; a dis-
agreement not exceeding 8 or 4 per cent, may be regarded
as unimportant : it is difficult to get within this limit with silk
suspension. The author has found the most satisfactory solution

Fig. 156.

to the difficulty is to make the determination in some place
where there is a draught, so that the current of air, by acting on
the "fin-plan" like a weather vane, keeps it in the desired
position.

When the tail-plane is not represented in the "fin-plan"
determination by ballast, it is taken account of by calculation,
its radius of gyration being assumed equal to the distance of
its geometric centre from the mass centre of the aerodone ; its

326

EXPERIMENTAL AERODONETICS §173

moment of inertia about its own mass centre is taken as
negligible.

The moment of inertia of the aerofoil may be calculated from
its mass and its radius of gyration in the usual manner ; the
latter may be assumed as approximately one quarter of the
fore and aft dimension of the aerofoil; it may sometimes be
a little more or less, but the error involved in making the
above assumption is in no case important.

The following is an example of the calculation of the moment
of inertia of the 12 gram model (Figs. 55, 60, and 61), of
which the "fin-plan" computation has already been given.

Grams.

A3

I gram, ft

Fin-plan

. 282

085*

•081

Tail-plane .

. "87

•075

•065

Aerofoil

. 5-40

•00174

•0094

Total t

8-59

•1554

Or, in lb./ft. 2 this becomes

' 1554 = -on.

1214

453

§ 173. Admissible Proportions of Models. — In designing an
aerodone for any given purpose, some of the data are supplied by
the conditions. If for example it be desired to verify the
equation of stability in a room of given size, and it is decided that
a minimum of four complete oscillations of the phugoid curve are
to be observed, the values of J/ tt and hence also V n can be at once
calculated and become part of the initial data. Now within
certain limits the size of the aerodone can be settled independently
of its \\ value, but owing to the considerations discussed in § 82,

■■'■ This is the augmented value of A-, i.e., the sum of the squares of the radius
of gyration about the centre of gravity of the fin-plan and the distance of the
latter from the centre of gravity of the aerodone.

t This is the total weight before addition of ballast : Vide § 174.

327

§ 173 AERODONETICS

a superior limit of size exists. Thus for ordinarily careful work-
manship it is difficult to give a stable flight path to an aerodone

* 2

r 2

whose tail length in feet exceeds —^

Likewise, since the fore and aft width of the aerofoil is limited
when the tail length is limited, and since the aspect ratio cannot
be extended indefinitely, there will be a limit to the total weight ;
this is found to take place when JF(poundals) is round about the

V 6
value, oQ?w)oo' w ^ ere ^ e aspect ratio is about 7 or 8, or in an

F 6
extreme case the value of IF may be made as high as - ^^ It

requires very careful design and exceptional workmanship to
exceed this figure.

If conversely, as is the case where the problem is that of
mechanical flight, the approximate value of W is given ; the same
equation may be employed to obtain the minimum permissible
velocity, thus,

F n 6 = 500,000 IF,

or, V n = 8-9 iTw\

or if W\ be the w T eight in lbs. the expression becomes

V n = 16 a/TFi (approximately).

Thus in the case of a man bearing aerodone, weighing in all
some 220 lbs. (the weight of the Lilienthal machine with
aeronaut), it will be difficult to obtain automatic stability at a
less velocity than 16 X 2*45 = 89 ft./sec. or approximately 27
miles per hour ; or, again, in the case of a flying machine of one
half ton gross weight, V = 16 X 8*22 = 51*5 ft./sec, or, say, 35
miles per hour is the minimum velocity.

There is admittedly much latitude in the value of the con-
stant in the above equations ; the exercise of skill on the part of
the designer and constructor, in combination with the highest

328

EXPERIMENTAL AERODONETICS §174

class material, may doubtless (if required) enable the constants
given to be materially improved ; a limiting condition must soon
be reached, however.

§ 174. The Design of an Aerodone. — In the design of an aerodone
there are two sets of data to be determined, namely, aerodynamic
and aerodonetic. The first, by the aid of the tables given in
Vol. I., may be settled at once ; the weight and velocity {IV and
V H ), being decided to come within the limiting condition of the
preceding section ; the area of the aerofoil for the conditions of
least resistance is given at once by Table X. The appropriate
angle of trail, ft, is then ascertained from Table IV. or V., and the
angle a is obtained from this by multiplying by the constant, «
(Table III.) ; the sum of ft and a determines the form of the
bolster from which the aerofoil derives its curvature.

The next step is to rough out the design and carefully calculate
the weights of the different components, from which a computa-
tion is made of the moment of inertia, and the stability is worked
out from the equation (§ 68).

Having ascertained that the degree of stability is that desired,
or having revised the design to correct the value of the stability
factor as may be necessary, the various parts of the aerodone
are made and separately weighed in order to ascertain that the
estimate is not being exceeded.

The parts are then put together and the results checked from
stage to stage by weighing, and the moment of inertia by the
method of suspension. The first putting together may be for a
trial flight, the parts afterwards being made a permanent job.

The initial design and calculations should be complete in every
detail, the results as recorded, and any alterations subsequently
made, being entered on the drawing for reference.

As an example of the foregoing the "12 gram" model of
§ 72 may be taken. The calculations of the aerodynamic data
may either be made for known mass or lor unknonn mass. The
present model was dealt with by the latter method, and the mass

329

§ 174 AER0D0NET1CS

was subsequently made good by ballast 1 to a value afterwards cal-
culated appropriate to the least stable velocity (presuming this to be
the value required), as determined from the equation of stability.

The aerofoil was taken arbitrarily as 2 feet by 2 inches, thus
n = 12. The plan form was approximately elliptical, 2 and the
grading parabolic. The tail and fin areas, tail length and other
proportions, were settled from rough preliminary calculations.

Aerodynamic calculation : —

Equivalent aerofoil area (Vol. I., § 192),

2 2

= - . x 7z = *22 sq. ft.

By Table IX., taking £ = -025, £ 2 = -0437, or

r

K = 4r 2 = -0487 X '222 = "0097.

W
V

By Table V. (Vol. I., § 181), taking value of £ = -025 as
before, for least resistance, = '27, or mean value for stated
plan form and grading, say, = *24 (radian).
Aerodonetic calculation : —

Tan y (estimated 8 probable value) = '14.
Tail data,

I = -28 ft.
a = -0625 sq. ft.
Constants,

P = '078 •
C = -7 ]

c = 2-27 [ From the Tables, Vol. I., §§ 177, 180.
€ = -75 j
Moment of inertia,

I = -00035 (§ 172).

1 The ballast is added as compactly as possible in tho region of the mass
centre, so as not to materially increase the moment of inertia.

2 The wing extremities were left square instead of being rounded off;
this is now the author's usual practice, see Figs. 60 and 62. The paper wing
tips (Figs, bo and 60) were added subsequently.

8 The value ascertained subsequently by measurement was '13,

330

EXPERIMENTAL AERODONETICS § 175

Now, as in § 79,

jr _ J "00035 (103 + 716) _

Hence least stable velocity = V n = 8 V 1'35 = 9'3 ft./sec.
Aerodynamic calculation : —

IF
Now K = T V- 2 = -0097

I*

.\ IF = -0097 X 9-8 2 = -84.
This is the weight in poundals that should give to the aerodone
in question its least stable velocity as its F tt . In grams this
becomes approximately 1 *84 x 14 = 11*75.

Now the actual weight of the components as designed and
made was 8*59 grams (§ 172), hence the model required 8*16
grams ballast to give it stability.

In the actual trial of this model, the ballast added to give
stability was approximately that calculated, the gross weight
as finally adjusted being 12*03 grams, but the natural velocity
was true to the calculation, the measured velocity being 9*3 feet
per second when the condition of stability was reached. 3

§ 175. On the Angle of the Tail-plane and the Position of the
Centre of Gravity.— Assuming that the aerofoil is of arc form of
section, its mean angle to its direction of motion, that is, the
angle formed by the chord of the arc to the line of flight, is given

PL — n n

by the expression, ^—x — (Vol. I., § 188), and since e = ^

this becomes — s — jS. Consequently if the tail be a purely

directive organ, that is to say, if, for the straight flight path, it
carry no pressure reaction, either positive or negative, then we

might suppose the tail-plane set at an angle = — ^ — /3 to the

1 One poundal is approximately equal to the weight of 14 grams mass.

2 Such exactitude of agreement between the calculated and experimental
values is unusual ; there is in fact a defect of about 7 cent., duo to the error
in the estimated value of tan y.

331

§ 175 AERODONETICS

chord of the aerofoil section (Fig. 157 (&)). But this presumes
that the tail-plane is situated where it will be unaffected by
the " wash " of the aerofoil.

If we take account of the latter we know (§ 68) that the
residual downward velocity of the wake is represented in terms
of the velocity of flight by an angle (1 — e) 0, so that if, as is
usual, the tail-plane be situated in the wake of the aerofoil, the
relative positions will be as represented in Fig. 157 (a), for the
conditions of zero tail reaction.

Such a setting as that indicated in Fig. 157 (a) possesses many
disadvantages in theory, and in practice is found to be unwork-

o-e P

(lj

«c

Fig. 157.

able. All actual settings of the tail-plane require to be made on
the basis of a negative (i.e., downward) pressure reaction.

It is of some interest to note that the positions of the tail-plane
as determined on the assumption that it is situated in the
"wash," or clear of the "wash," of the aerofoil, in both cases

make an angle = — - — /3 with the chord of the aerofoil section,

but in the one case this angle is of positive and in the other of

negative sign, 1 Fig. 157, for (1 — e) /3 — fi = — ^— 0.

We know that in practice the leading portion of the aerofoil
must be modified for the reasons stated in Vol. I., § 191,
otherwise the periptery may become unstable. This is illustrated

1 The acute angle made by the tail-plane to the chord of the aerofoil
section, in Fig. 157, measured positive in a counter-clock sense.

832

EXPEEIMENTAL AERODONETICS §175

in Fig. 158, where it will be seen that the effect is to increase the
angle of the tail-plane as defined.

The possibility of an uq stable periptery is a greater danger
when the amplitude of the phugoid oscillation becomes con-
siderable. In the extreme case, that of the semi-circle, when
the aerofoil is launched by being dropped from rest, it is
evident that the angle of the tail-plane must be positive, for
otherwise when the aerodone is released the aerofoil will experi-
ence a pressure reaction on its leading edge in the wrong
direction, and it will proceed on a flight path of opposite curva-
ture than that intended. The author has frequently made
models perform in this way, the same model being capable of

Fio. 158.

flight either right way up or upside down, in the manner of a
ballasted aeroplane. 1

In order to avoid irregularities of this kind, which apart from
the objection stated are liable to detrimentally affect the stability
of all phugoids wboee amplitude is considerable, the tail-plane
should be given a distinct upward trend, 2 its angle (as above

1 It is surprising sometimes how well models will fly the reverse way up
to that for which they are designed ; when the aerofoil is of true pterygoid
form the velocity of inverted flight is commonly higher than the normal
velocity, but the gliding angle is sometimes quite good.

9 It might at first sight be supposed that the sustaining of a pressure
reaction by the tail would invalidate the investigation of Chap. V., and thus
affect the Iruth of the equation of stability, since in this investigation the
tail was taken as a purely directive organ. As a matter of fact, the modifi-
cation of the conditions is without influence ; the law of pressure on the tail
is that of the small angle l'p = c Q 1\ M , and thus is a straight line law.
Hence the effect of an initial pressure reaction is merely to alter the zero
without altering the influence of a given angular departure.

333

§ 175 AERODONETICS

defined) being made nearly equal (1 — c) /3, Fig. 158, this being
a rough approximation to that which the author has found to
give the best results in practice.

§ 176. On the Angle of the Tail-plane and the Position of the
Centre of Gravity (continued). — Counterpart with the upward
trend given to the tail-plane is the position that must be assigned
to the centre of gravity. It is evident that if the function of the
tail-plane were purely directive, the centre of gravity would
need to coincide with the centre of pressure of the aerofoil. If,
however, the tail-plane is destined to sustain a pressure reaction,
the centre of gravity will need to be moved either forward or
backward to suit according as the pressure reaction on the tail-
plane is downward or upward. And since the pressure reaction
on the tail-plane acts downwards, the centre of gravity must be
displaced forward.

In practice the author has found that for his standard form
of aerofoil the appropriate position of the centre of gravity is
about *4 of the width of the aerofoil from its leading edge, that
is to say, the best results will almost invariably be obtained
when distance of the centre of gravity from the leading edge in
terms of the maximum width of the aerofoil is from 3/8ths to
7/16ths.

This displacement of the centre of gravity is approximately
that which would be expected from the conditions. It would
seem that the tail-plane must from its " setting " be moving at

an angle of at least — ~ — /3 athwart the stream, and that con-
sequently it must experience a pressure approximately — ~ —
times that on the aerofoil, 1 or a total pressure reaction

1 The value of the constant r is taken to be the same for both aerofoil and
tail-plane, an assumption that is onl} T justified by the absence of definite data.
The actual value of this constant must be different, not only on account of the
difference in the value of n, but also on account of the difference between the
aeroplane and the pterygoid form. Comp. Vol. I., § 172, footnote.

334

EXPERIMENTAL AERODONETICS § 176

= T ! t imes fche loa(1 - In Ml© case of fcbe 1 ^ 8 ram
model this will be —

( 1 — -75) X '0fr25 __ ^015(5
2 X *22 ~~ -44

= -0354.

Or the displacement of the centre of gravity requires to be
— *035 I where I is the tail length, or in the case in point,
•0354 X -28 ft. = -01 ft. or -12 in.

This is less than the actual distance between the centre of
gravity and the geometric centre, but the centre of pressure is
in all probability materially in advance of the latter. 1 Thus in
the actual model the distance between the geometric centre and
the centre of gravity is approximately a 2 in., so that the
difference to be accounted for by the displacement of the centre
of pressure is no more than *08 in. 2

1 The centre of pressure of an aeroplane of the proportions of the aerofoil
is very much in advance of the geometric centre (coinp. Vol I., § 148), but
this does not apply to the same extent to an aerofoil of pterygoid form. If
the arc section be that adopted, not modified as in Vol. 1., § 191, and of
uniform form of section as in Vol. I., § 120, then in theory the centre of
pressure should coincide with the geometric centre. Owing to the fact of
the modification of the section by which it becomes of flatter form towards
the extremities, and owing to the curtailment of the forward or dipping
portion in accordance with Vol. I., § 191, already cited, the centre of
pressure will in practice be situated in advance of the geometric centre, but
only to a moderate degree. lXT

* As a matter of fact, if any discrepancy existsjjs in the opposite direction
to that suggested by the above computation ; tne centre of gravity is in
practice not so far forward as might be anticipated from the conditions. The
displacement in the above computation has been based on a minimum
estimate of the inclination of the tail-plane to the direction of the stream ;
if we take the probable value in view of the influence of the wash of the
aerofoil (after due allowance for the curtailment of the dipping edge), it
is at least double that taken, or, say, = (1 — «) g, and the position of the
centre of gravity is more than accounted for without recourse to any allow-
ance for a displacement of the centre of pressure. The matter calls for
fuller investigation.

885

§177

AERODONETICS

§ 177. Methods of Steering. — One of the points in aerodonetics
that is most easily investigated by model experiment is the
steering of an aerodrome or aerodone.

It is often found that a model when first flown has a tendency
to steer either to the right or to the left, the flight path being the
arc of a circle instead of a straight line. So much is this the
case that some adjustment is usually required before it is possible
to obtain flights of sufficiently constant direction for the deter-
mination of the true natural velocity and gliding angle.

If a model that has a bias in the one direction or the other be
examined, it will be found that either the two fins are not truly

Fig. 159.

in line, or that the aerofoil has a twist or " wind." If the former
is the case, the fins should be first set truly in the same plane or
in parallel planes ; ] if after this the direction of flight is not con-
stant, the error is due to a want of perfect symmetry of the aerofoil
and can be corrected by twisting the aerofoil one way or the
other.

When the model is a small one, it is sufficient to breathe on the
glue by which it is affixed to the bolster and carefully apply the
twist by taking the " two wings " one in each hand, between the
forefinger and thumb. When the model is of any size, it is
convenient to arrange a tourniquet as shown in Fig. 159, by

1 It evidently is not important that the tins should bo accurately in the
same plane so long as they are truly parallel.

83G

EXPERIMENTAL AERODONETICS §178

tightening which any desired degree of twist may readily be
applied. 1

It is easy to remember which way the steering twist is required
from the fact that the rotary motion imparted to an aerodone in
flight by a twisted aerofoil is in the same sense as would be the
case if the aerofoil were mounted free to rotate about a vertical
axis, and allowed to fall through the air, acquiring a rotary
motion from its screw-like form ; thus an aerofoil that has a
corkscrew-liko twist 2 causes an aerodone to steer to the right and
vice versa.

