flbacmillan'0 Science 
 
 STABILITY IN AVIATION 
 
MACMILLAN AND CO., LIMITED 
 
 LONDON . BOMBAY . CALCUTTA 
 MELBOURNE 
 
 THE MACMILLAN COMPANY 
 
 NEW YORK . BOSTON . CHICAGO 
 ATLANTA . SAN FRANCISCO 
 
 THE MACMILLAN CO. OF CANADA, LTD. 
 
 TORONTO 
 
Stability in Aviation 
 
 An Introduction to Dynamical Stability as 
 applied to the Motions of Aeroplanes 
 
 BY 
 
 G. H. BRYAN, Sc.D., F.R.S. 
 
 PROFESSOR OF PURE AND APPLIED MATHEMATICS IN 
 
 THE UNIVERSITY COLLEGE OF NORTH WALES 
 FORMERLY FELLOW OF PETERHOUSE, CAMBRIDGE 
 
 MACMILLAN AND CO., LIMITED 
 ST. MARTIN'S STREET, LONDON 
 
 191 1 
 
RICHARD CLAY AND SONS, LIMITED, 
 BRUNSWICK STREET, STAMFORD STREET, S.E. 
 AND BUNGAY, SUFFOLK. 
 
 '. ' 
 

 PREFACE 
 
 UP to the present time the problem of stability has 
 received very inadequate attention in connection with 
 aviation. From the point of view of the practical 
 aviator, this is, perhaps, little to be wondered at. It 
 would scarcely be possible for him to devote months of 
 concentrated attention to long and laborious stability 
 investigations when it is no exaggeration to say that very 
 frequently his success or failure depends, above all things, 
 on the names of the towns at which he starts and lands. 
 If a prize is offered for a flight from Folkestone to 
 Flushing, it is useless for him to fly from Harwich to 
 the Hook, even on a much more stable machine than 
 that used by the winner of the prize. 
 
 It is hoped that the publication of this memoir will lead 
 to aeroplane stability being made the subject of much 
 more continuous study and investigation than has been 
 possible in the past. A general abstract of the present 
 investigation is contained in the introduction, which may 
 be read with advantage before proceeding to details of 
 a more mathematical character. The general conclusions 
 show that there should be no difficulty in securing in- 
 herent stability, both longitudinal and lateral, in an 
 aeroplane, by means of suitably placed auxiliary surfaces 
 rigidly attached to the machine ; but in order to achieve 
 success the conditions of stability must be very carefully 
 studied, and account must be taken of the effects of 
 
 272834 
 
vi PREFACE 
 
 the inclination of the flight path to the horizon and 
 other causes which may affect the result seriously. 
 
 There seems a general desire on the part of many 
 writers to minimise the dangers of instability or 
 defective stability, and to attribute accidents to other 
 causes. But in reading the accounts of accidents, 
 both fatal and otherwise, that appear every few days in 
 the daily papers, it is difficult to avoid coming to the 
 conclusion that much of this loss of life and damage 
 
 o 
 
 could be avoided by a systematic study of stability and 
 certain other problems regarding the motion of aero- 
 planes particularised in this book. 
 
 To the mathematician who is able to devote any time 
 to original work, a wide region of unexplored ground 
 is opened up, of which it has, in many cases, only been 
 possible to exhibit glimpses in these pages. 'There is 
 a freshness about this region which can scarcely be found 
 to the same extent in searches for differential equations 
 that have not been integrated, high primes, or further 
 additions to the large existing collection of properties 
 of triangles and circles. I do not think any mathe- 
 matician who cares to take up the subject now will 
 experience any difficulties comparable with those which 
 have been encountered in the preparation of the present 
 book. 
 
 There is abundant work, too, for the student who wishes 
 to undertake "research" for educational purposes; and 
 "aeroplane stability " seems to me a very useful alter- 
 native for some of the branches of applied mathematics 
 now taught in our universities. 
 
 Quite recently, much has been written regarding so- 
 called " automatic stability," depending on the use of 
 gyrostats, pendulums, or other movable parts. Apart 
 from the fact that movable parts are liable to get out 
 of order, it must be remembered that they increase the 
 number of degrees of freedom of the machine, thus further 
 
PREFACE vii 
 
 adding to the number of conditions which have to be 
 satisfied for stability a number quite large enough 
 already. I anticipate that the successful aeroplane of 
 the future will possess inherent, not "automatic" 
 stability, movable parts being used only for purposes 
 of steering. 
 
 The great difficulty in writing this book has been 
 the newness of the subject, and the fact that the further 
 one advances in it the greater appear the possibilities 
 of still further exploration. The selection and arrange- 
 ment of the text has thus been much more difficult than 
 it would be in a text-book compiled from existing 
 literature. I have tried, so far as possible, to avoid, 
 introducing any investigation the results of which were 
 certain to be only of academic interest ; but this attempt 
 has not made it much easier to draw the line. For 
 example, the transformations of 30, which are, of course, 
 particular cases of a more general transformation, were 
 placed on the excluded list, but their application to the 
 pendulum experiment suggested the desirability of 
 retaining them. 
 
 One pleasant duty remains, and this is to record my 
 indebtedness to Mr. E. H. Harper for the valuable 
 services he has rendered in connection with the investi- 
 gations here described, and to claim priority for him 
 in many of the results. It was Mr. Harper that directed 
 my attention in the first instance to the important 
 effects of the inclination of the flight path on stability, 
 and the investigations of the lateral stability of the 
 Antoinette type, and several other forms, are entirely 
 due to him. The whole of the formulae in this book 
 have been checked by independent working by 
 Mr. Harper and myself, and I hope that they are correct. 
 Besides Mr. Harper, Mr. A. Ferguson and Mr. Robert 
 Jones have revised the proofs and made a number of 
 corrections. It is impossible to be too careful in a 
 
viii PREFACE 
 
 matter where a mere slip of a sign might change stability 
 into instability, and for this reason it is desirable that 
 readers should, as far as possible, keep a further check 
 on the formulae, even in spite of these precautions. 
 
 Last but not least my thanks are due to Prof. R. A. 
 Gregory not only for his unfailing kindly advice and 
 assistance in the preparation of the text, but also for his 
 offer to include a volume on the investigation of stability 
 in the series of Science Monographs, of which he is the 
 Editor. There are probably many instances, of which 
 this is one, in which a volume of this kind presents a more 
 appropriate medium for the publication of results of 
 investigation than the proceedings or transactions of 
 a scientific society. Had it not been for Prof. Gregory's 
 timely invitation, the theory of stability would have been 
 developed much less fully than has now been done in 
 these page 5 . 
 
 G. H. BRYAN. 
 
CONTENTS 
 
 CHAPTER I 
 
 PACK 
 
 INTRODUCTION AND SUMMARY . , . . . . . . . . 1 
 
 CHAPTER II 
 
 FUNDAMENTAL PRINCIPLES . . . ' . ... ,< . 19 
 
 CHAPTER III 
 
 GENERAL CONSIDERATIONS REGARDING SYMMETRICAL DERIVATIVES . 38 
 
 CHAPTER IV 
 
 GRAPHIC STATICS OF LONGITUDINAL EQUILIBRIUM 57 
 
 CHAPTER V 
 
 LONGITUDINAL STABILITY OF SINGLE-LIFTING SYSTEMS . . . 75 
 
 CHAPTER VI 
 
 LONGITUDINAL STABILITY OF DOUBLE-LIFTING SYSTEMS. EXTENSION 
 OF RESULTS TO SYSTEMS OTHER THAN NARROW AEROPLANES 
 MOVING AT SMALL ANGLES . . . .... . . 95 
 
 CHAPTER VII 
 
 ASYMMETRIC OR "LATERAL" STABILITY STRAIGHT PLANES AND 
 
 VERTICAL FINS . 123 
 
x CONTENTS 
 
 CHAPTER VIII 
 
 LATERAL STABILITY BENT UP PLANES . 
 
 CHAPTER IX 
 
 GENERAL CONCLUSIONS 
 
 CHAPTER X 
 
 COMPARISON WITH OTHER THEORIES ... 
 
 CHAPTER XI 
 
 PROBLEMS . 
 NOTES . 
 
 NOMENCLATURE 
 
 NOTATION 
 
 178 
 181 
 
 185 
 188 
 
LIST OF ILLUSTRATIONS OF AEROPLANES 
 
 %* The following illustrations, distinguished by Roman numerals, will, it is 
 hoped, enable the reader to visualise the systems of planes described in the text, 
 or systems resembling them, better than would be possible from the diagrams 
 bearing Arabic numbers. They also suggest material for further research. 
 Under each figure will be found a brief statement of the particular points which 
 it illustrates in connection with stability. 
 
 I. A CHANUTE GLIDER 58 
 
 II. THE LATE M. E. LEFEBVRE FLYING AT RHEIMS ... 96 
 
 III. EDWARD MINES BIPLANE AT DONCASTER .... 110 
 
 IV. M. HENRY FARMAN FLYING AT BLACKPOOL .... 118 
 
 V. GLEN CURTISS BIPLANE AT RHEIMS . . , . . 130 
 
 VI. DIAGRAMMATIC VIEW OF A WRIGHT BIPLANE USED AT PAU 139 
 
 VII. ROUGIER FLYING AT BLACKPOOL 143 
 
 VIII. A MONOPLANE IN CONSTRUCTION BY BLERIOT IN 1907 . . 152 
 
 IX. LATHAM'S TRIAL FLIGHT . . 160 
 
STABILITY IN AVIATION 
 CHAPTER I 
 
 INTRODUCTION AND SUMMARY 
 
 1. The development of the aeroplane has opened up a 
 large number of interesting problems in both theoretical 
 and experimental mechanics. 
 
 One of these problems is the investigation of the 
 pressures on plane and other surfaces moving in a given 
 manner through a fluid medium such as air. This problem 
 had attracted the attention of mathematicians long before 
 applications to aerial navigation had reached the stage of 
 practicability. The earliest attempt at such an investiga- 
 tion was that due to Newton, who assumed that the 
 resultant pressure on a lamina could be measured by the 
 momentum or change of momentum imparted (per unit 
 time) by the lamina to the particles of air with which 
 the lamina came into contact. This led to the so-called 
 "sine squared" law, according to which the pressure on 
 a plane lamina appeared to be proportional to the square 
 of the normal velocity of the lamina relative to the sur- 
 rounding medium. On the Kinetic Theory of Gases 
 Newton's assumption might reasonably be expected to hold 
 good in the case of a rarefied gas, in which the mean free 
 path of the molecules was large compared with the breadth 
 of the lamina. The second method employed was the theory 
 of Discontinuous Fluid Motion, introduced by Helmholtz 
 
 B 
 
INTRODUCTION AND CHAP. 
 
 and Kirchhoff, which forms the subject of a memoir by 
 Sir George Greenhill, published by the Advisory Committee 
 for Aeronautics. 1 
 
 Divergences between theory and observation necessarily 
 arise from the character of the assumptions required to 
 bring the problem within the reach of analytical methods. 
 The failure of Newton's Law showed that it was impossible 
 to estimate the pressure without taking into account the 
 currents set up in the medium in the neighbourhood of the 
 lamina. In spite of this, many writers, even at the present 
 time, seek to establish formulae for the pressure on moving 
 planes by reasoning which is more or less Newtonian in its 
 fundamental characteristics, but it will be evident that 
 such methods should only be adopted with great caution, 
 the results being treated more or less in the light of 
 empirical formulae which must stand or fall on experimental 
 evidence. 
 
 In the case of the theory of discontinuous motion, we 
 are limited by mathematical difficulties to the considera- 
 tion of a perfect incompressible fluid ; and divergences 
 between theory and experiment necessarily arise from 
 the differences between the physical properties of such 
 an assumed medium and those of air, or of water as the 
 case may be. Moreover, the soluble problems in general 
 only refer to two-dimensional motions, so that in any 
 case the necessary conditions would only be satisfied 
 approximately in the case of aeroplanes the so-called 
 " aspect ratio " of which is small, and the theory would fail 
 in the neighbourhood of the extremities of such planes. 
 
 In this particular problem a suitable balance between 
 theory and experiment has been sought by the publication 
 
 1 Report on the Theory of a Stream Line past a Plane Barrier, and of the 
 Discontinuity arising at the Edge, with the Application of the Theory to an 
 Aeroplane. By Sir George Greenhill, F.R.S. Pp. 96 + 106 figs. 
 (Advisory Committee for Aeronautics Reports and Memoranda, No. 19.) 
 (London : H.M. Stationery Office : Wyman and Sons, Ltd.). Price 5s. 
 
i SUMMARY 3 
 
 of Sir George GreenliiU's report by the Advisory Com- 
 mittee which has been organising the construction of 
 apparatus for the experimental determination of air 
 pressures on moving planes. 
 