When for any reason a curvilinear flight path is desired, it
may be given either by setting the leading fin slightly in the
desired direction, or by giving an appropriate twist to the aerofoil,
or both methods may be employed simultaneously.

§ 178. On the Ballasted Aeroplane. — The ballasted aeroplane
is an article of such simplicity that but few words are necessary as
to its correct proportioning or constructional detail.

In general, much of that already written as to the preparation
and treatment of mica applies without alteration.

Formerly the author was in the habit of employing split shot
for the purpose of ballasting, but more recently this has been
given up in favour of the composite ballast described in § 168,
as being more convenient to adjust and less liable to become
detached.

There is a relation between the size of the mica used and its
minimum thickness; if it is too thin the aeroplanes become
unduly flexible ; if, however, a very low velocity model is required,
a great deal may be done to lessen the weight and retain stiffness

1 It is usually found that a model when first tried has a slight bias in
the one direction or the other, and the tourniquet may then be fitted on
whichever side it is required. If, however, a model is approximately
symmetrical, and it is desired to be able to steer it either to the right or left,
a second tourniquet may be fitted as shown in the figure.

9 The length being taken transverse to the line of flight The same twist
reckoned about the axis of flight is of the contrary sense.

A. 837 z

§ 178 AERODONETICS

by artistio scraping in the manner already described. For
unscraped plates the minimum thickness it is desirable to employ
is given roughly by the expression —

1000 t = V^

where I is the length of the plane, and t the thickness, in like
units of length.

§ 179. Some Vagaries of the Flight Path. — It must not be
supposed from the apparent completeness of the present work
that everything that relates to the flight path of an aerodone has
been thoroughly worked out or explored ; in making experiments,
or even in merely repeating experiments previously made, new
and unexpected points frequently arise, and the deportment of
some particular model under some particular conditions often
seems inexplicable until by careful and repeated observation
some clue is discovered to the behaviour observed. Usually
anything in the way of an anomaly in flight is eventually trace-
able to the conditions assumed for the purposes of theoretical
investigation not being complied with, either as touching the
initial hypothesis or the limitations imposed by the nature of the
investigations.

When a model is constructed to come well within the limits of
stability, both lateral and longitudinal, its behaviour in flight is
usually irreproachable ; it yields results as to time period and
phase length approximately in harmony with the equations of
§ 86, and its flight path is ordinarily quite consistent. It is
when dealing with models whose flight path borders on the
unstable, and whose lateral stability has but a very small margin,
that irregularities are met with and anomalous cases of flight
occur.

Thus, in experimenting with models Figs. 60 and 62, the
" 12 gram " and " 4 gram " aerodones employed for the verifica-
tion of the equation of stability, the time period of the phugoid

888

EXPERIMENTAL AEROPONETICS § 180

curve, when of considerable amplitude, was found to be greater
than that calculated from the natural velocity; consequently
the phase length also was greater. This anomaly is frequently
met with in large low velocity models, and it appears that
it is principally due to the size of the model in its relation
to the flight path, or more correctly the length of the tail
in relation to H n . The author has found that, generally
speaking, so long as the tail length is less than one- tenth of H nt
the agreement between the phase time and length and the
velocity is fairly good ; but if the tail length exceeds this value, an
appreciable error, for paths of sensible amplitude, begins to be
manifest, and when the tail length is made equal to | of H n9 the
error may amount to as much as 30 or 40 per cent, or thereabouts.
An alternative explanation of the abnormal phase length may
be founded on the fact already discussed (§ 115), that the air in
the periptery contains energy, and the influence of this is in
effect to somewhat lower the effective value of g in thri phugoid
theory (§ 20), but investigation shows that the extent of the
probable error from this cause is too small to account for the
observed difference. Beyond this, in watching the models to
which reference has been made, it is possible to see the hanging
up or partial suspension of the motion at and about the crest of
the flight path ; this is the point at which the radius of curvature
is least, and consequently the influence of tail length is most
marked. 1

§ 180. Vagaries of the Flight Path (continued). — When experi-
menting out of doors, the author has sometimes been astonished
by obtaining long, uniform, almost horizontal, glides; in one case
a drop of 2 ft. 6 in. only was recorded in a flight of over 50 feet
length ; this flight was obtained with the 12 gram model of § 72.
Now the actual gliding angle of this model, when carefully

1 The author is not thoroughly satisfied that every case of abnormal
phase length is fully explained as above ; it is possible that there are other
contributory causes. The investigations continue.

339 z 2

§180

AERODONETICS

ascertained indoors (§ 72), is given by the expression tan y = *13 ;
it is evident, therefore, that there is something abnormal about the
above flight in wbich tan y is less than *05.

Owing to the low velocity of the model in question and to the
fact that its flight path is on the verge of the unstable, we know
that this anomalous flight was not due (as in the author's 1894
models) to the launching velocity being in excess of J r n , for an
excess equivalent even to a few incbes of extra " head " (H)
would have been quite obvious from the amplitude of tbe resulting
phugoid-, if not sufficient to actually put an end to the stability.
The explanation is consequently to be sought elsewhere.

Fig. 160.

At tbe time of the experiment there was a very light air moving ;
the velocity, estimated from the rate of drift of the smoke from a
cigarette, 1 amounted to approximately tbree feet a second in the
direction of flight ; this at once accounts for some portion of the

diminution of gliding angle, for the length of the flight through

y
the air was probably but f/ * of its observed length relatively

to the earth— that is approximately f x 50 feet, or say 88 feet.

Now the correct fall where tan y = 13 for a 88 feet flight is
5 feet approximately, so that there is still a discrepancy of 2*5

1 The drifting of a smoke cloud is certainly the most convenient way of
estimating the velocity of very light aerial currents. With two observers
ami a slop watch it is probably also the most accurate; perhaps better results
could be obtained by timing the drift of a soap bubble.

340

EXPERIMENTAL AERODONETICS §180

feet between the actual value and that which would have been
observed under still air conditions.

The next point noted as affecting the conditions was that the
flight took place from an open space towards a building (Fig. 160),
the aerodone striking the wall of the latter about 4 feet above the
ground. Here we find two distinct factors tending to sustain the
aerodone in flight, both contributing to the observed result.

It is in the first place evident that the air in the region traversed
will have an upward component to its motion (comp. Figs. 114
and 115), and thus the apparent gliding angle will be diminished ;
secondly, since the aerodone automatically regulates its own
velocity through the air, and its launching velocity is from the
evidence of its flight path approximately correct, the initial
velocity relatively to the earth will be 3 ft./sec. higher than the
final velocity, 1 and the difference of kinetic energy due to this
velocity difference will appear as a gain in altitude. The exact
gain due to the first of these causes is difficult to assess, but the
second accounts for a gain of about one foot, so that between the
three different causes present to minimise the rate of fall we may
regard the total discrepancy as roughly accounted for. The exact
velocity of the air current might quite well have been as much
as 4 ft./sec. at the moment of launching; fluctuations of this
magnitude are constantly taking place in the velocity of
the wind, even when this velocity is as low as a few feet per
second.

The importance of the present example is as an illustration of
how easy it would be to an inexperienced observer to record, in
perfect good faith, fictitious and misleading results, and, further,
the desirability of conducting all quantitative experiments in a
suitable building. A motion of the air of one or two feet a

1 The launching velocity relatively to the wind was approximately = 7 n
= 9*3 ft./sec. ; to this must be added the velocity of the wind, i.e., 3 ft./sec,
to obtain the volocity relatively to the earth at launching. At the end of
the flight the air lias no horizontal motion, hence tho absolute velocity of
the aerodone becomes V n .

341

§ 180 AER0D0NETICS

second velocity is scarcely noticeable, and yet it may materially
vitiate the accuracy of experimental work.

§ 181. Vagaries of the Flight Path (continued). — In experiment-
ing with the 4 gram model of § 72, Fig. 62, an opportunity
occurred of studying a peculiarity in the flight path already
observed in other cases, but for which no explanation had
previously suggested itself.

In making the quantitative measurements of velocity and
gliding angle it is very important to obtain a directionally
straight flight path, and when adjusting a model in this respect,
the regular difficulty is of course that the flight path, instead of
being quite straight, will follow the arc of a circle of greater or
less radius. By two or three trial flights it is usually possible
(by twisting the aerofoil as already explained) to obtain a flight
sufficiently straight for the purposes of measurement. In certain
cases it may be observed that the flight path will be apparently
straight for some distance and then suddenly change its direction
without visible cause, and a semblance of regularity may often
be noticed in the behaviour of a model under these conditions,
the disturbance taking place at a certain definite point in the
flight path, or one of two definite points.

The first impression one receives is that there must be a draught
or other aerial disturbance at the point in question, but this
theory is soon disproved by the fact that when the position of
launching is changed the critical points in the flight path move
also.

In the trial flights of the 4 gram model it was observed that
the model would frequently make a perfectly regular flight
measuring approximately 88 feet, Fig. 161, b and c, and then
take a sudden turn to the right, but occasionally it would
continue on its flight path undisturbed.

Various theories were tested to account for the peculiarity, but
none would bear investigation. It was then noticed that occasion-
ally a slight change of course would take place at a point about

842

EXPERIMENTAL AERODONETICS

§181

17 feet from the point of launching in the same direction as
before, i.e., to the right, and that a further change would follow
at the usual 33 feet distance, Fig. 161, d ; this fact, coupled
with direct observation, gave the clue to the anomaly ; the
alterations of course took place when the aerodone was cresting,
and although the fact of the flight path oscillation was scarcely
visible (so small was the amplitude), yet the aerodone itself could

Fig. 161.

feel the change in its velocity, and, with a perversity usually asso-
ciated only with animate matter, took advantage of the moment of
its least velocity to behave in a crooked and disreputable manner.
If the foregoing account of the behaviour of the aerodone were
correct, it was evident that the model was being in every case
launched from the crest of its flight path, and that, if the velocity
of launching were sufficiently increased, the points of deflection
should be displaced to fall about halfway between the positions
previously observed ; the experiment was tried with the result
anticipated, Fig. 161, e andy.

343

§ 181 AERODONETICS

The explanation so far is evidently incomplete, for it has not
been made clear in what way the change in the velocity due to
the phugoid oscillation is able to effect a change in the direction
of flight. It seems evident that if the aerodone were rigid in all
its parts it would be impossible to account for a steering effort as
due to a change of velocity ; on the V 2 law the reactions of the
component parts, i.e., aerofoil, fins, etc., should vary precisely as
the square of the velocity, and so their directive value would
vary proportionally. The effect is evidently too pronounced to
be accountable on the basis of the defect of the V 2 law ; hence we
must either regard it as due to the change of velocity in
conjunction with some effect of elasticity, or else as due to
something apart from velocity, say the curvature of the
phugoid path.

Now the effect is ope that it is just possible might be accounted
for on the basis of the path curvature. Let us suppose that the
fins and the aerofoil are so set as to tend to steer in different
directions ; then for a straight phugoid they may exactly balance
each other, and so long as there is no oscillation the direction
will be constant. When, however, the aerodone is cresting, the
fins will be travelling through the air faster than the velocity of
flight, for they are moving at a greater distance from the centre
of path curvature than the mass centre of the aerodone ; conse-
quently their influence will be greater and their steering influence
will preponderate.

If the feature in the flight path under discussion were observed
only when the flight path curvature is considerable, then the
foregoing might be regarded as an adequate explanation, but
such is not the case ; the deviation of the flight path will some-
times make its appearance before the existence of the phugoid
oscillation is, by direct observation, at all obvious. It appears
to the author that the true explanation is most probably to be
found in the elasticity of the aerofoil, on the basis that the two
wing extremities are not equally flexible, and under changes of
apparent load, such as take place in the undulating flight path,

844

EXPERIMENTAL AEIiODONETICS §181

changes in the form of the aerofoil take place that determine the
alterations in the course.

It is evident that a very small change per cent, in the
apparent weight may be sufficient to determine the elastic yield
of the wing terminals, and since the apparent weight is for small
amplitude directly as the value of H> and since lT n is only of
about 16 inches magnitude, a fraction of an inch in the path
amplitude is a quantity that might make itself felt, and is one
that would certainly not be noticed by direct observation.
The yield may be an ordinary elastic yield that is greater in one
wing than the other, or it may be a sudden change like that of
the diaphragm of an oil can, owing to a kinking of the mica
where the approximately flat extremity merges into the curved
section. 1

It is evident that if the aerodone possess a stability coefficient
greater than unity, the deviation may be expected to take place
at the first crest (or trough) of the flight path, or not at all, for
the amplitude will diminish at each oscillation. The model on
which the present observations were made possessed a value of
* less than unity both by calculation and by observation ; hence
the uncertainty as to the point of deviation. The amplitude may
be insufficient to bring about a deviation on the first oscillation,
but it may have augmented sufficiently by the second or a
later oscillation; hence there is some uncertainty as to when
the departure from the rectilinear gliding path will manifest
itself.

When repeated deviations occur at successive crests of the
flight path, the latter becomes of polygonal form ; the most the
author has witnessed is three sides of such a polygon, but

1 In the case of the M 12 gram" model this sudden yielding of the
extremities of the aerofoil by the kinkiug of the mica was actually observed.
This model was originally designed and made with only two ties on each
wing, the outer one being added subsequently to give stiffness to the
extremities, and so remove the defect. In a low velocity model of this size
one can observe every detail in the deportment of the model in a way not
possible in an aerodone of small size or high velocity.

315

§ 181 AEROPONETICS

doubtless with a large enough arena, and a sufficient time spent in
alteration and adjustment, a model could be made to repeat the
performance with regularity. 1

In the case investigated by the author the normal flight path,
when not subject to deviation, curved somewhat to the left; this
is shown in Fig. 161 (a) ; the flight paths showing deviation are
shown dia grammatically, (/>),(c), (d), (e) and (/), the latter being
paths when the aerodone is launched in excess of its natural
velocity,

1 There would appear to be no reason why the change should not be made
to take place during troughing ; except that then the velocity is higher and
the stability is greater, it consequently takes more to divert the aerodone
from its course.

34<)

GLOSSARY

Worth additional to those already given in the glossary to Vol. I. are marked*.

Aerodone (Author), from the Greek d*po-SoVrjros, lit. tossed in
mid air; soaring. To denote a gliding or soaring model or
machine ; in particular, any gliding or soaring appliance
destitute of propelling apparatus or auxiliary parts.

Aerodonetics (Author, see aerodone). The science specifically
involved in problems connected with the stability or
equilibrium of an aerodone or aerodrome, or of birds in
flight, and with the phenomenon of soaring. Equivalent
to Aerodromics, as proposed by Langley.

Aerodrome (Langley), from the Greek aepo-bpopos, lit. traversing
the air; an air runner; originally proposed to denote a
gliding or soaring model or machine, or a flying machine
of any kind. Restricted by the author to the latter
signification ; a fully-developed flying appliance ; a power-
propelled aerodone, or an aerodone furnished with directive
apparatus.

Aerodromics (Langley) originally proposed to denote the
science concerned in the equilibrium, etc., of an aerodrome ;
equivalent to acrodoneties as used by the author.

Aerofoil (Author), from the Greek aipos and QvMov, lit. an
air deaf. Denoting the organ of sustentation of an aerodone
or aerodrome, or the spread wings of a bird. A supporting
member (or members collectively) of undefined form : thus
pterygoid aerofoil, an aerofoil of wing-like form ; plane
aerofoil, an aeroplane, etc.

847

GLOSSARY

Apteroid (Author), from the Greek a, mtpiv and eidos, the
converse of pterygoid. Thu3 apteroid aspect, with the
greater dimension arranged in the direction of flight ; the
reverse to that which obtains in the wing plan-form of birds.

Aspect (Diet.), proposed by Langley in its present usage to
denote the arrangement of the plan-form of an aero-
plane, or other aerofoil, in relation to the direction of
flight.

•Attitude (Diet.), used to denote the position of an aeroplane
or aerofoil about a transverse axis relative to the direction
of flight ; analogous to the term aspect. The attitude of a
given aeroplane or aerofoil is thus defined by its angle :
a change of attitude involves a change in /3.

Ichthyoid (Diet.), fish-shaped, here applied to denote a body of
practical stream-line form.

Peripteral. See Periptery.

Peripteroid. See Periptery.