 2. In the problem of aeroplane stability here discussed, 
 we are concerned with the motions or changes of motion 
 set up in the flying machine itself by these pressures and 
 by the other forces acting on it when equilibrium in 
 steady motion is disturbed. This problem, unlike the 
 previous one, is a direct application of the principles of 
 theoretical mechanics involving the use of the equations 
 of Rigid Dynamics. About the validity of these, no 
 question can at present arise. The only further assump- 
 tions are the expressions for the air pressures on the 
 planes and other parts of the machine ; and the closeness 
 of agreement between theory and observation in any 
 particular calculation will depend on the degree of 
 approximation to which these pressures are represented 
 by the formulae assumed for them. 
 
 The present subject has hitherto not received so much 
 attention as it deserves, one reason, probably, being the 
 complexity of the algebraic formulae which present them- 
 selves at the outset, and another, doubtless, the success 
 with which aviation has been accomplished on machines the 
 stability of which has not been studied. When the Wright 
 Brothers made their first flights, the question at issue was 
 one of motive power and weight, and an important con- 
 dition for success was the reduction of weight and head 
 resistance by avoiding the introduction of any more 
 auxiliary surfaces than were absolutely necessary. If this 
 involved instability its effects could be overcome by skill 
 and experience, and in particular lateral control was 
 obtained by warping the wings. Since then many 
 attempts have been made to circumvent the Wright 
 patents, with the result of diverting attention from the 
 question of inherent lateral stability. 
 
 B 2 
 
4 INTRODUCTION AND CHAP. 
 
 The Bryan-Williams paper of 1903 1 was, I believe, 
 the first attempt to direct attention to the mathematical 
 aspect of the question, and it was intended to be suggestive 
 rather than conclusive. The late Captain Ferber, whose 
 papers were published soon afterwards, investigated inde- 
 pendently the equations of motion of an aeroplane, and he 
 subsequently applied the method of small oscillations to 
 the problems of both longitudinal and lateral stability. 
 This constituted the first attempt at a theory of lateral 
 stability ; unfortunately, however, this last is defective in 
 several respects, and the results probably fall very wide of 
 the mark when applied to aeroplanes the main surfaces of 
 which are of considerable span, as is usually the case. 
 
 In Mr. Lanchester's second volume on Aerodonetics, 
 which appeared in 1908, the problem of stability, both 
 longitudinal and lateral, is treated from a highly original 
 point of view, the use of the equations of Eigid Dynamics 
 being practically avoided. 
 
 Of more recent contributions we have the papers of 
 Prof. Marcel Brillouin and Dr. H. Reissner, the latter 
 of whom has considered the problems of both lateral 
 steering and lateral stability ; while the Government Blue 
 Book (Cd. 5282) contains abstracts of papers by Lieut. 
 Crocco on the steering and stability of dirigibles, and by 
 M. Soreau on similar problems for an aeroplane, with 
 especial reference to longitudinal stability. This group of 
 papers arrived after the investigations here discussed had 
 been completed, and only a brief comparison of the results 
 has been attempted near the end of the present work. 
 
 3. In this book the six equations of motion are first 
 written down, and then applied to the small oscillations 
 about steady motion of a body, such as an aeroplane, 
 moving in a resisting medium. 
 
 I first intended to measure forces in dynamical units, 
 and perhaps the equations would have been a little simpler 
 
 1 Proc. R. S., June, 1903. 
 
i SUMMARY 5 
 
 if these had been retained, but for the sake of convenience, 
 gravitation units of force have been substituted, the equa- 
 tions being based upon the "homogeneous" system, 
 
 17 -rcr i j. acceleration 
 
 b orce = Weight x - , 
 
 u 
 
 which is now recommended by teachers of elementary 
 mechanics, and avoids both the poundal and the slug. 
 The advantage arises when air pressures are measured in 
 pounds or grammes weight per unit of area instead of 
 poundals or dynes. 
 
 In the case of an aeroplane which is symmetrical with 
 respect to a plane which is vertical in steady motion, the 
 small oscillations fall into two groups each determined by 
 three equations of motion, and no further separation is at 
 the outset possible. 
 
 The first group represents motions in the vertical plane 
 on which longitudinal stability depends. They may be 
 better described as " symmetrical " oscillations. 
 
 The second group determines sideways rotations and 
 displacements to one side or the other of the plane of 
 symmetry. These should be called " asymmetric oscilla- 
 tions." On the nature of these depends the lateral 
 stability of the machine. Some writers attempt to 
 separate the three motions here considered, and to treat 
 separately the stability for rolling motions and " direc- 
 tional" stability, a mistake which Lanchester avoids by 
 using the term "rotative stability." It is important that 
 the interdependence, not only of the two rotational oscilla- 
 tions, but also of sideways displacements, should be taken 
 into account. It is only through the latter displacements 
 that gravity can have any effect on the direction of an 
 aeroplane. Without such an effect there can be no 
 tendency on the part of an aeroplane to right itself after 
 it has heeled over sideways, and in the absence of such a 
 tendency the aeroplane, if it has a tilt to one side, will be 
 
6 INTRODUCTION AND CHAP. 
 
 .liable to " side slip " with uniform acceleration. In most 
 existing aeroplanes this tendency is counteracted by 
 warping, but it does not follow that because this plan 
 can be successfully used it is necessarily the only or the 
 best plan. 
 
 For the second kind of stability the best name would 
 be " asymmetric " stability. We shall sometimes, for 
 shortness, use the term lateral stability to denote the 
 stability dependent on this group of displacements in 
 spite of the term having been used in a more restricted 
 sense by other writers. 
 
 4. The effects of the air resistances in each group of 
 equations are represented by nine coefficients (or eighteen 
 in all) which maybe called the "resistance derivatives." 
 On elimination, the frequencies and logarithmic decrements 
 (or in the case of instability, logarithmic increments) of the 
 symmetrical oscillations are determined by an equation 
 of the fourth degree in \ where the displacements are 
 assumed proportional to e x *, this equation involving the 
 "symmetrical" resistance derivatives. The conditions of 
 stability require that the coefficients of the biquadratic, as 
 well as Routh's discriminant, should be positive. 
 