Periptery (Diet.), proposed by the author in its present usage
as denoting the region round about the wing or in the
vicinity of the aerofoil (Greek, it*pi and irrepor), § 107.
Hence peripteral, as in peripteral theory, peripteral area;
peripteral zone ; peripteral motion, comp. Vol. I., footnote 2,
p. viii. (Preface). Hence also peripteroid motion (Greek,
wept, irrepoV and ei8oy), the form of flow proper to the inviscid
fluid in a doubly connected region, resulting from the
superposition of a cyclic motion on one of translation.
Resembling the motion in the jwiptei'y, lit- round-ubout-
the-wing-like.

•Phugoid Theory (Author), from the Greek <f>vyrj and ei6os, lit.
flight-like. 1 The theory dealing with longitudinal stability
and the form of the flight path. Hence also Phugoid chart,
Phugoid curve, Phugoid oscillation, etc. (Ch. II.)

1 The appropriateness of the derivation is perhaps diminished by the
fact that the word <pvyn means flight in the sense of escupe rather than the
act of flying in the present signification.

348

GLOSSARY

Pterygoid (Diet.), iving-likc. Hence piayguid aspect, wilh the
lesser dimension in the direction of flight, as in the wing
plan-form of a bird.

Sweep (Diet.), proposed by the author in its present usage to
denote the cross-sectional area of the stratum of fluid,
supposed by hypothesis to be that to whose inertia the
supporting reaction is due, § 160.

349

APPENDIX I

THEORY OF STABILITY— PENAUD, 1870

In § 65 reference has been made to the work of Mons. Penaud
in connection with the stability and equilibrium of his flying
models, and in particular to an argument advanced by him
touching the theory of longitudinal stability.

In an article by him published in "L' Aeronaut/ 1 tome v.,
p. 2, Mons. Penaud not only describes two types of flight model,
a " h&icoptere " and an aerodrome (the latter already given in
Figs. 9 and 10), but gives also a full statement of his views of
the theory of automatic stability. The author feels that, in spite
of the rather inadequate nature of the explanation which Penaud
was able to offer, the matter is of sufficient historical interest to
demand its being quoted in extenso. Whatever faults may exist
in Penaud's theory, this much is certain : the man himself had a
clear understanding that an aerodrome could be made possess-
ing complete stability within itself, without any adventitious aid
in the way of equilibrating devices, a fact that since his time
has been almost forgotten.

The portion of Mons. Penaud* s article that has been thought
worthy of reproduction is as follows : —

" Apres avoir varie de toutes fa<jons les proportions de mon
heliocoptere ; apres Tavoir vu voler sur place, planer oblique-
ment, s'elever comme un trait, la pensee me vint tout naturelle-
ment d'appliquer mon mSchanisme a la propulsion d'un
appareil du genre aeroplane, de fa?on a montrer la possibility
de ce systeme comme Thelicoptere d6montrait la puissance

850

APPENDIX App. I.

de Thelice. . . . Heureusement, apres quelques recherches,
j'imaginai un organe tres simple, remplissant le but desire.

" C'est un petit gouvernail horizontal incline vers le dessous
de plan sustenteur derriere lequel il se trouve. Ce dispositif
peut, en definitive, se rattacher a la propriety que possedent
les surfaces convexes vers le bas, de tomber verticalement,
sans chavirer, et a l'aide desquelles M. Joseph Pline reussit
meme a construire de jolis petits papillons qui planent oblique-
ment vers le sol. Mais dans mon aeroplane il-y-a comme chez
Toiseau, separation complete de la surface supportante et de
la surface dirigeante qui, des lors, ne peuvent plus se nuire,

^ — ? - - nv

s
s
s

I 1

I I I

I \c' \o'

rS;

4

Fig. 162.

la translation est rendue sensiblement plus facile; enfin mon
gouvernail represente mieux Torgane evidemment mobile dont
serait pourvu un navire a£rien conduit par un homme.

" Quoi qu'il en soit, la theorie d'un gouvernail fixe regularisant
automatiquement le niveau de Tappareil n'ayant ete exposee ni
meme etudiee par personne de ma connaissance je vais la traiter
ici en quelques mots.

" Soit un plan horizontal c d charge d'un poids place au
dessous de son centre de pression, et lie a un gouvernail g k
incline vers le dessous de sa surface.

" Supposons qu'il se meuve horizontalement avec une certaine
vitesse qui Tamene en c" d" au bout d'un temps donne. Pendant
ce meme temps il sera tombe, en vertu de la pesanteur d'une

851

App. I. APPENDIX

certaine quantite c" c' dependant de sa surface et du poids qui le
charge en sorte qu'en definitive, c d se trouvera au bout du
temps considere en c' d' par suite g k g ( k' ; et Ton con?oit quen
donnant une valeur convenable soit a la vitesse de translation,
soit a la chute verticale, soit enfin a Tangle du gouvernail et du
plan g k puisse se trouver enfin de compte sur le prolongement
de g k comme sur la figure, en sorte que dans le mouvement
reel, il n'aura fait que fendre Fair par sa tranche.

" Supposons maintenant que c d s'incline legerement vers le
sol, sa vitesse augmentera, puisque a la force de propulsion se
joindra une composante de sa pesanteur ; mais alors pour une
meme chute c" c' le chemin parcouru horizontalement sera plus
grand, la position finale de c d et par suite de g k sera a droite
de c d' g k', le gouvernail recevra done le choc de Pair sur sa
partie superieure, il tendra a s'abaisser, et comme Pappareil
tend a tourner autour de son centre de gravity Tavant d se
relevera, et le plan c d reprendra sa position horizontale.
Si le plan tend au contraire a se relever ; sa vitesse se ralentit
aussitot, le gouvernail recjoit Fair sur sa partie inferieure ce qui
ramene encore c d dans sa position naturelle en sorte que
Tappareil se trouve force de descendre suivant la direction g g f .

" Si Ton suppose que c d au lieu d'etre absolument horizontal,
soit legerement incline vers le haut ou vers le bas dans sa position
normale, on peut voir avec un peu d'attention, que les memes
effets du gouvernail se produiront encore de fa9on que Ton peut
obtenir ainsi, soit un vol horizontal soit une montee ou une
descente isochrones sous un angle determine. 1 On peut meme
obtenir tous ces effets en faisant porter au gouvernail lui-meme
une partie du poids en placant le centre de gravity un peu vers
Parriere. La surface supportante est ainsi augmentee, ce qui
est avantageux."

1 In this passage Mons. Penaud appears to overlook the fact that an
upward course requires a supply of energy or a relatively downward course
an increased resistance.

The adjustment suggested would alter the value of V n9 but not the gliding
angle, as Penaud supposed.

352

APPENDIX II

THEORY OF STABILITY; AUTHOR, 1897

The explanation of the automatic longitudinal stability of an
aerodone or aerodrome given by the author in his Patent 8608
of 1897, to which reference has been made in the present work, is
as follows : —

" When a machine constructed as hereinbefore described
travels through the air with a sufficient velocity its weight is
supported dynamically by the reaction of the air on the upper
and under wing surfaces, the curved form of section developing
a region of pressure beneath and rarefaction above in the same
manner as a simple inclined plane, but with the advantage of
greatly reduced resistance in the line of motion. Now if the
velocity of travel is insufficient the air reaction will be deficient,
and if the velocity be excessive the air reaction becomes greater
than the weight of the machine, and consequently there is for
any particular adjustment of a machine a certain speed at which
its weight will just be buoyed up so that (so long as this critical
speed be maintained) its centre of gravity will continue to move
in a straight line ; that is to say at this speed (which has already
been referred to as the ' natural velocity ') there is no tendency
of the course of the machine to change. Now let us suppose
that the machine without motor be launched horizontally at its
' natural velocity/ then at first no change in its direction of
motion takes place, but as its velocity falls owing to the resistance
of the air (frictional and otherwise), it gradually assumes a course
inclined downwards, and after some oscillation finally settles down

a, 853 A A

App. II. APPENDIX

to such an inclination as will by its descent enable it to be
supplied with the energy necessary for its propulsion, the steep-
ness of the descent under given conditions depends upon the rate of
dissipation of energy by the resistance of the machine ; this angle
may be termed the angle of ' recuperation.' If now we suppose
the machine to be launched at a speed considerably above its
natural velocity, then in the first instance it takes an upward
course of gradually increasing inclination till its excess of kinetic
energy has been absorbed as potential, the inclination of its
course then gradually diminishes till it again becomes horizontal,
when, its velocity having become considerably deficient, its course
takes a downward trend, and thus it proceeds to describe curves
in the air of approximately trochoidal form of gradually diminish-
ing amplitude, slowly settling down to its angle of ' recuperation.'
If the launching velocity be excessive (greater than one and a half
times the natural velocity in most cases) the machine runs serious
risk of being capsized, and the limit of longitudinal stability
may be said to be reached somewhere about this point."

" The natural velocity of a machine may be modified at will
by altering the relative angle between the wings and tail-plane,
thus causing the former to meet the air at a greater or less
angle."

" If in the course of its evolutions a machine (constructed as
hereinbefore described) heel over sideways one way or the other,
or if a rolling motion be set up, the first effect is for the machine
to begin to slide down, so to speak, in the direction in which it
is for the time being inclined, this motion is very quickly arrested,
however, by the resistance of the ' fins ' whose centre of pressure
is arranged above the centre of gravity of the machine, and
equilibrium is thereby restored. A similar result might be
brought about by inclining the wings or the tips of the wings
upwards to the right and left; but an arrangement of fins is
specially valuable owing to its damping action on any side
oscillations that may be set up."

" It is easy to understand when a motor is employed for the

354

APPENDIX App. II.

purposes of propulsion that according to the magnitude of the
thrust transmitted to the machine the ' angle of recuperation '
of the machine is diminished, and if the thrust be sufficient the
machine will retain its natural velocity with a horizontal course
or even an upwardly inclined course, and it will moreover
automatically adapt itself to whatever thrust may be applied,
so that the effect of an increased thrust is only to increase the
velocity of the machine in quite a transitory manner, the final
result, after the trochoidal oscillations due to the disturbance
have settled down, being a change in the course of the machine
in an upward direction."

" Thus, in order to effect a change in the course of the machine
in a vertical plane, I operate on the propelling mechanism by
either increasing or diminishing the supply of working fluid or
by such other means as may be appropriate to the motor em-
ployed. I may in certain cases introduce a counter thrust,
instead of operating on the motor mechanism, by erecting a
small plane or other obstruction perpendicular to the direction
of motion of the machine so as to cause it to take a downward
course."

" When only a temporary change of direction is required, as
when evading an obstacle, the tail-plane may be used with effect
for vertical steering, but only within the limits of the safe velocity
of the machine."

355 A a 2

APPENDIX III

THE AUTHOR'S 1894 AERODROME

In Chap. I. mention has been made of an aerodrome constructed
by the author in 1894 ; a representation has been given in Fig. 24,
and particulars of a successful flight in § 14. A photograph of
this machine is given in Fig. 163, taken at the time of the
experiments, and a scale drawing in Fig. 164.

The general construction is evident from the drawing, and
follows closely on the lines of the aerodones fully described and
figured in § 11. Owing to the greater size and weight of this
machine, it is stayed by piano-wire guys in the manner shown,
in order to provide longitudinal stiffness both vertically and
horizontally.

The propulsion is, as figured, by twin propellers driven by
twisted " elastic " (indiarubber), the data of the propelling
mechanism being as follows : —

Propellers, 17£ inches diameter ; two blades of approximately
6 square inches each ; pitch recorded as 16 inches, but on
remeasurement it appears to be nearer 20 inches. The construc-
tion of the propellers is shown in detail in Fig. 165 ; the blades
are of sheet aluminium '08 in. thick, ( ='75 m.m.), mounted
on arms formed by a single steel wire of '1 in. diameter. The
propeller blades are so fitted as to " feather " automatically if
the aerodrome overruns the range of the propulsion; that is to say,
when the twisted rubber is spent, the blades swivel approximately
into the line of flight and do not act as a drag on the machine.
This feathering of the blades is accomplished by mounting them

357

App. III.

APPENDIX

o

358

APPENDIX App. III.

to pivot on the wire support (the aluminium being bent to form
a hinge), and the provision of a stop consisting of a short strip
of brass soldered in position to limit the angular motion and at
the same time to locate the blade longitudinally.

Energy of Propulsion. — The energy of propulsion stored in the
two indiarubber skeins amounted in all to about 1,000 ft. lbs. (load-
ing energy) ; the total number of propeller revolutions being
500, representing an average of one foot pound of energy per
revolution. This requires to be multiplied by a coefficient to give
the energy available at the propeller shaft ; in all probability,
after allowance also for the propeller efficiency, not more than
50 per cent, of the loading energy is usefully employed in
propulsion.

The total weight of rubber in the two skeins is recorded as
•7 lb., each skein being composed of six strands.

General Notes. — The aerodrome weighed with rubber complete
2} lbs. ; the angle made between the flat face of the aerofoil and
the tail-plane as first adjusted was 8 degrees, but there is a
record to the effect that the angle was subsequently increased to
twice this amount, i.e., 6 degrees. The author has no note as to
which of these angles was employed when the flight recorded in
§ 14 was made.

The range of flight should theoretically amount to about 250
yards before the energy of the indiarubber is expended, and there
would be probably another 50 or 60 yards covered while the
aerodrome is coming to earth. Unfortunately, owing to obstacles,
the full range of flight was never realised ; the most satisfactory
flight is that recorded, but here the machine finished its career
prematurely in an elm tree, the propulsion energy being only
abouf half expanded.

An attempt made to photograph the aerodrome in flight proved

abortive.

359

T

App. in.

APPENDIX

360

APPENDIX IV

SOLUTION OF THE CUBIC EQUATION

Equation (14), where cos 0=1, i.e. for the highest and
lowest points of a phugoid, is,

o-^ 1 -.?,.)

in which C is known and it is required to find the value of II ;

C 1 II

v^ = 1 -r// n

C _ _ II

^(^irfe)

3i/» -

Let = x ; the equation becomes

SH,

n

= 1 -tf

V 8 H n x

which can be solved on a slide rule of the " Gravel; " type as
follows : —

Using the ordinary nomenclature for the four scales,

IA
B

C
D

(1) Set the cursor to C on scale D.

(2) Set 8 II n on scale B to C a , as indicated on scale A.
(8) Move cursor to 1 on scale B (this divides C 2 by 8 H n ).

8G1

App. IV. APPENDIX

(4) Set the slide to read the quantity x on scale B at the
cursor, so that scale D reads (opposite 1 on scale C) 1 — x.

(5) Move the cursor back to its original position, and read //
required on scale B.

The whole operation of thus solving the equation occupies
hardly more than half a minute, and is accurate to within
1 in 500.

When O is negative — in the case of tumbler curves — the
quantity {x -+- 1) is negative, x being greater than unity. This
fact does not complicate the slide rule manipulation, but it is
convenient to dispose of the anomaly by modifying operation (4)
to read : — (4) Set the slide to read the quantity x on scale B, so
that scale D reads x — 1 (opposite " 1 " on scale C).

It is evident that since different values of II are given by
values of C which differ only in their decimal point, such as
C = 15, C = 1*5, C = '15, etc., it is necessary to watch the
change involved in every operation, in the position of the
" unit 1 " on scale D.

It may be also noted that the inflected curves have two
solutions for the value of H, the tumbler curves have each only
one ; this agrees with what is demanded by the equation.

In order to further demonstrate the method and manipulation of
the slide rule in the solution of the cubic equation, the following
examples are given : —

(a) Inflected Curve. C = 2*71 H n = 64.

(1) Set cursor to 2*71 on scale D. (Note : The left-hand "1"
is " unit 1.")

(2) Divide 2'71 2 , as indicated by cursor on scale A, by
8 H n = 192 (on scale B). (Note : The left-hand "1" on
scale I) becomes '1 since 1*92 is used for division instead
of 192.)

(3) Move cursor to the left-hand " 1 " on 3cale B.

(4) Set slide to read simultaneously x = '0417 on scale B at
cursor, and 1 — x = '9583 on scale D opposite " 1 " on scale C
(remember right-hand " 1 " is " unit ").

302

APPENDIX App. IV.

(5) Replace cursor at 2*71 on scale D and read H on scale B.
//= 8.

Again, process (1), (2), and (8), as before : —

(4) Set slide to read simultaneously x = *7788 on scale B at
cursor, and 1 — x = *2217 on scale D opposite 1 on scale C
(remember still that right-hand " 1 " is unit).

(5) Replace cursor at 2*71 on scale D and read second value
of H on scale B. II = 149.

In the curve in question the two values of II when cos = 1
are : — II = 8 and H = 149, the highest and lowest points on
the curve.

(b) Tumbler Curce. C = 15'21 H n = 64.

(1) Set cursor to 1521 on D. (Note : Left-hand " 1 " is
unit.)

(2) Divide 15'21 a on scale A by 192 on scale B. (Note: The
right-hand " 1 " becomes unit.)

(3) Move the cursor to left-hand 1 on scale B.

(4) On moving slide to obtain a coincidence, we find but one :
we read x = 1*815 on scale B at cursor, and x — 1 = '815 on
scale D opposite " 1 " on scale C.

(5) Replace cursor at 15*21 on scale D, and read H = 848 on
scale B.

The relationship between II and C when cos = 1 may also
be expressed graphically. In Fig. 166 ordinates = H, and
abscissae = C ; the curve is plotted to half the scale of Fig. 42,
II n = 64.

This curve not only gives at once the relationship between
II and C when cos = + 1 for phugoids of all amplitudes
(within the limits of the plotting), but also exhibits clearly
the existence of the two values of II for every positive value
of C (that is, for values of // between zero and 3 //J, and the
single value only when C becomes of negative sign.

3G3

App. IV. APPENDIX

A curve is also plotted for cos = — 1, (dotted line) ; here C
is of negative sign throughout.

Let us draw, for any value of C, an ordinate cutting the
curves at two points, then : —

-15

-10

-5

^

-»»

.

'

*

s

'

s r

\

•

s*

^

S*

s

•r*—

}00

j

1

/

/

/

r

•

MO

399

s

i _ .

Fig. 166.

(a) If C is positive, we have, cut off on the ordinate by curve
cos = 1, the maximum and minimum value of II for an
inflected curve.

(b) If C is negative, we have, cut off by curve cos = 1, the

364

APPENDIX App. IV.

maximum, and by curve cos = — 1, the minimum value of H
for a curve of " tumbler " type.

These curves may be initially plotted by assuming values for
II> and calculating the corresponding value of C, thereby
evading the necessity for solving the equation in the opposite
sense, the curve may then be used in the manner indicated, and
the solution of the cubic equation as such is avoided.

865

APPENDIX V

CALCULATIONS FOR THE PLOTTING OF THE

PHUGOID CHART

(The reference numbers correspond to those given in Fiy. 42 and the

Frontispiece,)

Curve No. 1 is the special case of the straight flight path.
Curves Nos. 2 and 3 are plotted by the approximate method of
§31.
Curve No. 4. (Inflected type.)

II n = 64. 7/i = 25. Cos 0i = 1.

C = V7/ 1 (c O8 1 -i^-)
= 5X ( 1 -S) = 4 - 35 -

For highest and lowest points on the curve, cos 0i = 1, and

II 435 _
192 + Vli ~~

the solution of which gives alternative values of II (compare
§28).

H = 25, highest point of curve (already known).

II = 118"5, lowest point on curve (value required).
Point of Inflection. — This is given by § 28, Equation (15).

". -(*%*)

2

56

_ / 192 X 4-35 \ _
306

APPENDIX App. V

And by Equation 16, § 28,

„ 56 . 485
cos 0, = rw + 7 - = -873

or ©, 29° 10'.

Curve No. 5. (Inflected type.)

H n = 64. J/i = 16. Cos ©i = 1.

Vide § 28.
Curve No. 6.

//„ = 64. Hi = 8. Cos ©i = 1.

8 X = 2-71.

C = V 8x(l- 19l2 ) = 2-

Lowest point on curve,

// sm __ -

192 + V 11 ~"

whence //(max.) = 148*7.

Point of Inflection.

2

//, = ( 192 * 2 ' 71 Y = 4Q-75

, n 40-75 , 2-71 .,„

and co* 0, = ., nn H — . = -636

192 ^ V 40-75

or 0j = 50° 80'.

Curve No. 7. Special case. Semicircle.
Curve No. 8. (Tumbler.)

//„ = 64. //i = 8. Cos 0! = — 1.

C = V8x(-l- 1 Jg) = - 2-946.

Lowest point on curve, 1

// , - 2-946 __
192 + V7T ~

whence // == 229.

1 The lowest point on the tumbler curve, i>., when cos = + 1, is
calculated as a check on the plotting.

367

App. V. APPENDIX

Cnrre No. 9. (Tumbler.)

//„ = 64. Ih = 16. Cos @i = — 1.

C = 4x(-l-^) = -4-888.

Lowest point on curve,

II - 4-333 _
192 + </H
whence 77 = 245.

Curve No. 10. (Tumbler.)

II n = 64. H x = 36. Cos X = - 1.

= 6X (- 1 -1) = - 7 ' 125 -

Lowest point on curve,

It — 7-125 _

192 ' </Tl
wlience II = 275.

Curve No. 11. (Tumbler.)

Jf„ = 64. Ih = 50. CosQi = — 1.

C = V~50 X (-1-^) = - 8-91.

Lowest point on curve,

H - 8-91 _
192 + V II ~
whence II = 292.

Curve No. 12.

//» = 64. Ih = 100. Cos &! = — 1.

C = 10x(-l-^) = -15-21.

Lowest point on curve,

IL -u - 16*21 _ i
192 + VI?

whence 77 = 848.

The calculated values of r for the preparation of trammels
for the plotting of the curves as above are appended,

868

APPENDIX

App. V,

4.

If

25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
45

56

70

to

80

85

90

95
100
110
1135

82-1
89*3
971
105-5
1150
124*3
135-1
146-7
158-9
173-2
188-9
206
224
245
269
295
500

00

668
538
461
411
376
350
330
301
293

ll

*•

1

r

16

42-5

18

511

20

65-2

22-5

830

25

105-2

275

132*6

1 ?0

167*6

32-5

2220

35

274

37-5

358

40

490

45

•

1155

49-8

co ,

OO

1408

60

794

65

585

70

482

75

419

80

378

85

348

90

328

95

310

100

296

110

276

120

262

129-9

i

1

252

0.

IF

r

8

18-2

9

22-2

10

26-6

11

31-3

12

36-5

13

421

14

48-4

15

55*1

16

62*6

17

70-7

18

79-8

19

89-7

20

100

25

177

30

329

40-75

oo

OO

529

60

435

65

382

70

345

to

320

80

302

85

286

90

276

95

267

100

260

no

2-18

120

239

130

233

140

227

148-7

1
i

i

224

8.

//

1

r

8

14-2

9

16-7

10

193

11

220

12

246

13

27 3

14

300

15

328

16

35 4

17

38-1

18

40-8

19

435

20

461

.22

51-3

24

563

26

61-3

28

660

30

70-5

32

74-8

34

79-2

36

83-2

38

87-0

40

90-7

45

99-1

50

106-6

55

1134

60

119-3

65

124 6

70

129-5

75

133-7

80

137-5

85

1410

90

1441

95

147

100

149-7

110

1541

120

158-0

130

161-2

140

164-0

150

166 4

160

1685

170

170-1

180

171-8

190

1731

200

1744

210

175 8

220

176-8

229

177-4

A,

869

B B

App. V.

APPENDIX

9

H

r

16

256

17

1 276

18

i 29*8

19

319

20

34

22

377

24

42-3

26

46-4

28

1 504

30

54-3

35

63-8

40

72-6

45

80-8

50

, a«:2

00

950

60

101-3

65

107-0

70

112-2

to

1170

80

121-3

85

125-3

90

1290

95

132-5

100

135-5

no

1410

120

H5-7

130

149 8

140

153-2

150

156*4

160

159-3

170

161-5

ISO

163-6

190

165-5

200

1672

210

168-8

220

170-2 '

230

171-5 .

240

172-6

245

i
1

i
i

10.

H

36

38

40

42

44

46

48

50

55

60

65

70

75

80

85

90

95

100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

275

46-1

48-9

51*9

54-6

574

601

62-7

65-4

71-8

777

83-3

88-5

93-5

98-0

102-5

106-5

110-3

1140

120-4

126-2

131-4

135-9

1400

143-5

146-6

149-5

152-1

154-5

156-7

158 7

160-5

162-2

1636

1669

11.

12.

E

50

52

54

56

58

60

65

70

75

80

85

90

95

100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

260

270

280

290

56-2

58-4

60-8

630

65-3

67-7

730

78-0

830

876

91-7

96-0

99-8

103-5

110-3

116-3

121*8

126-6

131-0

1350

138-5

141-7

144-8

147'5

149-8

1520

1540

1561

158-0

159-2

161-0

162-4

163*6

//

100
104
108
110
115
120
125
130
135
140
145
150
155
160
165
170
175
180
185
190
195
200
210
220
230
240
250
260
270
280
290
300
310
320
330
340
348

78-1

80-7

83-4

84*8

88

91-0

93-9

96*8

99-4

102

104-5

107

109-25

111-5

113-65

115-65

117*65

1196

121-5

123*3

125

126*6

129-8

132-6

135-3

137*9

140- 1

142-3

144-4

146-4

148-2

149-8

151-4

152*9

154-3

155-6

156*7

370

APPENDIX VI

MOMENT OF INERTIA
THE METHOD OF DOUBLE SUSPENSION

In certain cases, as for example when determining the moment
of inertia of the body of a bird, it is not convenient to employ
the method of § 172 owing to the difficulty of accurately locating
the centre of gravity. In such cases the method of double suspen-
sion may be employed, the value of A 2 being deduced from two
determinations of the time period, made with two different points
of suspension a known distance apart.

When a single determination of the time period is made, the
length of the pendulum (i.e., the I of § 172) being unknown, a
curve may be plotted for a series of assumed values of I, the
ordinates representing the values of A 2 corresponding to values
of I as abscissae. The resulting curve is a parabola passing
through the origin and through a point distant from the origin
by an amount equal to the length of a simple pendulum corre-
sponding to the observed period.

By plotting two curves from origins 0\ and 2 , Fig. 167,
separated by the distance apart of the two points of suspension,
the value of A 2 is given by the point of intersection ; this is the
only value of A 2 consistent with both observations.

In Fig. 167, the line Oi O* represents the distance between
the two alternative points of suspension, and the two curves
representing possible values of A 2 are found to intersect in the
manner shown, giving the required value of A 2 . The plotting of
these two curves may be effected over just the necessary portion

871 b b 2

/

/

/

/

/

/

/

NT

*

\

\

\

\

\

\

\

\

\

K>

$5~>

T

\

\

\

\

\

P0- £0

ZO

"• X

APPENDIX

App. VI.

of the curves (where they overlap) as shown. In the example
given the calculation is as follows : —

Suspension from origin 0\ gives h = -29.

Now, A 2 = 1 1 g ( - J 2 — I V from which it is evident that A

falls to zero when l\ = o (i.e., at origin), and when h = g I -J
that is

Zi = 32 ' 2 ( *i§) a = ' 274 -

Suspension from Oa, t = 86

' ( ; )' = 3a ' 2 ( m ) = - iM -

These values give h and Z 2 respectively for A 2 = o laid off
from 0\ and O2. The intersection evidently lies between these
points. We proceed to plot this portion of each curve as
follows : —

/l.

A 2

•25
•24
•23

A 2 = -25 X (-274 - -25)
A 2 = -24 X (-274 — -24)
A 2 = -23 X ('274 — -23)

= -0060
= -0081

= -oioi

And similarly for the curve of / 2 . The ordinates representing
these plottings are shown in the figure.

873

APPENDIX VII

THE GYKOSCOPE

The importance of the gyroscope in connection with the present
subject is due to the fact that in certain actual cases, as in the
flight of a projectile from a rifled gun, and in the flight of a
boomerang, gyroscopic action plays an important part ; and
further, the gyroscope has actually been proposed as a means
of securing stability in the flight of an aeronautical machine.

In view of these facts, and in view of the contingent possibility
of the employment of gyroscopic action as a means of improving
the stability of a flying machine, it has been thought desirable
to include in the present work a discussion of the theory of the
gyroscope, and an account of some of the more important of its
present applications.

It is usual to treat the gyroscope and problems involving
gyroscopic action exclusively by mathematical methods, in which
the physical reality of the problem is partially obscured. One
result of this is that, in spite of the unimpeachable nature of the
said methods, mistakes are not infrequently made, even the
direction of the gyroscopic torque is sometimes misstated. 1

The author has for many years employed a direct method of

1 In his "Dynamo Electric Machinery" (-Uh ed.), p. 388, Silvanus P.
Thompson, discussing the gyroscopic couple, says : — ". . . Then F = 30*6 lbs.
on each bearing alternately acting up and down at each roll, if the axis of
the dynamo lies athwart the ship." The actual direction of the gyroscopic
couple is about an axis at right angles to the forced precession ; in the case
in point the resulting stresses on the hearings are horizontal, not vertical as
stated.

874

APPENDIX App. VII.

dealing with the gyroscope which has many points in its favour.
The account that follows forms a portion of a paper read before
the Institution of Automobile Engineers. 1

" The fundamental principle employed in the following exposition
is termed the principle of the conservation of angular momentum.
This is the rotational analogue of the conservation of (linear)
momentum, wbich is corollary to the third law of motion — When
force acts on a body, the momentum generated in unit time is propor-
tional to the force. The rotational form of this law is — When a
couple or ' torque ' acts on a body about a given axis the angular
momentum generated per unit time about that axis is proportional to
the magnitude of the torque.

" In both cases the term 'body* may be construed as including
not only a rigid body, or body in one piece, but a complex body
of any kind whatever consisting of an indefinite number of parts
associated amongst themselves by any known or unknown laws
of attraction or repulsion. In other words, for body we may
substitute the term self-contained system — such a system being
one not associated with its surroundings in any way whatever
except as regards the specified applied force, or torque, as the
case may be.

"As it is important that the signification of the above principles
should be made quite clear, an illustration will not be out of place.
A well-known application of the principle of the conservation of
momentum is the case of the gun and projectile: when a gun is
mounted so that it is quite free to recoil, in fact, so that we may
regard it as constituting with its charge and projectile a self-
contained system, the firing of the charge has no influence on
the motion of the mass centre of the system, so that when the
shot and powder gases are projected forward the gun recoils
backward, the minus momentum of the gun exactly neutralising
the plus momentum of the charge. In order to reduce this
problem to a nice simple form, it is usual to suppose that the

1 " Some Problems Peculiar to the Design of the Automobile," Proc. Inst,
A.E., read March, 1908.

875

2
/

{

App. VII. APPENDIX

powder gases are without mass, to imagine, in fact, that the
projectile is discharged by a repulsive force instead of by gaseous '

pressure. As so presented the problem is reduced to simple
arithmetic : thus, if the projectile weigh 1 oz. and the gun
100 ozs., and if the muzzle velocity of the shot be 2,000 ft./secs.,
the velocity of recoil is t Jq X 2,000 = 20 ft./secs.

" We may extend the above to deal with the spin of the projectile
and the influence of the said spin on the motion of the gun.
Thus, when the gun is fired, since there is no torque applied from
without, the angular momentum of the self-contained system can
undergo no change, and the angular momentum received by the
bullet by virtue of the rifling will be exactly neutralised by the
opposite angular momentum imparted to the gun. Thus, if the
moment of inertia of the gun be 500 times that of the bullet, the 1

angular velocity of the bullet will be 500 times that of the gun,
the rotation being, of course, in opposite directions. It may be
remarked that the question of the powder gases does not sensibly
affect the rotational problem.

"Now in the foregoing illustrations the motions are confined to
one line in the case of the linear problem, and about one axis in
the analogous case of rotation, but the principles apply equally
if the motions involve movements simultaneously along or about
the other two co-ordinate axes.

*' We will now apply the principle of the conservation of angular
momentum to the study of the gyroscope. Taking the case of
the ordinary lecture model (Fig. 168), in which the fly-wheel is
mounted freely in gimbals, it may at once be noted that the
mounting is such as to exercise no kind of rotational restraint
on the spinning wheel ; the latter is arranged, rotationally
speaking, as a self-contained system. Thus, a movement of the
stand which forms the mounting, although taking effect trans-
lationally on the gyroscope, has no effect rotationally, except so
far as may be due to the friction of the gimbals, which is of the
nature of a mechanical defect.

" If a torque or couple be applied to the axis of the wheel, or to

376

APPENDIX

App. VII.

the hoop A, which in effect is the same, so long as the wheel is
not in rotation, an angular velocity will be communicated about
the axis of the applied torque in accordance with the law, the

3-

Fio. 168.

angular momentum I o> (where I is moment of inertia and o> is
angular velocity), being equal to the applied torque multiplied
by the time of its application ; let us, however, suppose that the
wheel be set in rapid rotation and examine the conditions that
supervene.

877

App. VII.

APPENDIX

SS

□

Fig. 169.