 Exactly similar conditions apply to asymmetric oscilla- 
 tions, which again lead to a biquadratic in A. involving 
 the nine " asymmetric " resistance derivatives and corre- 
 sponding conditions of lateral stability. 
 
 The actual co-ordinates of the centre of gravity, as well 
 as the azimuth, do not occur in the equations of motion, 
 though the corresponding velocities do so. The reason is 
 that the conditions of equilibrium of an aeroplane, like 
 those of a ship, do not depend on its position or compass 
 bearing. For variations of these, equilibrium is neutral. 
 
 The resistance derivatives are proportional to the 
 velocity of the aeroplane when in steady motion. 
 
 The problem as now before us resolves itself into (l) the 
 determination of the expressions for the resistance deriva- 
 
i SUMMARY 7 
 
 tives of the planes based on the available formulae or data 
 regarding the pressures on them ; (2) the evaluation and 
 simplification of the coefficients in the biquadratics and 
 their discriminant ; (3) analysis of the general character 
 of the oscillations ; (4) separate discussion of the effects of 
 various subsidiary causes which may affect the stability of 
 an actual flying machine. 
 
 The first object to be kept in view is to evolve something 
 like order out of the chaos of algebraical symbols that 
 present themselves at the outset of the inquiry. This 
 can best be done by a method of successive approxima- 
 tion and building up, the problem being at first treated 
 subject to the simplest possible assumptions, but the 
 methods adopted being sufficiently general and elastic to 
 allow of subsequent modifications, corrections, and exten- 
 sions of the most general character. In this book attention 
 is concentrated on the mathematical aspect of the problem 
 for several reasons. In the first place, there is no obvious 
 alternative between developing the mathematical theory 
 fairly thoroughly and leaving it altogether alone ; any 
 attempt at a via media would probably lead to erroneous 
 conclusions. In the second place, the formulae arrived at, 
 even in the simplest cases, are such that it is difficult to 
 see how they could be established without a mathematical 
 theory. In the third place, there is probably no lack of 
 competent workers interested in the practical and experi- 
 mental side of aviation, and under these conditions it is 
 evident that the balance between theory and practice can 
 be improved by throwing as much weight as possible on 
 the mathematical side of the scale. Lastly, it is hoped 
 to advocate the claims of aeroplane equilibrium and 
 stability as an educational subject suitable for study in our 
 Universities alongside with such branches of applied 
 mathematics and mathematical physics as hydrodynamics, 
 and particle and rigid dynamics. 
 
 5. Starting with longitudinal or symmetric stability, the 
 
8 INTRODUCTION AND CHAP. 
 
 available experimental data refer to the air pressures and 
 positions of the centres of pressure of plane and other 
 areas moving through air with given relative velocities of 
 translation. If a lamina has in addition a motion of 
 rotation, as when an aeroplane begins to pitch, the effects 
 of this rotation appear to be at present unknown. In all 
 existing theories of stability they have been neglected, 
 and at the time of writing no indications have reached us 
 of the matter receiving attention at the Government 
 Laboratory. The effects referred to introduce two constants 
 or coefficients into the equations, and these I call rotary 
 derivatives. 
 
 Apart from mathematical considerations, determination 
 of these effects would be a matter of common prudence, 
 as rotations may easily put an aviator into an awkward 
 predicament, especially when near landing. The experi- 
 mental problem may if desired be reduced to the 
 determination of two points which are here shown to 
 exist in the plane of the lamina, such that rotation about 
 one does not affect the resultant pressure, while rotation 
 about the other does not displace the centre of pressure. 
 Two methods of experimenting are suggested. 
 
 The difficulty is meanwhile obviated by the use of narrow 
 planes for which the rotary derivatives are negligible, 
 stability being secured by auxiliary surfaces, such as a tail. 
 But in this case the variations in the centre of pressure 
 will also, in general, be small. We are thus led to the 
 theory of narrow planes flying at small angles. This 
 theory forms the best starting-point for the systematic 
 study of aeroplane motions, in just the same way that the 
 definition of a perfect fluid forms the starting-point in 
 hydrodynamics, the theory of refraction of narrow pencils 
 at small angles of incidence in geometrical optics, or the 
 theory of elliptic motion in planetary theory. According 
 to this hypothesis, not only are rotatory derivatives and 
 variations of the centre of pressure neglected, but the 
 
i SUMMARY 9 
 
 pressure is assumed proportional to the sine of the angle of 
 attack, or what is the same thing, to the product of the 
 resultant and the normal velocity. 
 
 A conclusion described as the Principle of Independence 
 of Height shows that, subject to certain limitations, the 
 stability of an aeroplane is but very slightly affected by 
 raising or lowering its planes ; for this reason there is 
 practically nothing to choose between monoplanes and 
 biplanes in this respect. 
 
 6. The section on " Graphic Statics of Longitudinal 
 Equilibrium " stands by itself apart from the rest of the 
 book, and will, it is hoped, be found instructive. It is 
 sometimes a little difficult to see how the conditions of 
 equilibrium of a flying machine are altered by varying 
 the propeller thrust or altering the inclinations of the 
 planes, and the object is to exhibit the results by means 
 of geometrical constructions. This section might very well 
 be taught to university students. 
 
 7. The simplest case of longitudinal stability is that of a 
 single lifting plane with a " neutral " auxiliary tail propelled 
 horizontally by a constant horizontal thrust passing through 
 the centre of gravity. The condition of stability agrees 
 with Lanchester's result. By the use of the equations of 
 equilibrium this condition may be put into several 
 alternative forms, one of these being independent of the 
 velocity of propulsion. By the further use of approximate 
 methods the roots of the biquadratic are separated, the 
 short and long oscillations discussed, and their general 
 characteristics described. It is shown that Lanchester's 
 method only applies to the long oscillations, but, fortunately 
 for Lanchester, these are the oscillations on which stability 
 mainly depends. 
 
 In a non-horizontal path, the stability varies very greatly 
 with the inclination of the flight path to the horizon. 
 The important part played by this inclination was found 
 out, both for longitudinal and for lateral stability, by 
 
io INTRODUCTION AND CHAP. 
 
 Mr. E. II. Harper. A glider has greater longitudinal 
 stability (cceteris paribus) than a horizontally propelled 
 aeroplane ; and when an aeroplane rises the longitudinal 
 stability rapidly falls off. Under simple assumed conditions, 
 longitudinal instability sets in when the tangent of the 
 elevation has some value less than twice that of the angle 
 of attack. 
 
 I find, however, that longitudinal stability is in general 
 increased by head resistance, and a further increase occurs 
 when the propeller thrust is a function of the velocity 
 which decreases with an increase of velocity. Both these 
 effects are conveniently represented as being equivalent, in 
 their effects on stability, to a depression of the direction 
 of the flight path, through an angle for which a formula is 
 given. The machine can therefore rise at a steeper angle 
 than before without losing stability. 
 
 By a doubly lifting system, I mean two lifting planes 
 placed tandem, or, what comes to the same thing, a front 
 lifting plane and a lifting instead of neutral tail. (In 
 stability investigations it must be remembered that two 
 superposed planes practically count as one of the same 
 total area.) Here a property is proved, which may be 
 called the principle of equivalent systems, which enables 
 the combination of two planes in question to be replaced 
 by an "equivalent" single lifting plane with neutral 
 tail, the two systems having the same conditions of 
 equilibrium and also of stability and, in addition, the 
 same total combined area of their planes. 
 
 Without this substitution the stability of doubly lifting 
 aeroplanes involves the simplification of very unwieldy 
 formulae. For the mathematical student the properties 
 of equivalent systems and their invariants afford some 
 interesting applications of determinants. From a technical 
 point of view, the conclusion to be drawn is that there 
 is no advantage or disadvantage in a lifting tail as com- 
 pared with a neutral one except that the neutral tail will 
 
i SUMMARY ii 
 
 have to be the longer of the two. In an actual aeroplane 
 where the pressure on the rear planes is affected by the 
 wash of the front ones, the machine with the neutral tail 
 would, probably, have the greater lifting capacity. 
 
 In the following sections, the necessary modifications are 
 considered for propeller axis not passing through centre 
 of gravity, variations in the position of the centre of 
 pressure as the angle of attack varies, the principal 
 corrections for cambered planes, effects of tangential or 
 frictional resistances, raised planes, deviations from the 
 sine law of resistance, "wash" on tail plane produced by 
 front one, and rotary derivatives. The discussion of these 
 corrections might be extended almost indefinitely, and, in 
 order to draw the line somewhere, I have in some cases 
 thought it sufficient to indicate the methods only. 
 
 The effect of variations in the position of the centre 
 of pressure calls for a little comment in view of the 
 discussions that have already taken place on this subject'. 
 In the sections dealing with this effect two cases have 
 been considered ; one is the case where stability is mainly 
 dependent on the action of an auxiliary tail plane (which 
 in this case must not be parallel to the front plane) and 
 the effect of the variations in the centre of pressure takes 
 the form of a small correction in the result. The second 
 case refers to the stability of a single plane, which is 
 entirely dependent on the variations in question. (Of 
 course the latter solution is not altogether satisfactory, 
 owing to the fact that the unknown rotary derivatives 
 have been neglected.) Now, when the conditions of 
 stability are taken in conjunction with the conditions of 
 equilibrium, it is shown that the stability depending on 
 the tail action is independent of the velocity of the aero- 
 plane, but the stability of the tailless aeroplane decreases 
 as the velocity increases. This confirms the view that the 
 use of a tail affords the more efficient means of securing 
 inherent longitudinal stability. 
 
12 INTRODUCTION AND CHAP. 
 
 As has been observed, the conditions of longitudinal 
 stability can be exhibited in various forms by making 
 use of the equations of equilibrium, so that two appa- 
 rently widely different conditions may in this way be 
 equivalent. 
 
 A " three plane system," such as a lifting plane with 
 auxiliary surfaces fore and aft, is shown, in certain 
 circumstances, to possess advantages over a two plane 
 system in respect of stability, especially when rising in 
 the air. 
 
 It will thus be seen that the mathematical method is 
 not limited in its application to the simplest case of an 
 ideal aeroplane, but that account can be taken of prac- 
 tically all the modifications occurring in an actual flying 
 machine as soon as the necessary data become known. 
 And in any case we find out which modifications increase, 
 and which diminish, the stability. At the outset of the 
 investigation a certain determinant was found to vanish, 
 and this fact was of great importance in simplifying the 
 analysis which, but for this, appeared of great difficulty. 
 In the discussion of the " three plane system," the fact 
 that this determinant does not now vanish is important 
 in indicating the greater stability of such a combination. 
 
 8. Lateral or Asymmetric Stability presents a 
 difficult problem possessing many interesting features. 
 In this case the couples set up by rotations of the main 
 planes depend on the span instead of the breadth of the 
 planes, and instead of being negligible play a very important 
 part in the stability. To obtain expressions for them 
 which may at least be regarded as approximate, a further 
 assumption has to be made regarding the distribution of 
 pressure on them, this assumption being regarded as a 
 further property which is to be implied in the definition 
 of " narrow planes gliding at small angles." The couples 
 in question are the " turning moments " which are so 
 well known in aviation, and are usually counteracted 
 
i SUMMARY 13 
 
 by operating on " ailerons " or warping. In lateral stability, 
 however, we have to consider separately in turn the effects 
 of " straight " main planes, auxiliary vertical surfaces or 
 rudders which may be called " fins," " bent up " wings, and 
 slanting up planes fixed at the extremities of the main 
 planes which may be called "stabilizers." 
 
 A.n aeroplane having straight (i.e. unbent up) planes 
 (without fins or stabilizers) is devoid of lateral stability. 
 Moreover, a horizontal auxiliary tail plane, such as is used 
 for maintaining longitudinal stability, does not enter into 
 the equations of lateral motion unless its span be con- 
 siderable. 
 
 If a single vertical fin is added, there is great difficulty 
 in reconciling two of the stability conditions. If the fin is 
 placed in front and near the level of the centre of gravity 
 one condition is satisfied but not the other ; if placed 
 behind and near the same level, the second condition is 
 satisfied but not the first. Stability might possibly be got 
 by placing the fin above the centre of gravity and slightly 
 in front, the height above being large compared with the 
 distance in front, but a small change in the motion of the 
 aeroplane would be liable to render it unstable. The 
 objections to a fin raised considerably above the C.G. are 
 evident. That most existing aeroplanes are laterally 
 unstable is a necessary further deduction. 
 