" In Fig. 168, three co-ordinate diagrammatic projections of the
gyroscope are given, which will be referred to as the A", Y, or Z

view, according to the axis along
which the view is supposed to
be taken. A similar conven-
tion applies to the further
diagrammatic Fig. 169.

" Let us suppose now a clock-
wise torque about the axis of z,
and let us assume provisionally
that the effect of this torque
is, as before, a movement in
like direction about that axis
(Fig. 169) . Then in the Y view
we have evidently an immediate
development of angular momentum about the axis of //, an axis
about which no torque or couple exists; but we know this is
impossible, hence the provisional
assumption is at fault, and the torque
applied about the axis of z cannot
give rise to motion about that axis.

" We may term the angular momen-
tum that makes its appearance about
any axis owing to the movement of
a rotating wheel, as in the above
example, angular momentum of aspect
owing to its being due to a change of
aspect of the spinning wheel; thus,
if in Fig. 170, (a) represents a spin-
ning wheel in edge aspect, it will
require the application of a clockwise torque to bring it into
the successive positions (b), (c), (d), etc., this torque being, as
indicated, about an axis at right angles to the plane of the
paper.

" Now the motion supposed to take place in Fig. 169 corresponds

878

p*

r£CTIO/*
TQttQUC

O/o

I

fit

I

n

m

i

j

(W

(c)

y

f

1

\tl

Fio. 170.

APPENDIX

App. VII.

in every way to the requirements of a counter-clock torque
applied about the axis of y ; it is evident that if a torque be so
applied, the angular momentum of the spinning wheel (in the
motion depicted) makes its appearance in a manner that will
fulfil the requirements of the principle of conservation. This
reasoning may be extended to constitute a definite proof.

" Let a torque be applied about the axis of y, then since this
torque has no component about the axis of z, and since there is
no independent torque about that axis, there can be no change
of angular momentum about that axis ; consequently the motion
of the gyroscope must be such as will not display momentum of

(*J

j

Fig. 171.

aspect in Z view (Fig. 169), and therefore it can only take place
about either the axis of x or the axis of z. But the former is
the axis of rotation, and change of momentum about litis
axis requires a torque about this axis which does not exist;
therefore the only axis about which motion can take place is that
of z, so that when a torque is applied about the axis of y, either
there will be motion about the axis of z, or there will be no
motion at all. But an unbalanced torque cannot act on a self-
contained system without giving rise to angular momentum
about the axis of its application, hence there must be motion, and
it will be about the axis of z.

" We will now deal with the problem quantitatively. Let us
first examine the question of angular momentum as due to
aspect, with a view to expressing it in terms of the angular
momentum of the wheel about its axis of rotation.

879

App. VII.

APPENDIX

" In Fig. 172, let us represent the torque by which the angular
momentum of the wheel may be imparted in unit time by the
couple A B, and let each of the forces constituting the couple A B
be resolved on to any two co-ordinate axes x and y into the two
couples fli 61 and a 2 fc 2 » this is merely a matter of resolving each
of the two forces A and B by the usual parallelogram. Then
the resolved couples are together in every respect the equivalent
of the parent couple, and the angular momentum generated by

/;' I

A

£

Fig. 172.

the couple a x b\ will be to that generated by A B as the magni-
tudes of the individual forces a x b\ are to the individual forces
A B ; therefore the angular momentum about the axis x is to the
angular momentum about the axis of spin as the force «i is to
the force A, or as the force b\ is to the force B, that is, if be
the angle made by the axis x to the plane of rotation, and if I be
the moment of inertia and &> the angular velocity of spin of the
wheel, so that I o> is the angular momentum, we have the angular
momentum of aspect = !<*> sin 0.

880

APPENDIX

App. VII.

.1

" Let us represent the rate of change in the angle 6 the symbol

d0

12, that is, 12 = — - and is the angular velocity of precession.

" Now rate of communication of momentum due to change of

, . d (I o> sin 6) , . , . . d sind

aspect is lx -, which, since I o> is constant = I co

dt , , dt

At the instant when = zero (that is, when the wheel is in edge-
wise aspect), this expression becomes I «— =- (6 being expressed in

circular measure), which substituting for -j- becomes I co 12.

" But if we are employing absolute units,
the gyroscopic torque, which we will denote
by the symbol r, is equal to the angular
momentum communicated per second,
that is,

T = I CO 12,

which is the well-known expression.

"The above deals immediately with the
case of a precession about an axis at right
angles to the axis of spin ; it is easily
extended, however, to include precession
about an axis inclined at an angle. Thus
in Fig. 173, let A B represent the axis of
precession, and let A C\ A C 2 represent two
successive positions of the axis of spin ; let the angle B A C
be denoted by a. Now while the precession takes place through
the angle C\ B C2, the angular motion of the axis of spin is
denoted by C\ A C 2 , and since C\ L\ is common, these angles
are in the relation A C : C B, that is, as 1 is to sin a, hence
expression becomes : —

t = I co 12 sin a.

" There is one little difficulty still to discuss. In Fig. 169,
the precessional motion about the axis of z exhibits direct
angular momentum about that axis, and although this is
constant after the motion is once initiated, it is at first

881

.*

App. VII. APPENDIX

sight difficult to see how, under the conditions, it can arise.
It is quite certain that if such angular momentum exists its
initiation must involve a torque about the axis of z } which
the conditions of free mounting definitely preclude. Hence
we may boldly declare that the momentum that apparently
must be there cannot exist. The precession of the wheel
in Z view we cannot deny ; such precession evidently involves
angular momentum about z which is absent when the
torque about y is removed and the precession ceases ; and
there is no torque at any time about the axis of z ; the only
solution is evidently that the system contains equal and opposite
momentum about z somehow or other ; and the only form in
which such can exist is momentum of aspect. Here we have the
solution : the commencement of the precession is marked by a
slight yielding directly to the applied couple, and the termination
of the precession is marked by the reverse effect. 1 Thus, in
Fig. 171, (a) represents the wheel before the application of the
torque ; (b) represents the conditions during the application of
the torque ; and (c) after the torque is removed.

" We thus have some insight as to the mode of development of
the precessional motion about the axis at right angles to the
torque ; there is an actual yielding when the torque is applied
during a short period in which the precessional motion is
generated, the extent of this yielding being just that neces-
sary to give the equal and opposite momentum of aspect
about the precessional axis. It is thus very little more than
a matter of ordinary arithmetic to calculate the extent of the
yielding that will take place under any stated conditions, a
problem that as a matter of analytical dynamics can scarcely be
regarded as simple."

1 It is assumed that the couple is applied and withdrawn gradually,
otherwise a vibration is set up.

382

APPENDIX VII U

THE RIFLED PROJECTILE

In the flight of a projectile fired from a rifled gun two influences
are at work of an entirely different kind tending to modify the
form of the trajectory l ; these are gyroscopic action and the aero-
dynamic action dealt with in Vol. I. § 30, as an effect of
discontinuity in the fluid motion on the flight of a ball in
rotation.

Before the days of rifling the second of these influences alone
took effect on the flight of the spherical bullets then in use, and
as the accidental rotation that the ball acquired in the barrel of
the gun was an unknown quantity, so was the resulting aero-
dynamic reaction, and the accuracy of the weapon was little
better than that of the crossbow it superseded. About the
middle of the 18th century Robins showed that a bend in the
barrel would produce a rotation of the ball owing to the
friction due to its centrifugal force, and that this rotation,
after a comparatively short flight, would more than compensate
for the direct effect of the bend ; he further advocated rifling (a
well-known expedient even then) as a remedy for the defects of
the musket of that time.

When a spherical bullet is fired from a rifled gun the axis of
rotation remains sensibly constant in direction, for the aero-
dynamic forces are initially symmetrical ; the result of this is
that at first the flight path is a simple trajectory, and thus
the bullet is very soon travelling at an angle to its axis of

1 In addition of course to the direct aerial resistance.

88

o

App. Villa. APPENDIX

rotation and then the considerations of Vol. I. § 80, begin to
have weight, and the ball moves laterally under the influence
of the modified aerodynamic system. If the rifling is right-
handed, t.c, like a corkscrew, the wake current will be thrown
to the right, and the reaction on the ball and its consequent
drift are to the left. Conversely, if the rifling is left-handed
the reaction on the ball and drift will be to the right.

In the case of elongate projectiles of the forms commonly
employed, the conditions are different, and gyroscopic considera-
tions require to be taken into account. Thus after a short
distance has been traversed, owing to the curvature of the
trajectory the direction of motion no longer coincides with the

axis of rotation (Fig. 174),
there is then a couple
tending to turn the pro-
jectile into a broadside-on
position, 1 and owing to the
rotation this produces a
precessional effect. It is
Fla 174 evident that for an elongate

projectile the end-on atti-
tude is one of unstable equilibrium, just as in the case of a
spinning top standing upon its peg ; and that stability is given
in both cases by the rotation. As in the case of the spinning
top also, the effect of the gyroscopic precession is to make the
axis of spin describe a cone about the axis of equilibrium, Le.,
about the axis of flight.

The condition of a steady state evidently requires that the
projectile shall move in a spiral path, for the reaction of the air
due to the inclination of the projectile to the line of flight causes
a continual change in the direction of motion, and as this force
is in a direction passing through the axis of precession, the equal

1 This we know from aerodynamic considerations, the centre of lateral
resistance is in front of the centre of figure, and so is in front of the centre
of gravity.

884

APPENDIX

App. Villa.

and opposite inertia force will be the centrifugal force of a
circular motion ; the latter superposed on the motion of trans-
lation becomes a spiral. This condition is illustrated diagram-
matically in Fig. 175, in which the plane of the paper contains
the axis of the projectile for the time being; the motion will be of

Jp~~ ^

^

Fia. 175.

spiral form, but owing to the fact that the centrifugal force must
be truly radial, the axis of rotation will not intersect the. axis of
flight, but will be tangent to a concentric cylinder in the manner
shown.

In Fig. 175, x x is the axis of flight, and p p pi pi the spiral
flight path. The axis of the projectile a a ib directed inwards at
an angle as shown is tangent to a cylindrical surface, 8 8 8, whose

a. ~ 885 c c

App. Villa. APPENDIX

trace on the plane a x a h containing the axis of the projectile, is

If there are no forces tending either to retard or hurry the
precessional motion of the projectile in its spiral flight path, then
that path will preserve its initial amplitude (except so far as the
conditions may be affected by the falling off of the velocity owing
to the resistance to flight). If forces exist tending to retard the
precession, then as in the simple gyroscope the projectile will
slowly yield to the applied torque and the amplitude of the spiral
path will increase, in such a case the flight path would be " wild."
If, on the contrary, there are forces tending to hurry the preces-
sion, the projectile will, as in the case of a common spinning top,
move in opposition to the applied torque, and the amplitude of
the spiral will diminish, the flight path settling down gradually
to a closer and closer approximation to a smooth trajectory. 1

1 The damping of the amplitude of the spiral flight path is only a definite

consequence of the diminution of the angle a by virtue of the law correlating

'J the lateral pressure on the projectile and the angle o. If, in Fig> .2QQ, a be

the inclination of the axis a a to the line of flight at any instant, and if the

centre of lateral resistance for small angles be assumed constant in position,

i\^ and situated a distance I front of the centre of gravity, then the torque will

A be proportional to the lateral force and the latter will be related to the

angle a according to some law.

If for example we assume the Pp oc sin law (the law of the aeroplane
for small angles), we have: —

F oc sin a (1)

And the equation to the precession (App. VIL),

t = I«Q sin a
\ F oc n sin a (2)

And writing r for the radius of the circular component of the / flight path,
the equation giving the centrifugal force,

F = mr n a or, Focrfl 8 (3)

where m is the mass of the projectile.
By (1) and (2),

n sin a oc sin a
or n is constant

.-. by (3) F oc r

or by (1) r oc sin a

Consequently as sin a diminishes, due to the hurrying of the precession,
r diminishes in like ratio.

886

APPENDIX App. VJHa.

It is evident that since the want of alignment of the axis of spin
and the axis of flight in the case of a spherical projectile is a thing
of gradual growth, in the case of the elongate projectile the spiral
of the flight path does not ordinarily acquire sensible magnitude,
for the individual increments of change are infinitesimal ; the
whole effect is merely that the axis of the projectile follows the
line of flight, the projectile flying point foremost, as is the case
with a feathered arrow (Fig. 176), in contrast to the manner it is
frequently depicted, as in Fig. 177. The spiral flight path only
arises when some abrupt disturbance takes place, such as when
at the time of firing the muzzle of the gun is situated in a cross-
wind, or as when the bore at the muzzle end is damaged or
irregular, when it is well known considerable disturbance of the
trajectory results.

Since, in a well-conducted projectile, any initial "wildness"

We may similarly obtain a more general solution; thus for (I) we may
write,

F x *i'w» a (4)

where according to the foregoing example n = 1, or according to the
Newtonian law (i.e., the law of the plane of extreme proportion in apteroid
aspect), where n = 2. We have,
By (2) and (4),

fl sin a x sin n a
or ft x sin »-l a

and by (2) and (3)

r ft* x ft sin a
sin a

or A oc

r

sin a

x sin*- 1 a

r x «»*-* a

Thus in the case of the Newtonian law, n = 2, r becomes constant, and
the diminution of the angle does not result in any change in the radius of
the circular component of the spiral flight path.

In practice it may be taken as almost certain that the index n is less than
2 and probably greater than unity, consequently we may conclude that the
reduction of sin a due to the hurrying of the precession has the effect of
diminishing the value of r, although the said diminution may take place at a
less rate than the simple ratio of the change in siu a.

887 c c 2

App. Villa. APPENDIX

in the flight diminishes, so that the shooting may some-
times be relatively more consistent at 200 yards than at
100 yards, it is evident that there must be some influence at
work tending to hurry the precession. The nature of this
influence is in all probability that discussed in Vol. I. § 80, i.e.,
it is due to the effect of " cut " or " side " ; the projectile in its
spiral flight and converging attitude (Fig. 175) experiences a

- E

Fig. 176.

transverse reaction due to the want of symmetry of the lines of
flow (comp. Fig. 22, Vol. I.) ; and this acting always tangentially
to the cylinder containing the spiral flight path, tends to hurry
the precession al motion. The effect so deduced is at least in the
right direction, though in order to hurry the precession it would

Fig. 177.

appear not to be sufficient merely to chase the projectile side-
ways as a whole, but rather it is necessary to act upon it by
a torque. This torque may conceivably arise as due to the
streamlines acting more fully on the after body of the projectile
than on the forward portion.

Apart from the explanation of the spiral flight path, a
phenomenon that has frequently been observed 1 (but so far as

1 The late Mr. E. H. Hausman informed the author that in bright
sunshine it is frequently possible to actually see the bullet describing a cork-
screw path, even at as high a velocity as 2,000 ft/sec., apart from
e\ idence obtained by paper screens.

888

APPENDIX App. Villa.

the author is aware not previously explained), the main result
that has been deduced is the " tangent-to-trajectory " flight.
Taking this as a basis, the author in 1904 succeeded in giving a
quantitative solution to the "drift" problem, showing that the
drift of an elongate projectile is almost entirely gyroscopic in
origin.

The author's investigation on the " Drift of Projectiles " was
contributed to Technics (No. 11), and the quantitative work
related to the military '808 Lee-Metford. By permission of
Messrs. Geo. Newnes & Co., this article is here given in
extenso.

The Drift of Projectiles.

"The cause of the deflection of a rifle bullet or projectile
in its flight, to the right or left of the line of sight according
to the direction of the rifling, is due to gyroscopic action.
One of the conditions necessary to the proper flight of a rifle
bullet is that its mass-centre (centre of gravity) shall not be too
far forward ; to be exact, it must be well behind the centre of
wind pressure when the bullet has a slightly oblique motion.
This is, in effect, how all bullets and projectiles for rifled guns
are made. As a result, if the projectile were not spinning its
equilibrium would be unstable, and it would tumble over in its
flight.

"A parallel may be drawn between the rifle bullet and the
ordinary spinning top, of which also the equilibrium is unstable
in the absence of rotation. Just as the spinning top acquires
stability by rotation, so does the projectile.

" This familiar parallel enables one to accept at once the well-
known fact 1 that the axis of a projectile lies at all points of the
flight in the direction of the trajectory ; for, if we suppose this
axis to maintain its original direction, so that after a time it
becomes oblique to the direction of flight, we see that its condition

1 The author has more recently come to the conclusion that this fact is not
so well known as he at the time of writing supposed.

889

App. Villa.

APPENDIX

is that of a spinning top, resting permanently in an inclined
position without precession, which is a condition we know to be
impossible.

" If we suppose a bullet to be launched obliquely (as when the
muzzle of the gun is in a strong wind) its subsequent career is
of corkscrew form, owing to ' precession ' such as shown by a

• gC

*r*K

|0>

(Side View)

c'-

/

/

/

/

/

/

/

X

Fig. 178.

spinning top having an inclined axis; but if the change of
condition is slow, such as when the course of the bullet is
changed by gravity, the axis simply adapts itself to the new
direction.

"I have so far dealt with the problem by analogy rather than
by the more rigid method, in order to avoid becoming tedious
and taking too much space ; I propose now to take ' tangent-to-
trajectory ' flight of the projectile as an established hypothesis

890

APPENDIX

App. Villa

(it is generally admitted as an experimental fact), and show how
the drift follows as an immediate consequence.

"Fig. 178 represents a projectile and trajectory in side
elevation.

" Fig. 179 gives same in plan.

"Fig. 180 shows the change of aspect in plan after a brief
interval of time.

" Let r be the instantaneous radius of the trajectory, then

lErs

fP/rtnf

B

a

(Plan)

Fig. 179.

Fio. ISO.

the projectile may be regarded as moving for the moment
geometrically about the centre C (Fig. 178).

"Taking three co-ordinate axes as 'X* in the line of flight,
'Y' vertical, and 'Z' at right angles to both, we see that in
plan Fig. 179, there is no angular momentum about the axis of
' Y * ; but in plan Fig. 180, a small increment of time later, we
find, due to the change of ' aspect/ the presence of angular
momentum in a counter-clockwise direction when viewed from
above (the rifling is taken as left-handed).

" Now, we know from first principles this signifies a ' torque '
or ' couple ' acting on the bullet in the direction of the angular
momentum received ; and we know that from the conditions of
the problem, the elements of this couple can be resolved as a

891

App. Villa. APPENDIX

lateral wind pressure acting at a point A (Fig. 181), and an equal
and opposite inertia resistance at B (the centre of mass).

" This signifies that the condition of flight necessary to the axis
' following the trajectory ' is that this axis is slightly oblique
laterally, as indicated in Fig. 181. It is to this obliquity that
the ' drift ' is due.

" Of the two elements of the couple, a and b, it is seen that the
force a, representing wind pressure, is the only one acting from
without, and it is under the action of this force that the drift is
produced : if we know the magnitude of the couple and the
distance A — B, we can assign a definite value to the force a, and

A

_ AjrjS

m ?c^&-^?~ zf » e > ° r

FUffht
{Plan)

Fig. 181.

knowing the mass of the projectile, we can at once determine the
acceleration of drift.

" Now, I know of no direct way by which the position of the
point 'A* can be determined. This point may be described
as the centre of lateral wind pressure for small angle; it
might be found by experiments conducted on a carefully
suspended model, but it is doubtful whether complete reliance
could be placed on experiments at any speed that would be
practicable. Probably the only way of arriving at a value for
the distance A B is to deduce this from an actual measurement
of the drift itself, thus inverting the problem in order to obtain
the constant of any particular projectile.

"It is not actually certain that the position of point A is
absolutely constant in respect to velocity, but this can probably

392

APPENDIX App. Villa.

be relied on, provided that the range employed does not involve
a critical velocity such as that of sound. In the absence of
contrary data, it may be assumed as constant for our present
purpose.

" Now, let I be the distance between A and £.
v be the velocity of the bullet.
r be the radius of the trajectory.
" Let 12 be the angular velocity of bullet about axis of Z (due
to curvature of trajectory),
la) the avgxdar momentum of spin.
vi the mass of the bullet.

/ the lateral acceleration (i.e., along the axis of Z) to
which the drift is due.
"Then

r = - and X2 = - = v ^ = -
g r ir v

And couple = Iw 12 = — -

But also couple =fvil

fml = —?-

or,

v

r 1*> <l , 1<«> Q

f = 1 — Or I = jr— -

I m v fin v

"In the above expression it will be seen that the quantities
I and m are constants depending upon the bullet alone, the
quantity Io> depends upon the moment of inertia I, the initial
velocity, and the pitch of rifling. I estimate the value of
Io> for the -808 Lee-Metford at 2,000 ft./sec, as "038 lb.-ft.
ft./sec. 1

1 The moment of inertia I = '000,0022 and « (the angular velocity at
2,000 ft. /sec.) = 15,000 radians/sec. giving 1* = '033. In the original
investigation an arithmetical error appears to have been made in the com-
putation of the moment of inertia, the angular momentum being given as
■046 instead of 033 as above. If the quantitative results had depended
upon a prior knowledge of the value of / they would have been in error to a

893

App. Villa. APPENDIX

" The value of g is taken as 82*2 ; the error introduced by
neglecting the effective reduction in g caused by the ' soaring '
of the bullet is discussed subsequently.

"If the range is sufficiently short, so that the velocity does not
vary materially, then the value of / may be taken as constant,
and the extent of drift bears a definite relation to the drop due
to gravity ; but in practice the velocity falls very rapidly, and
the ratio of drift to drop increases with the range.

" Thus in Fig. 182, if the point * ' be the axis of X and line
O P the projection of the path of the projectile on a vertical
target, the curve will be of the form shown.

"The exact computation of drift necessitates plotting the velocity
curve as in Fig. 188, where the absciss® represent time. Curve
v, v 9 v shows the velocity as determined by experiment or known
data. Curve/,/, /gives acceleration of drift calculated from our
formula. Curve w, iv, w is the first integration of the curve/,/,/,
giving the velocity of drift at successive intervals of time ; and
finally, in Fig. 184, curve h, h, h gives the further integration
showing the actual drift displacement.

"I have taken the standard Lee-Metford rifle and service
ammunition as the basis of calculation in the curves plotted in
Figs. 182, 183 and 184, the former of which gives at a glance a
comparison of drop to drift for ranges up to 2,600 yards, the
ratio of scales being 10 : 1.

" The data as to velocity, plotted as a curve in Fig. 188, were
supplied to me by Mr. E. H. Hausman, and, I understand, calcu-
lated from Mr. Bashforth's tables. The ' constant ' on which,
the scale of drift curves (Figs. 182 and 188) is based was deter-
mined by careful experiment at 1,000 yards range, the mean
amount being found to be 80 inches. I am aware that this figure

like extent, but since the value of I was itself deduced from an actual
measurement of the drift (30 inches at 1,000 yards), the drift values are
correct, but the value of I originally given, i.e., 445 inch, is in error, the
corrected value being *31 inch or approximately jf^ths. The only effect of
this is to throw Fig. 185 out of scale.

394

App. Villa. APPENDIX

differs from that frequently adopted, and that the exact deter-
mination of drift for some known range is at present wanting ;

but pending further experimental determination (which should,
if possible, be made at about 1,400 or 1,600 yards range) I think

Fig. 184.
the value may be accepted as being certainly within 10 per cent.
of the truth.

" The value of I for the Lee-Metford bullet on the above basis
works out at 0'81 inch, that ib to say, the effective centre of
lateral pressure for evanescent angle is situated at about five-
sixteenths from the nose of the bullet. It follows from our
39 ti

APPENDIX

App. Villa.

equation that the amount of drift varies inversely as this quantity
l t and consequently any new determination may affect the value
here given ; conversely, if the value 0*81 inch appears in any
way improbable, it will for the present cast doubt either on the
method of investigation, or on the accuracy of the experimental
determination of the actual drift referred to above.

" It is of interest, therefore, to examine the question of the
lateral wind pressure at small angle, with a view of testing the
probability or otherwise of the figure we have obtained.

" Fig. 185 illustrates a supposed case of the resolution of forces
required by the conditions of the problem. It will be seen that

Fig. 185.

the pressures on the bullet must be analysed into a force in the
axis of flight passing through the centre of gravity, and a force
at right angles. From the construction shown it would appear
to be possible that the point * A ' might be situated clear in front
of the bullet, although this seems at first sight somewhat para-
doxical; the actual interpretation is, of course, that the real
couple acting on the bullet is in part due to the main resistance
to flight not being aligned to the centre of gravity of the bullet
as indicated by e, the resolution of forces shown being necessary
to obtain the position ' A ' as required by the assumption that
the forces constituting the couple are transverse.

" It would thus appear that although at first sight the amount
0*81 inch would seem excessive, as placing the centre of lateral

897

App. Villa. APPEN DIX

pressure too far forward, closer examination puts quite a
satisfactory interpretation on the matter.

" It is impossible in a brief article to deal fully with secondary
effects, but attention may be called to one point in respect of
which the above figures may require a correction.

"It is an experimental fact that rifled projectiles in flight
' soar ' to a certain extent ; that is to say, they do not drop as
much as calculation would show, taking g as 32*2. One cause
of soaring is without doubt the effect of ' cut/ owing to the axis
being slightly oblique (sideways) to the line of flight. By 'cut'
is meant the action which takes place when a golf ball soars, or
a tennis or cricket ball swerves in the air. In the above

numerical work the value of a in the equation / -= — — has been

taken as 32*2, whereas, though commencing at this value,
it diminishes considerably as the range increases. The effect of
error introduced from this cause will be to make the values given
by the curve at long range too high. An error in the same
direction, but of very small magnitude, will be introduced owing j

to the falling-off of the value of Ico owing to the slowing down
of the velocity of spin ; it is believed that error from this cause
is negligible.

"The experimental determination of drift is not so easy a
matter as might be expected; a range of 1,000 or 1,200 yards is
the least that is of real value, and almost perfectly still air is a
sine qua non. The Hon. T. F. Freemantle, in his ' Book of the
Bifle,' confesses himself at a loss to judge between the different
expert estimates, which vary from that of Mr. L. B. Tippens,
who gives it as 4 feet at 1,000 yards, to the ' Musketry
Regulations,' in which it is given as 11 inches for the same -*

range.

" I venture to think that the drift diagram, Fig. 182, can be
made of material assistance in bringing these widely different
estimates into approximate harmony. Referring to the figure,
it will be seen that the true drift is given by the departure of the

398

APPENDIX App. Villa.

curve P horn, the axis of Y ; but in order to exhibit this a
match rifle, with spirit level and backsight set exactly at right
angles, is necessary. If such a rifle, instead of having the level
and sights set geometrically, were adjusted to shoot accurately
at short range, say at 200 or 800 yards, the observed deflection
at longer range will be an apparent drift given on the diagram
by taking the resultant line Ii as datum.

" Now the apparent drift scaled from this diagram is about

I foot at 1,000 yards and 2 feet at 1,200, but as the scale is very
small I have calculated these quantities, and they come out as

II inches at 1,000 yards and 23J inches at 1,200 yards, and
these figures are in remarkably close agreement with the
'Musketry Regulations,' in which the amounts are cited as
11 inches and 28 inches respectively.

"Before concluding, it is worthy of remark that spherical
bullets fired from a rifled gun drift in the reverse direction to
that indicated. This is owing to the points. 'A' and *B' being
coincident, and the axis of rotation consequently retaining its
initial direction; then as the bullet drops, the effect of 'cut'
makes its appearance as a force acting laterally, and the direc-
tion of drift is negative. The negative drift sometimes reported
with bullets of unusual shape or proportion is probably due to
the same cause."

399

APPENDIX VIIIb

THE BOOMERANG

The boomerang originated as a weapon amongst the aborigines
of Australia, both for use in war and for the pursuit of game. It
must be looked upon from its origin as a discovery rather than
a premeditated invention.

The Australian boomerang (Fig. 186) is commonly made from

Fig. 186.

a naturally bent piece of hard wood, whittled down roughly to
the form of section given at (a); the shape is that of two
approximately straight limbs malting an obtuse angle with one
another, but the angle is sometimes a right angle. 1 The
boomerang, when correctly thrown, has a spin imparted to ifc,
the throw consisting in great part of a wrist motion, and the
flight-path may be made to vary considerably, depending in part
upon the form of the particular boomerang chosen, and in part
upon the manner in which it is thrown. The most characteristic
peculiarity of the flight is the well-known " return," though this
feature is not invariably present.

1 The form, has been sometimes more accurately described as an approxi-
mate hyperbola.

400

APPENDIX App. VHIb.

It is believed that the boomerang was evolved by the natives
from a wooden club or sword, which it became the fashion to
throw at an adversary or quarry when otherwise out of reach ;
the bent form may in the first instance have been merely an
accident, or possibly it was the result of a cunning endeavour

Fig. 187 (± full size).

to make a sword that would overreach the guard of an
enemy ; this explanation almost suggests itself when one handles
a boomerang of the Australian type.

A convenient and handy size of boomerang termed the
" Payne-Galway," Fig. 187, has been placed on the market, 1

Fio. 188 (full size).

the section being as given in Fig. 188. This boomerang is of
English ash, steam bent ; it is of comparatively light weight and
takes but little practice to throw successfully. The author has
made boomerangs of sheet celluloid of three-limbed form
(Figs. 189 and 190); these, when of the sizes and thicknesses
shown, are suitable for indoor throwing or for use in a garden
or other restricted area ; the reason for and advantages of the
three-limbed form will be explained later.

1 Made by Messrs. Buchanan, of Pall Mall, London.
a. 401 D D

App. vnib.

APPENDIX

The theory of the flight of a boomerang has not been fully
worked out. Probably the most advanced work that has been
done on the subject is contained in a memoir by G. T. Walker, 1
in which an attempt is made to deal with the problem by
mathematical analysis. It appears to the present author that
this memoir, in spite of its unquestionable value, is not altogether
sound in its initial premises; the results, however, speaking
generally, are of the right kind.

In the discussion that follows no attempt is made to deal with
the question quantitatively, but rather to elucidate the principles

S~\

Fig. 189 (J full size).

on which the peculiarities of the flight of the boomerang depend,
regarding the latter as an example of the practical application
of gyroscopic action for the maintenance of stability in flight.

Let us first examine the case of a circular flat disc, Fig. 191,
projected edgewise, horizontally, with a rotary motion. Then the
disc will begin to fall, and after a short time we may regard its
weight as sustained by the aerodynamic reaction. But we know
that the centre of pressure will then be in advance of the centre
of gravity ; consequently there will be a couple or torque tending
to lift the leading edge, and this couple, counter-clock in
Fig. 191, will give rise to angular momentum of like sense.

1 Phil. Trans, cxc, 1897.

402

\

APPENDIX

App. vmb.

Owing to the spin of the disc this will be angular momentum of
aspect, and the successive positions of the disc are as represented
in the figure ; as a secondary consequence the flight path is
diverted from its original direction, owing to the lateral com-
ponent of the pressure reaction. Thus if we throw a disc in the
manner illustrated so that the direction of the spin viewed from

Fig. 190 (J full size).

above is counter-clock, it will first tilt to the right and then
undergo a change of course in like direction.

Let us now examine, under similar conditions, the behaviour of
a boomerang. Instead of the simple disc, a boomerang, Fig. 192,
is projected horizontally with a counter-clock spin about a
vertical axis.

It is evident that the centre of pressure may still be in advance
of the centre of gravity, but in addition to this the centre of
pressure is displaced laterally. If we adopt the treatment of

403 i> d 2

r

App. vmb.

APPENDIX

§ 107 and suppose that each element of the area of the
boomerang experiences a reaction proportional to the square
of its velocity, we shall have the centre of pressure displaced as

w

n

Fig. 191.

represented diagrammatically in Fig. 193 (a). In this figure it
is evident that the motion of the boomerang viewed from above
being counter-clock, the centre of pressure is displaced to the

><«*>

wpf

T~> («)

■-• (b)
(a)

Fig. 192. Fig. 193.

right, and the resulting torque, from the point of view of the
thrower, is counter-clock. Consequently the angular momentum
imparted to the boomerang about the axis of flight will nlso be
counter-clock, and owing to the rotation it will be angular
momentum of aspect; it is therefore the upper face of the

404

APPENDIX App. Vinb.

boomerang that will come into view as the result of the pre-
cession in the manner indicated in successive positions shown,
(b) t (r), and (d). As a consequence the path of the boomerang
will take an upward trend, Figs. 194-5, and it will continue to
rise until its kinetic energy is absorbed as potential, when it will

(a) ^ ' /

/

s
/
/
s

Fig. 194.

slide back along a similar path, and fall at the feet of the
thrower, Fig. 194 (a).

When we take account of the forward displacement of the
centre of pressure as well as its lateral displacement, we have

Fig. 195.

superposed on the motion as above depicted a further motion
partaking of that of the previous example, the disc, and the
flight path, instead of being in plan a straight line (Fig. 195),
becomes a curve (Fig. 196).