 This difficulty can be got over by the use of two fins 
 instead of one. We have here to investigate the difference 
 between the effect of several fins and that of a single fin 
 of the same total area placed at their centre of pressure, 
 and as the expressions involved are the moments of inertia 
 of the areas of the fins, we transform these by the " theory 
 of parallel axes/' We find that the horizontal distance 
 between the fins has much more effect on stability than a 
 difference in height. 
 
 Probably the best arrangement is one in which the 
 aeroplane is provided with two fins placed fore and aft, and 
 
i 4 INTRODUCTION AND CHAP. 
 
 raised somewhat above the centre of gravity, so that their 
 centre of pressure is vertically above the centre of gravity, 
 or nearly so. This arrangement is shown to possess a wide 
 range of stability which is practically unaffected by the 
 angle which the direction of flight makes with the horizon. 
 The condition for stability requires that the height of the 
 fins should not exceed a certain limit. 
 
 If instead we place the fins at the same level as the 
 centre of gravity, their centre of pressure must be in 
 front of it, and the stability is secured by making the 
 distance between the fins large and the distance of the 
 centre of pressure in front of the centre of gravity small. 
 This arrangement will be stable in horizontal or ascending 
 flight, as also in a gentle descent, but the aeroplane will 
 become laterally unstable and liable to turn round side- 
 ways if it dives downwards at an angle to the horizon 
 equal to or greater than twice the angle of attack. 
 
 A third arrangement, which, like the last, was in- 
 vestigated in the first place by Mr. Harper, has a vertical 
 tail fin and a second vertical fin directly above the centre 
 of gravity. For stability the tail must not be less than a 
 certain length, and in this case it is found that stability 
 decreases when the aeroplane is ascending, thus leading to 
 a limit to the angle of elevation. 
 
 The general character of the lateral oscillations is 
 discussed, also the general effects of couples due to 
 head-resistance, twin screws, friction, and camber. 
 
 We next consider the effects of "stabilizers'" (or bent 
 up planes attached to the tips of the wings, or, in fact, 
 wings with bent up tips), and bent up wings in general. 
 It is natural to suppose that a pair of stabilizers is 
 equivalent to a vertical fin ; this is, however, shown to 
 be only true in certain conditions, such as when they are 
 placed at grazing incidence with the line of flight, or are 
 neither before nor behind the centre of gravity. The two 
 solutions which follow are due to Mr. Harper. The first 
 
i SUMMARY 15 
 
 discusses the stability of an aeroplane fitted with stabilizers, 
 the second the " Antoinette " type of aeroplane, furnished 
 with bent up wings and a tail-fin. Mr. Harper has shown 
 that lateral stability in this case can be obtained in two 
 ways. Neglecting the tail, it is sufficient to raise the 
 dihedral angle of the wings above the centre of gravity, 
 but in this case there is a superior limit to its height. Or 
 again without raising the wings the tail must be made not 
 less than a certain length. In the first method stability 
 falls off when the aeroplane is descending ; in the second it 
 falls off when rising, so that by a judicious combination of 
 raised wings and tail, stability may be made practically 
 independent of the elevation of the flight path. 
 
 9. Under the heading of " General Conclusions," no 
 attempt has been made to give a detailed summary of 
 the work, this having been done in the present intro- 
 duction. The discussion refers chiefly to the diversity of 
 opinion which now exists as to the advantages and 
 disadvantages of inherent stability. 
 
 There is, however, an element of indeterminateness in 
 the arguments relating to this particular point, arising from 
 the fact that a complete investigation of aeroplane motions 
 involves the solution of other problems besides that of 
 stability. A list of some of these problems is given at 
 the end. While some of them afford materials for experi- 
 mental research, the list should afford the most sceptical 
 reader evidence as to the necessity of a large amount of 
 further original work of a purely mathematical character, 
 in which all that is required is the deduction of conclu- 
 sions from definite stated hypotheses. At present there 
 is widespread belief that the methods of exact science 
 cannot be brought to bear on the study of aeroplane 
 motions owing to the uncertain conditions to which they 
 are subjected. But after all the cause of the uncertainties 
 probably lies in the fact that the study of aeroplane 
 motions has not received so much attention on the part of 
 
1 6 INTRODUCTION AND CHAP. 
 
 mathematicians as has been given to other problems of a 
 similar character. 
 
 10. A brief discussion is given, dealing with investiga- 
 tions of an allied character by other writers. It will be 
 seen that in several cases erroneous conclusions have been 
 arrived at through not starting with sufficiently general 
 hypotheses, a frequent cause of error being failure to 
 recognise the interdependence of the three components of 
 lateral motion. In this respect Lanchester's work forms a 
 conspicuous exception, although he has approached the 
 subject from an entirely different direction from that here 
 adopted, and has avoided the use of the actual equations 
 of motion. 
 
 11. If we consider only the problem of stability, in which 
 it has been attempted to arrive at some approach to a definite 
 understanding in this book, it will be seen that this problem 
 presents many complexities which it would have been very 
 difficult to foresee from experimental evidence alone. In- 
 deed, I should be inclined to think that the difficulty 
 was almost as great as that which would be experienced 
 by anyone who endeavoured to explain the motions of 
 the solar system without the assistance of the theory of 
 gravitation and the fundamental equations of dynamics. 
 The dependence of the stability of aeroplanes on the 
 inclination of their flight-paths to the horizon is very 
 difficult to explain from first principles, but an attempt 
 at such an explanation has been given in a note. 
 
 .These difficulties and the large number of problems 
 still awaiting solution must be my justification for repeating 
 the plea for the recognition of motions of aeroplanes as 
 a subject for study, research, and original work by teachers 
 and students of mathematical and physical science in our 
 Universities. 
 
 As a branch of higher applied mathematics the study of 
 aeroplane motions has been sadly neglected. The vaguest 
 notions still prevail even as to the very meaning of 
 
i SUMMARY 17 
 
 stability. The subject presents claims of the most urgent 
 and pressing character for systematic study and investi- 
 gation on similar lines to those which have been followed 
 in the development of other branches of mechanics and 
 mathematical physics, and it is only by the co-operation 
 of University teachers in mathematics and physics that 
 the present anomalous conditions can be remedied. No 
 less strong are the claims of this study from an educational 
 point of view. It will be seen that the subject matter 
 contained in these pages contains several interesting 
 methods, including applications of properties of deter- 
 minants, the use of approximations, and the " principle 
 of equivalent systems " as applied to double-lifting 
 aeroplanes ; further, the introduction, step by step, of 
 corrections which need in no respect stop short of practical 
 requirements, and may be possibly carried to a stage 
 even beyond what is sufficient for ordinary needs. 
 