If the plane containing the boomerang at the instant of pro-
jection is not horizontal, the flight path undergoes considerable
modification. The form of trajectory given in Figs. 195-6 is not

405

App. VHIb. APPENDIX

good, because owing to the boomerang coming (translationally)
to rest at the highest point of its flight path, its efficiency from
an aerodynamic point of view becomes very poor, and the
duration of the flight is correspondingly shortened. By initially
inclining the plane of the boomerang when thrown as in Fig. 197,
the flight path becomes (in plan) a loop of approximately circular

Fig. 196.

form, Fig. 198, and its velocity is nowhere unduly lowered. If
the boomerang be supposed to lose none of its velocity, either
rotational or translational, and neglecting the "disc effect,"
it would be possible to find conditions such that the flight
path would be truly circular, the precession taking place about
a vertical axis, as in the case of a spinning top, or more exactly

a gyroscopic conical pendulum, Fig. 199. It
is commonly the object of the thrower to
approximate as closely as possible to such a
circular path. 1

In making the comparison between a
Fio 197 boomerang in flight and a gyroscope pen-

dulum it is necessary to suppose the said
pendulum loaded with a heavy counterpoise, in order to simulate
the conditions as to the direction of torque and of precession ;
thus any influence acting to hurry the precession reduces the

1 Experimenting in a large room with the boomerang shown in Fig. 189,
the author has frequently obtained beautifully regular flight pathe of con-
verging spiral form, just as might bo represented by the motion of the wheel
of the gyroscopic pendulum with a torque applied to hurry the precession,
and with a superposed motion of the whole instrument vertically down-
ward, the latter being due in the case of the boomerang to the unavoidable
dissipation of energy.

406

APPENDIX

App. vnib.

angle of inclination of the plane of rotation, and any influence
acting to retard the precession has the opposite effect.

Now a displacement forward of the centre of pressure, as in
the case of the disc or a simple aeroplane, manifestly tends to
hurry the precession, for it is the equivalent of a force acting
parallel to the pendulum rod but some little distance in advance
of it in its motion; 1 hence the effect of such displacement is to
reduce the inclination of the plane of rotation. This is actually

found to take place when a
boomerang is thrown ; it be-
comes during its flight less
and less inclined, that is to
say, the axis of rotation be-
comes more and more nearly
vertical, until after a complete
loop has been traversed the
boomerang will sometimes
describe one or more simple
oscillations of the kind already
described with reference to
Fig. 196. A diagram of a
typical flight path is given in
Fig. 198.

For the same reason, in order
to obtain the longest possible
flight, the initial angle should be very steep ; in some cases, in
fact, the boomerang is thrown in a nearly vertical plane.

The extent to which the disc effect is felt depends primarily
upon the form of the boomerang ; if the limbs are broad and the
proportions we generally " chubby," the torque about the trans-
verse axis will be relatively considerable. When, on the contrary,
the limbs of the boomerang are thin and spider-like, the disc
effect will not be so noticeable.

1 As seen in plan it is evident that such a force exerts a turning effort
about the vertical axis.

407

Fig. 198.

I

App. VHIb.

APPENDIX

The dissipation of energy in the flight of the boomerang in all
probability is not so very much different from that of a simple
aerodone, and thus we may take the loss of energy as represented
by the weight multiplied by y times the distance traversed. The
value of y under the exceptional conditions in question we have

V

Fia. 199.

no means of calculating, but evidently if the boomerang is pro-
perly formed it will be of the same order of magnitude as the
simple aerodone, or say, = '2.

Now, since the initial and final altitudes are approximately the
same, we may take it that the whole of the energy expended is
taken from that of initial velocity or initial rotation. Taking a

408

APPENDIX App. VHIb.

flight of 500 feet traversed, we have, if W be the weight, energy
= 100 W. Now, if this be taken wholly from the energy of trans-
lation, and if, as is sometimes the case, the velocity be approxi-
mately nil when the boomerang alights, the initial velocity will
be that corresponding to a fall of 100 feet, or say 80 ft./sec, and
the time of flight will be between six and seven seconds. As a
matter of fact, the time for a flight of this length is usually about
eight seconds, so that probably the velocity of projection is some-
what lower and some of the energy is derived from the rotation.

The author does not believe, in the case of a good boomerang,
that much of the energy comes from the rotation, for after a
successful flight the boomerang commonly comes to earth spin-
ning briskly. In some cases it has even been observed that the
speed of revolution is greater at the end of a flight than at the
beginning, both as a matter of simple observation and when
counted against a stop-watch.

The relative absorption in flight of the energy of translation
and that of rotation depends upon the form of the boomerang.
Thus if it be quite flat it is evident that both sources of energy
will be utilised, whereas if it be of appropriate screw form the
rotation may be maintained by the pressure reaction, when the
energy of translation will alone be drawn upon. Again, if the said
screw form be exaggerated the energy of rotation may actually
be increased at the expense of that of translation. If, on the
contrary, the screw form be of opposite "hand," i.e., of cork-
screwlike form, the energy of rotation may be drawn upon to
any desired extent.

It is not always easy to appreciate from mere inspection of a
boomerang whether it is, in effect, of right-handed or of left-handed
screw form ; thus the flat or under face may be markedly a right-
hand screw and yet the whole form be in effect left-handed.
The reason for this is to be found in the explanation given in
Vol. I., § 80, of the aerial tourbillion ; the flow round the convex
leading edge of the boomerang ejects the dead water in that region
and causes a suction by the centrifugal force of the adjacent

409

App. VIIIb. APPENDIX

stream tending to maintain the rotation. This explanation in
the present case is to some extent a conjecture, but the author
believes that it will in due course receive confirmation. 1

There is a further difficulty in judging the effective form of a
boomerang, owing to its want of symmetry. If we suppose a
two-limbed boomerang rotating truly about its principal axis, it
is evident that there is nothing to balance the unsymmetrical
distribution of the pressure reaction, and in practice, there-
fore, the boomerang will not spin exactly about its principal
axis, but about an axis making a small angle with same, so
that the resulting dynamic and aerodynamic couples 2 will be
in equilibrium. It is for this reason that the author has
recently adopted the three-legged form; such a form appears
to be in every way equal to the Australian type, and has the
advantage of being perfectly symmetrical so that it will spin true,
and measurements of angles, etc., relatively to the containing
plane may be relied upon.

The torque that gives rise to the gyroscopic precession is a
variable depending upon the relation that exists between the
velocity of rotation and that of translation, and upon the total
pressure reaction. The latter is of necessity equal to the
apparent weighty i.e., the resultant of the weight and the centri-
fugal force of the boomerang at every instant ; for any given
value of the apparent tveight the torque is proportional to the
distance separating the centre of gravity and the centre of
pressure, and hence depends upon the position of the latter. It

1 Certain screw forms are very capricious in their behaviour. The aerial
tourbillion described in Vol. L (§ 30) may be looked upon as a windmill
capable of behaving either as right or left handed. There is a certain form
of screw that, mounted in a similar manner, will act as a windmill until a
critical speed of revolution is reached, and then it will suddenly change its
effective pitch and become a screw propeller eating its way up to windward
until its velocity is reduced to a lower critical limit when the windmill
conditions again supervene.

9 The aerodynamic couple here referred to is not that to which the pro-
cessional motion is due, but one due to the want of symmetry of the
boomerang ; this couple is mobile, its axis rotutes with the boomerang.

410

APPENDIX

App. vnib.

is evident that if the velocity of translation be zero, and that of
rotation finite, the position of the centre of pressure will coincide
with the centre of gravity. If, on the other hand, the velocity of
rotation be small in comparison with that of translation, there
will on the whole be but little lateral displacement of the centre
of pressure, and in the limit, if we suppose the rotation to be
extinguished, although the centre of pressure will be displaced
forward, as in the simple aeroplane, there will be no lateral
displacement, and the torque that produces the precession about
*he transverse axis will vanish.

Let us represent the boomerang by a straight narrow aero-
plane as in Fig. 200, and let us suppose that the instantaneous

Fig. 200.

centre be moved from the geometric centre away to infinity,
so that in the first instance the motion is pure rotation and in
the latter the motion is of pure translation. It is evident that,
as above stated, for these extremes there is no displacement of
the centre of pressure ; on the basis of equal increments of K
(com p. § 107), the pressure curve in the two cases is as repre-
sented respectively in Figs. 201 and 202. Now, for intermediate
positions of the instantaneous centre of motion there will be
displacement of the centre of pressure, as indicated in Fig. 200,
in which the displacement of the centre of pressure is repre-
sented by the ordinates of a curve where abscissae give the
position of the instantaneous centre. As plotted, this curve is
based on the V squared law and on the assumption of a uniform
value of K per unit length. If the aeroplane is other than flat,

411

App. Vlllb.

APPENDIX

as when it is, in effect, of screw form, or if the distribution of A r
is not uniform, the form of the curve is different. In the former
case, for example, the curve does not approach the line of zero
torque asymptotically as the instantaneous centre runs to
infinity, and the form of the curve is otherwise modified.

It is manifest that, from the considerations stated, the problem
is far too complex at present for complete mathematical solution,
that is for the flight path to be plotted from given data, etc.,
though it would appear possible to effect some approximation if
the conditions are restricted artificially to a sufficient degree.

The author believes that under ordinary circumstances the

Fig. 201.

Fig. 202.

portion of the torque curve utilised is that in Fig. 200 in
the region about and on the right-hand side of the maxi-
mum, so that the falling off of the velocity of translation,
with that of rotation fully conserved, results on the whole
in a diminution of the torque, and thus a reduction in the
rate of precession, so that the flight path, instead of being a
converging spiral, as would he the case on the basis of the
gyroscopic pendulum, becomes of approximately circular
form, so that the boomerang returns to its point of departure.
This effect is probably helped by the fact that as the plane of
rotation becomes more and more nearly horizontal the total
pressure reaction becomes less and less, since the total pressure
reaction may be computed by a simple resolution of forces.

412

APPENDIX App. Vlllb.

Hence the torque is diminished from this cause as well as that
stated.

It appears to the author that the above facts point to an
opening for the quantitative treatment of the problem. Thus,
for example, if we confine ourselves to the portion of the curve,
Fig. 200, where the displacement of the centre of pressure is
sensibly constant, 1 the torque that gives rise to the precession
will, under changes of inclination, vary directly as the effective
weight, that is inversely as cos a, where a is the angle made by

the axis of rotation to the vertical
(or by the plane of rotation to the
horizontal). But by the equation
to the gyroscope, App. VIL,

t = Io> 12 sin a
hence we can correlate changes in
the angle a and the rate of pre-
cession. Thus in the particular
case where a is 45° the torque and
sin a have the same relative rate
of change, and at this critical
value the rate of precession il is
constant in respect of a.
' It is well known that the flight

of a boomerang is greatly assisted by an appropriate wind. In
still air considerable skill is required to make sure of a clean
return, but flights of the form given in Fig. 208 are quite easy
even to a novice. When such flights are made with a wind
blowing from the " left front " a defective throw will become a
good return. If, for example, a given throw in still air would
give rise to a path of roughly cycloidal form, Fig. 204 (a), we
know that with an appropriate superposed translation this
becomes an approximate circle, Fig. 204 (/;), and likewise for

1 From an inspection of Fig. 200, and from the observed behaviour of a
boomerang, it would appear that, for a considerable portion of the flight
path at least, this assumption, as a first approximation, is justified,

418

App. Vlllb. APPENDIX

other curves. A further effect of wind is that if opposed to the i

direction or projection it increases the energy of flight, so that )

the total time of flight may be greatly prolonged. A well-thrown
boomerang in a high wind will return at a considerable height
over the thrower's head with a velocity like that of driven grouse.
The whole question of wind effect, unless the weather is exceed-
ingly boisterous, is a mere matter of relative motion, but even as
such it possesses some subtleties. Thus, as remarked, the
initial energy is enormously increased, in some cases being more

Fig. 204.

than doubled by the boomerang being thrown in the teeth of a
gale. Also, the greater the head wind the more important it
becomes to give an effective initial spin to the boomerang, for
the wind adds to the velocity without adding to the rotation,
and thus, unless the additional spin be given by the thrower, the
behaviour of the boomerang in a high wind is not satisfactory.
The author has found that in the case of boomerangs copied
from Australian models, in which the angle is very open, and
the principal axis consequently ill-defined, it is often easy to
throw down wind (although it will not then return), but
impossible to throw against even a moderate breeze, owing to
the insufficiency of the initial spin.

414

APPENDIX VIIIc

THE SCHLICK MAEINE GYBOSCOPE

The Schlick marine gyroscope is a recent invention for pre-
venting or minimising the rolling of ships at sea This apparatus,
illustrated diagrammatically in Fig. 205, consists of a fly-wheel A,
whose bearings B, B, are carried by the frame C, which in turn
is mounted on gimbals or trunnions D, D, arranged transversely
to the vessel, the axis of rotation under normal conditions being
vertical. When the vessel attempts to roll in a sea-way the
gyroscope rocks or precesses on its gimbals, and by its precession
it counteracts the turning moment, which would otherwise give
the vessel angular momentum about a longitudinal axis — in other
words, would produce a rolling motion.

The action of the gyroscope is illustrated in the series of
diagrams given in Fig. 206. The lettered diagrams (a), (b), (c),
etc., represent the vessel in the various phases of a passing wave,
the direction of the torque due to the slope of the wave being
clockwise when the water is higher on the left-hand side and
vice versa. The angular momentum communicated by this
torque appears in the momentum of aspect (comp. Appendix VII.)
of the gyroscope, instead of appearing in the rolling of the
vessel, as would under ordinary conditions be the case. Thus,
referring to Fig. 206 (a), we have a condition of maximum
clockwise torque, and the gyroscope is precessing in such a
manner as to exhibit clockwise momentum as at (ft), the
maximum condition being reached when, as at (c), the applied
torque has fallen to zero. The wave surface then begins to slope

415

App. vine

APPENDIX

in the opposite direction, and the torque becomes counter-clock;
the result of this is a precession of the gyroscope in the opposite
sense and a diminution of the clockwise momentum of asjwct,
until, as shown at (d), nearly the whole of the clockwise momen-
tum is given up, following which the momentum of aspect

Fig. 205.

becomes counter-clock, (c), and continues to increase until the
counter-clock torque falls to zero, when the precession, having
reached its maximum value, ceases also, (/). The applied
torque then again becomes clockwise, and the condition of the
gyroscope returns to that shown in diagram (a) and the whole
cycle is repeated.

Trials of this device have shown it to be highly efficient, and it

416

APPENDIX

App. vmc

a

^

/

d

>K'77r:

f

Fig. 206.

A.

417

£ £

App. VIIIc. APPENDIX

will doubtless come into more general use. An account by Sir
William White has been published in Engineering, Vol. LXXXIIL,
p. 448.

In order to ensure the mean position of the gyroscope being
that shown, i.e., axis vertical, its mass centre is arranged slightly
below the girubal centres. 1

It has been suggested by some that the force necessary to hold
a vessel steady in a sea-way must be such as to cause an undue
strain on the vessel, but such is by no means necessarily the case.
Where no gyroscope is fitted and the rolling of the vessel is
largely due to synchrony, the stresses and strains may some-
times be very much greater than would be the case were the
vessel prevented from rolling altogether by brute force, whether
supplied in the form of a gyroscope or otherwise. Of course,
under all conditions the gyroscope should be fitted in the
immediate neighbourhood of a transverse bulkhead, and where
a vessel is of great size it would be obviously expedient to employ
more than one gyroscope distributed along the length of the
vessel in approximate proportion to the displacement.

1 The above exposition, although fundamental and sufficient for the present
purpose, is not altogether complete ; complications arise under certain cir-
cumstances, as for example when the gyroscope is set in motion in a vessel
already rolling. (Ee fere nee should be made, for more complete detail, to
the article by Sir William White already cited.)

418

APPENDIX VIIId

THE GYROSCOPE FOR DIRECTION MAINTENANCE—

THE WHITEHEAD TORPEDO

The gyroscope in the Whitehead torpedo differs as to its mode
of employment from the preceding examples ; its function here
is purely directive, it is not in any way concerned with the
maintenance of equilibrium.

In brief, the gyroscope is mounted in a similar manner to a
lecture model, on double gimbals, so that it is unaffected by the
rotational movements of its surroundings ; the torpedo may
change its course upward or downward, or to the right or left,
without the axis of rotation of the gyroscope undergoing any
change of direction. The outer gimbal of the mounting is
arranged to actuate a small rotary valve which controls the
admission of air to a pneumatic cylinder ; thus the gyroscope is
utilised as if it were an accurately suspended mass of great
moment of inertia.