 To see how matters stand, let us compare the study of 
 aeroplane motions with some subject which is already 
 largely studied and taught in our Universities, say hydro- 
 dynamics. We have here a branch of applied mathe- 
 matics which has been developed on the theoretical side 
 quite independently of the requirements of the hydraulic 
 engineer. Many of its problems refer to hypothetical 
 motions in an indefinitely extended ideal medium, and are 
 far removed from any possibility of practical application, 
 yet no one questions the educational value of the training 
 afforded by their study, even to engineering students. 
 
 Now any honours student who devotes a year or more to 
 the systematic study of hydrodynamics will, at the end of 
 that time, have little prospect of finding any problem, 
 practical or unpractical, which has not already been solved, 
 and which he can approach with any reasonable chance of 
 obtaining a solution or result worth publishing. The 
 present subject, on the other hand, literally bristles with 
 unsolved problems, and the difficulty is not so much that 
 
i8 INTRODUCTION AND SUMMARY CH. i 
 
 the problems are insoluble as that their solutions are long 
 and tedious. For this reason it is important that the 
 number of persons interested in the subject should no 
 longer be nearly countable on the fingers of one hand. 
 12. There still remains the question of " Examples" 
 An almost unlimited number of these can be obtained 
 by applying the formulae and methods of this book to 
 existing aeroplanes the dimensions of which will be found in 
 the principal aeronautical journals and manuals. Possibly 
 readers of a practical turn of mind might have preferred 
 to see some of the calculations worked out here. This, 
 however, is quite unnecessary as the calculations are easy 
 and well within the powers of a fair student of B.Sc. 
 standard. In the case of numerical calculations there is 
 no need to limit the work to substituting values in the 
 conditions of stability. It is easy to obtain the coefficients 
 in the biquadratic and to determine approximate values for 
 the roots. In this way the periods and logarithmic decre- 
 ments of the oscillations can be found, or in cases of 
 instability their logarithmic increments. Thus, even if it 
 is decided to sacrifice some of the conditions of stability in 
 the construction of an aeroplane, the effects of the sacrifice 
 can be made the subject of calculation. As different new 
 forms of aeroplane are constructed fresh examples are con- 
 tinually suggesting themselves. For this reason it would 
 have been futile to take up any of these pages with calcu- 
 lations which would soon be out of date. Examples on 
 longitudinal stability were worked out by Mr. Williams in 
 our joint paper of 1903 referred to above. 
 
CHAPTER II. 
 
 FUNDAMENTAL PRINCIPLES. 
 
 General Equations of Motion. 
 
 13. In investigating the motion of aeroplanes we start, 
 of necessity, by writing down the general equations of 
 rigid dynamics. 
 
 We take as origin the centre of mass of the aeroplane, 
 and choose three axes mutually at right angles, fixed 
 relatively to the aeroplane and moving with it in space. 
 We use the following notation : 
 
 JF, weight of aeroplane. 
 
 A t Bj C, moments of inertia about the axes. 
 
 Dj E, F, corresponding products of inertia. 
 
 u, v, w, components of translational velocity. 
 
 jp, q, r, components of angular velocity. 
 
 h lt h 2 , 7i 3 , components of angular momentum. 
 
 Then we have the following equations of motion (using 
 homogeneous gravitation units) : 
 
 W ( + qw - rv ] = Accelerating force along axis of x (1) 
 \gdt T g gJ 
 
 and two similar, also 
 
 dhi + qh> 3 _rhz = Accelerating torque about axis of x . (la) 
 gdt g g 
 
 while the expressions for the angular momenta are in the 
 most general case : 
 
 \ = Ap - Fq - Er 
 
 h 2 = Bq - Dr - Fp (2) 
 
 7i 3 = Cr - Ep -Dq 
 
 c 2 
 
20 FUNDAMENTAL PRINCIPLES CHAP. 
 
 14. Suppose, in the first place, that the aeroplane is 
 flying steadily in a horizontal straight line. We take this 
 line as our axis of x, and shall call it the line of flight (the 
 centre of gravity being origin as postulated), a line drawn 
 vertically downwards as axis of y, and a horizontal line 
 perpendicular to these as axis of z. In the case of a sym- 
 metrical aeroplane, such as commonly exists in practice, 
 the plane of x, y will be the plane of symmetry, so that 
 D = and E = 0. It is here assumed that gyrostatic 
 effects due to the rotatory inertia of the propeller are 
 neglected. If it be necessary to take account of them 
 this must be done in a subsequent investigation, by insert- 
 
 FlG. 1. 
 
 ing additional terms in the equations of motion. For an 
 aeroplane with two propellers rotating in opposite directions 
 the gyrostatic effects annul each other. 
 
 If the aeroplane be turned into any other direction this 
 latter direction could be specified by Euler's angular co- 
 ordinates, but as these are not well suited for the study of 
 small oscillations, the following scheme is preferable. 1 
 
 Starting from an initial position, imagine the aeroplane 
 first rotated about the axis of y, through an angle ^ (this 
 
 1 In the system as specified inRouth's " Rigid Dynamics " and elsewhere, 
 the axes are first rotated about the axis of 2, then about the axis of y, then 
 again about the axis of z. The objection to this specification is that if the 
 system receives a small rotation about the axis of ', this cannot be represented 
 by small values of the angular co-ordinates. 
 
II 
 
 GENERAL EQUATIONS 
 
 21 
 
 merely changes its orientation), then about the axis of z 
 through an angle 0, lastly about the axis of x through an 
 angle <. The cosines of the angles between the old axes 
 
 Horizontal 
 
 Direction \ 
 of flight \ 
 
 FIG. 2. 
 