In order to obtain a clear idea of the action of the above
combination it is convenient to regard the rotary valve as
being rigidly attached to one of the outer ring trunnions of the
gyroscope, and so as being fixed in space, whilst the valve case
is fixed to the torpedo and shares with the latter any change of
course; then, every time the torpedo from any cause tries to
depart from the ordained course, it moves the valve case round
relatively to the valve (the latter being held stationary by the
gyroscope), and so admits air under 250 lbs. pressure to the

419 e k 2

illlllll

Pig. 207.

APPENDIX App. Vmd.

steering cylinder and almost immediately rectifies the error of
direction. As a matter of fact, the course of a torpedo con-
trolled by a gyroscope is not a straight line, but rather a just
perceptible zig-zag whose mean direction is the course set by the
gyroscope.

Beferring to Fig. 207, in which the more essential parts are
shown dissected from their surroundings, and to the photograph,
Fig. 208, the gyroscope wheel A is mounted on ball-bearings
in the ring 7?, which is pivoted in the outer ring C, which in
turn is mounted on fixed centres in the usual manner. The
rotary valve is not actually mounted on the outer ring trunnion
as described, but is operated by a pin P projecting upward from
the outer ring arranged to engage in a slot in the lever L fixed
to the valve V, the end of this lever being so formed that the
valve is capable of being moved through a certain definite range
and no more, any further motion of the torpedo away from its
prescribed course being without effect. The air is controlled to
and from the steering piston, contained in the cylinder S, by
passages in the cylindrical valve V; this valve is of small size,
and but little force is required to move it one way or the
other.

The effect of the little resistance that the valve offers is to
cause a precession of the gyroscope about the trunnion axis
of the inner ring li, without any appreciable yielding of the outer
ring C ; consequently the course is not in the le.ist affected by
valve-friction. The course could only become involved if the
precession were to continue until it could go no further, as when
the ring 7? reaches the limit of its motion, or, if no stops exist,
until the axis of spin becomes coincident with the mounting of
the ring (7. Under the conditions of service such a contingency
never arises, the total "flight " of a torpedo occupies but a few
minutes, and as it has at the worst but little " bias " one way or
the other, the reactions the gyroscope experiences are sometimes
in one direction and sometimes in the other ; consequently the
precessional movements in great part correct each other, and

421

App. vmd.

APPENDIX App. Vind.

there is no continued precession such as would be necessary to
put an end to the efficiency of the apparatus.

The gyroscope is put in action automatically immediately
after launching, the initial spin being given by a toothed seg-
ment D (Fig. 208), actuated by a spring, and initially in gear
with the pinion teeth on the axle of the fly-wheel. The friction
on the ball-bearings is so small that the initial spin imparted
in this way lasts for upwards of 20 minutes.

428

APPENDIX VIIIe

OTHER APPLICATIONS OF THE GYROSCOPE

Othkk applications of the gyroscope exist that are of interest
from the present point of view. Thus in the Brennan mono-rail the
equilibrium is maintained by a gyroscope combination, in which
the precession is hastened by the action of a small roller mounted
on an extension of the spindle of the revolving wheel. The
equilibrium thus has some analogy to that of a spinning top, in
which the precession is hastened by the factional contact with
the ground. 1

In 181)1 Sir Hiram Maxim proposed to secure the longitudinal

APPENDIX App. VIHe.

through a ball-and-socket joint at the lower end of the spindle
and escapes through apertures arranged tangentially at the
periphery and through an axial hole at the upper end of the
spindle ; the former acts by recoil, after the manner of the well-
known "Hero" engine, to maintain the velocity of rotation,
and the latter acts in a very ingenious manner to operate one or
another of a series of hydraulic cylinders by which the " plat-
form " (on which may be mounted a searchlight or a light gun)
is held in a horizontal plane. It is well known that water, escaping
by an orifice, is capable of generating a pressure little short of
that by which it is impelled. Thus in close proximity with the
axial nozzle four apertures are arranged, each communicating
with one of four hydraulic cylinders arranged to act on the plat-
form. The latter is mounted on trunnions to swivel universally
and carries the gyroscope mounting, so that if by the rolling or
pitching of the vessel the platform begins to move rotationally
in any direction, the jet of water issuing from the axial nozzle
passes into one or other of the orifices, and there generates pressure
in whichever of the hydraulic cylinders is in a position to correct
the motion of the platform, and so maintain its constancy of
direction. The platform is thus made to copy at all times the
position of the gyroscope, and so partakes of its steadiness.
The above form of relay is a singularly beautiful contrivance,
inasmuch as it acts without any contact, frictional or otherwise,
so that the gyroscope is unconstrained, and is not influenced by
the mechanism on which it operates.

The weight of the gyroscope wheel is borne in most part
by the water pressure at the socket joint, which serves as a
" foot-step " bearing. Any remaining friction is utilised to
retard the precession and so to bring the wheel into its position
of static stable equilibrium ; the centre of gravity of the rotating
mass is arranged slightly below the point of support.

425

V

i

INDEX

A.

Acentric aerodrome, type of, § 16

Active flight, as affecting problem of longitudinal stability, § 75

Aerodone, the, as a means of studying flight, § 1 ; curly forms of, §§ 8, 9 ;

author's 1694 model, § 10 et seq. ; used for verification of theory, § 68

et seq.
Aerodrome, author's 1894 model, §§ 13, 14, and App. HI. ; stability

investigated, § 80
Aerodromic radius, §§ 102, 108, 114
Aerodynamic radius, §§ 102, 106, 107, 114
Aerofoil, §§ 160, 170 ; flexibility of, §§ 116, 170, 181 ; steering by means

of, § 177
Albatros, position of feet in flight, §§ 76, 70 ; stability of the, § 70 ; soaring

flight of, § 143
Attitude, meaning of, § 4 and glossary ; changes due to moment of

inertia, § 53
Author, aerodones, §§ 10, 70, 71, 165 et seq. ; theory of stability, App. II. ;

aerodrome 1894. §§ 13, 14, and App. III. ; investigation of stability of

birds in flight, §§ 75, 76, 77, 78, 70
Author's experiments, 1894, § 10 el seq. ; in confirmation of phugoid

theory, §§ 68, 60 ; in verification of equation of stability, §§ 70, 71,

72, 73, 74 ; method, § 165 et seq.

B.

Ballast, distribution of, § 5 ; composition, § 168

Pal lasted aeroplane, the simplest form of aerodone, §§ 2, 8 ; launching of,

§ 3 ; facts demonstrated by, § 3 ; longitudinal stability of , § 4 ; lateral

stability, § 5 ; directional stability, § 6 ; symmetry of the, § 6 ; thickness

of mica for, § 178
Baste, observations on dynamic soaring, § 157
Bazin, " montagnes russes " soaring model, § 152
Biids, stability of, §§ 1, 75, 76, 77, 78, 70 ; stability complicated by con-

ditions of active flight, § 75 ; action of tail in flight, § 75 ; effect of wing

flexure, $§ 75, 77, 116

427

INDEX

Boomerang, App. VIH. B.

Brennan, gyroscope used in mono-rail apparatus, App. VIH. E.

0.

Catapult, launching device, § 12

Centric, meaning of, as applied to aerodone, § 16

Condor, soaring flight of, § 143 ; conditions of existence of, § 148

Constants, C, §§ 37, 41 ; K, §§ 37, 42 ; C and K, relation between, § 54 ;

changes in value of K, § 55
Continuation, of flight path, § 17
Corresponding speed, theory and laws of, § 126 et seq. ; in relation to the

phugoid theory, § 129
Course, § 93
Cubic equation, solution by slide rule and by graph, App. IV.

D.

Damping, of phugoid oscillation, §§ 46, 48, 49 ; law of, § 121 ; examples
and plottings, § 122 et seq. ; of lateral oscillation, § 91

Danger zone, § 39

Departures from elementary type, § 136

Diomedea exulans. See Albatros.

Directional stability, § 93 et seq. ; a study in, §§ 95, 96, 97 ; change of
course, conditions governing, § 97

Double suspension, method of determining moment of inertia, App. VI.

Drift of projectiles, App. VIII. A.

Dynamic soaring, § 150 et seq. ; more than one kind, § 150 ; available
energy, § 151 ; historical development, § 152 ; Mouillard's theory,
§ 152 ; the " switchback " model, § 153 ; quantitative investigation of,
§ 154 et seq. ; table of relative wind pulse velocity, § 155 ; on basis of
harmonic wind pulsation, § 156 ; form of orbit, § 157 ; in its relation to
the phugoid theory, § 158; energy available and utilised. § 159; efficiency
of, §§ 160, 161 ; as determined by different kinds of aerial disturbance,
§ 162 ; Langley's observations on wind fluctuation, § 162 ; as dependent
on dead-water regions, § 163 ; mixed conditions, § 164 ; hypothetical
case of, § 164

Dynamic stability, illustrative cases, § 1

E.

Energy, of fluid motion in periptery, § 115 ; required for dynamic soaring,
§159

Equation of stability, §§ 61, 62, 63 ; basis of, discussed, § 111 ; un-
accounted factors, § 112 ; effect of " wash " of aerofoil, §§ 112, 113

Equation to flight path, § 20

Equilibrium, two methods of maintaining known to nature, § 1 ; indefinite-
ness of problem, § 2 ; longitudinal, two kinds, § 4 ; condition of, not
necessarily stable, § 50

428

INDEX

Experimental aerodonetics, § 165 et seq:

Experimental verification, of theory of longitudinal stability, § 64 et seq.

P.

Fin resolution, investigation, §§ 99, 100

Fins, employed for direction maintenance, §§ 11, 98, 99

Flexure, of wing or aerofoil, §§ 76, 77, 116, 181

Flight models, materials employed, §§ 167, 168 ; space required for model
experiment, §§ 70, 166 ; nnca for, § 167 ; ballast for, § 168 ; the aero-
foil, §§ 169, 170 ; fin-plan and tail-plane, § 171 ; moment of inertia,
measurement of, § 172 ; admissible proportions, § 173 ; example
calculation, § 174 ; tail-plane, angle of, §§ 175, 176 ; unstable periptery,
§ 175 ; steering, § 177 ; vagaries of the flight path, §§ 179, 180, 181

Flight of boomerang, App. VIII. B.

Flight of projectiles, App. VIII. A.

Flight path, undulations, § 8 ; equation of the, § 20 ; stability of the, § 50
et seq. ; plotting, § 24 et seq. ; abnormal, §§ 179, 180 ; sudden changes
in, § 181

Freedom, degrees of , § 2 ; limited by artificial means, § 2 ; interaction of
motions in the, § 7

Froude, J. A., observations on soaring flight, § 148

Froude, W., on soaring flight, § 149 ; law of corresponding speed, § 126

G.

Gliding, the equivalent of a constant force of propulsion, §§ 48, 50 ; in

soaring flight, § 148
Gulls, soaring flight of, § 142
Gusts. See Wind.
Gyroscope, the, App. VII. ; theory of, App. vli. ; applications of,

App. VIH. ; of Whitehead torpedo, App. VIII. D. ; proposed as a

means of maintaining stability in flight, App. VIII. E. ; Schlick marine,

App. VIII. C.
Gyroscopic action, theory of, App. VH. ; in the flight of projectiles,

App. VIII. A. ; in the flight of the boomerang, App. VIII. B. ; in

Brennan mono-rail, App. VIII. E. ; in Tower Bteady platform, App.

VIII. E.

H.

Hargraves, flight model mentioned. § 8

Hele Shaw, ballasted aeroplane, glider, or aerodone, § 9

Hi r undo apus. See Swift.

Historical note, on theory of flight, § 140

I.

Inflected curves, on the phugoid chart, § 26 ; plotting the, §§ 27, 28
Inflection, point of, calculating, § 28 and App. V.

429

INDEX

L.

Lanchester. See Author.

Langley, aerodrome, § 8 ; observations on wind fluctuation, § 162

Lateral and directional stability, § 83 et seq. ; . mutual relationship,

§§ 84, 101 ; basis of theory discussed, § 113
Lateral stability, oscillations in the transverse plane, §§ 86, 87, 88;

damping influences, § 89 ; effect of moment of inertia on, § 90 ; form of

oscillations, §§ 90, 91 ; oscillations in the transverse plane in practice,

§92
Launching staff, §§ 3, 166

" Leading plane," employed in lieu of tail-plane, § 136
Lilienthal, gliding machine, §§ 8, 81 ; stability investigated, § 81 ; relied

on skill to preserve equilibrium, § 81 ; fatal accident to, § 81 ; instability

due to insufficient velocity, § 82 ; suitable velocity for, § 173
Limitations and unaccounted factors, § 115 et seq.
Longitudinal stability, §§ 4, 38 et seq., 50 et seq., 118 et seq., 158,

173, and App. II.

M.

Marey, observations on the form of the flight path, § 67 ; observations on

the motion of the ball in Bazin's model, § 157
Marine gyroscope, Schlick, App. VIII. C.
Materials used for flight models, §§ 167, 168 '
Maxim, captive flying machine, acentric, § 16

Mica, properties and defects of, §§ 6, 167 ; used for flight models, § 167
Model experiments. See Experimental aerodonetics ; also Scale

model experiments.
Models. See Flight models ; also Aerodones
Moment of inertia, effect of, §§ 48, 49, 52, 53, 54 ; influence on flight

path, §§ 59, 60, 62 ; influence always detrimental, § 92 ; measurement

of, § 172 and App. VI. ; of birds, expression for, § 79
Mouillard, observations concerning flight path, § 66 ; conditions of soaring

dependent on aerial motion, § 145 ; on the variable sustaining power of

wind, § 162

N.
Natural gliding angle, §§ 3, 61
Natural velocity, §§ 3, 21

O.

Olton, author's flight experiments at, § 13

Orbit, in dynamic soaring, form of, § 157

Oscillation, vertical, in the flight path, §§ 4, 15, 17 ; phugoid oscillation,
§§ 26 et seq., 30 et seq., 47 ; damping of phugoid oscillation, §§ 46, 48,
49, 121, 122 et seq. ; secondary effects of, § 181 ; lateral, §§ 5, 83
et seq. ; damping lateral, § 91

480

INDEX
P.

Penaud, flying model, or aerodrome, § ; oa undulatory flight path, § 65 ;

theory of stability, App. I.
Periodic disturbance, effect of, § 47
Phase curve, § 32

Phase length, of phugoid curve, §§ 32, 36
Phugoid chart, the, § 32, also Fig. 42, and frontispiece ; plotting the,

§ 24 et seq. ; plotting data, App. V. ; employment of, § 43 ; changes

in scale of, § 53
Phugoid curves or phugoids, equation to, § 20 et seq. ; special cases of,

§ 23 ; semi- circular, § 23 ; straight line, § 23 ; " tumbler " curves, § 26 ;

inflected curves, § 27 ; plotting the, § 24 et seq. ; of small amplitude,

§ 29 ; form of orbit, § 30 ; trammel, use of in plotting, §§ 25, 26 ; time

period, §§ 33, 34, 36 ; unstable, § 39 ; form of nearly straight, § 56 ;

considered as sine curve, 56 ; damping of the, §§ 46, 48, 49, 121, 122

et seq.
Phugoid oscillation. See Oscillation.
Phugoid theory, the, § 18 et seq. ; hypothesis, § 19 ; investigation, § 20 ;

the key to the theory of flight, § 82 ; aerodynamic basis of the, § 110
Pline, glider or aerodone due to, § 9 ; flight path of, § 67
Plotting data, of phugoid chart, App. V.
Projectiles, flight of, App. VIII. A.
Propeller, automatic feathering used by author, App. III.
Propulsion, by twisted indiarubber, §§ 9, 13, and App. III. ; investigations

relating to effect of, §§ 119, 120, 121 ; influence of mode of, on stability,

§ 118 et seq. ; the screw propeller as means of, § 118 ; curves of torque

and h.p., §§ 118, 119

*

R.

Rayleigh, dictum defining conditions essential to soaring flight, § 144

Relative motion, wind in relation to flight path, § 42 ; in soaring flight,
5 153 et seq.

Resistance, influence of on flight path, §§ 48, 51 ; effect of in quantitative
investigation of stability, §§ 57, 58, 61 ; and propulsion, combined
effect, § 118 et seq.

Rotative stability, § 101 et seq. ; investigation, § 103 et seq. ; verifica-
tion, § 137 ; application of theory, § 138

S.

Santos Dumont, on upward motion of air, § 146

Scale model experiment, § 131 et seq. ; allowances, § 134 ; moment of

inertia of, §§ 132, 133. See also Experimental aerodone tics.
Schlick marine gyroscope, App. VIII. C.

Sea breeze, rationale of, and examples as evidence of up- current, § 145
Size of aerodone, effects of, §§ 39, 82, 173

481