 #o, 2/o> ZQ, and the new axes x lt y l9 z lt are given by the 
 following scheme : 
 
 
 *, 
 
 Vi 
 
 % 
 
 X Q 
 
 cos 6 cos ^ 
 
 sin sin >// 
 
 cos sin ^ 
 
 
 
 
 - cos cos ^ sin 6 
 
 + sin cos ^ 
 
 / sin 6 
 
 i/o 
 
 sin 6 
 
 cos $ cos 
 
 cos ^ sin (f> 
 
 
 Z Q 
 
 cos 6 sin ty 
 
 sin cos ^ 
 
 COS COS >/^ 
 
 
 
 
 + cos (j) sin ^ sin 6 
 
 - sin sin ^ 
 
 / sin 6 
 
 (3) 
 
 If we assume, however, that the old axis of x is in the 
 same vertical plane as the new one, so that the change of 
 orientation ty is zero, the scheme assumes the simpler form, 
 
 
 % 
 
 l/i 
 
 >, 
 
 x 
 
 cos 6 
 
 - sin 
 
 6 cos 
 
 sin sin 
 
 1/0 
 
 sin $ 
 
 cos 
 
 COS 
 
 - cos 6 sin 
 
 
 
 
 sin 
 
 * 
 
 COS 
 
 (3o) 
 
22 FUNDAMENTAL PRINCIPLES CHAP. 
 
 The angular velocities, p, q, r, are given in terms of 
 i/r, 0, <, by the scheme, 
 
 p =. (j) + \jr sin 6 
 
 q = B sin < + ^ cos cos 
 
 r = 6 cos < ^ cos # sin </> . . (4) 
 
 reducing, as they should, in the case of = 0, $ = to 
 
 P = <t>, q = ^, r = 0. 
 
 It will be observed that 6 is the inclination of the line 
 of flight to the horizon, taken positive ivhen the aeroplane 
 
 is flying downwards, and $ is the angle through ivhich 
 the plane of symmetry is turned out of the vertical position 
 when the aeroplane turns over sideways. 
 
 15. The impressed forces and couples are due to 
 (i) gravity, (ii) propeller thrust, (iii) air resistance. 
 
 The components of gravity along the axes are 
 
 W sin 0, TTcos 6 cos </>, - JFcos 6 sin . . (5) 
 
 the corresponding moments all vanishing. 
 
 For the components of propeller thrust, we notice that 
 in steady horizontal flight this acts usually in the direction 
 of motion, and we are therefore justified in choosing its 
 direction as our axis of x. (The more general case is 
 
ii SMALL OSCILLATIONS 23 
 
 discussed later). Let the thrust be H, and let it act at a 
 perpendicular distance, h, below the origin, then its 
 components are specified as below : 
 
 Point of application . ... h 
 Force . . . . . - . . H 
 Torque . . . . . . .0 -Hh 
 
 (6) 
 
 For the components of air resistance we assume that 
 these reduce to forces X, Y, Z, and couples L, M, N, and 
 that these are taken positive when they tend to retard the 
 corresponding motions of translation and rotation. Thus, 
 e.g., the force acting along the the positive direction of 
 the axis of x instead of being represented by X as is usual 
 in text-books, is represented by X This convention is 
 made because the forces and couples are of the nature of 
 resistances, and it really appears to work out more satis- 
 factorily than the ordinary convention. 
 
 We now write the equations of motion : 
 
 If the axis of x is a principal axis of inertia, F= and 
 the equations are greatly simplified. 
 
 Steady motion and small oscillations. 
 
 16. Let the aeroplane be descending with uniform 
 velocity U in the direction of the axis of x, and let this axis 
 make a constant angle, , with the horizon. It will follow 
 from conditions of symmetry that the plane of x, y is 
 vertical, and therefore = 0. In this case u is constant 
 
24 FUNDAMENTAL PRINCIPLES CHAP. 
 
 and equal to U, whereas v y w, p, q, r, the other velocity 
 components, are all zero. 
 
 Let the components of resistance and thrust in this case 
 be denoted by the suffix o, thus X 0) Y , Z , L 0) M , N ot and 
 H . The equations for steady motion become 
 
 = W sin 6 + H - X 
 
 = W cos S - Y 
 
 = - Z 
 
 = - L 
 
 0= - M 
 
 = - H h - N . . ... (8) 
 
 These equations merely state the fact that the forces 
 are in equilibrium among themselves. The conditions of 
 equilibrium may be studied by any methods that are most 
 convenient, for example, by the use of graphical methods. 
 
 If the thrust, H, instead of being parallel to the axis 
 of x is inclined to it at an angle rj y the first two equations 
 are replaced by 
 
 = W sin + H cos 17 - X and = W cos 6 + H sin 17 - Y 
 
 If now the aeroplane begins to perform small oscilla- 
 tions in still air, we assume that in this case the velocity 
 along the axis of x changes from U to U+u, where u is 
 small ; further, that the other velocity components, 
 v t w, p, q, r, are all small. In the " theory of small 
 oscillations " of dynamics we suppose that squares and 
 products of u 9 v, w, p, q, r are negligible. 
 
 The resistances, X, Y, Z, L, M, N, are functions of the 
 velocity components, U+ u, v, w, p, q, r, and the further 
 assumption in dealing with small oscillations is that to a 
 first approximation these resistances are expressible in the 
 form 
 
 X = X + uX u + vX v + wX w + pX p + qX q + rX r . (9) 
 
 i.e., that they are linear functions of the small quantities 
 
 u, v, w, p, q, r (the coefficients being denoted by X u , X v . . . 
 
 Y ut Y v . . . etc.). This assumption, being commonly made 
 
 in treatises on theoretical mechanics, as a first approxima- 
 
ii SMALL OSCILLATIONS 25 
 
 tion where small oscillations are concerned, need not be 
 here discussed. 
 
 If the aeroplane is moving in a turbulent atmosphere, 
 there will be additional forces and couples caused by the 
 gusts of wind ; in this case, we must add to the expres- 
 sions, terms X lt Y lt Z lt L^ M^ N l} varying with the time, 
 representing these disturbing forces and couples. It is 
 clear that the first step in the discussion is to investigate 
 what happens to an aeroplane when left to itself, i.e., when 
 these components are all zero, and the aeroplane is said to 
 be performing free oscillations. 
 
 Separation of the two groups of oscillations. 
 
 17. The six components X, Y, Z, L, M, N, being 
 each expressed as a linear function of u, v, w, p, q, r (for 
 purposes of approximation), there would be altogether 
 thirty-six coefficients, such as X UJ X V) Y u , if we knew 
 nothing about the symmetry of our aeroplane. But if 
 this is symmetrical, the coefficients reduce to eighteen, the 
 other eighteen vanishing. Thus X, Y 9 N do not occur 
 with suffixes w, p, q, and on the other hand Z, L, M do 
 not occur with suffixes u, v, r. It is easy to verify these 
 statements, as the following instances will show : 
 
 If, for example, X w were different from zero (say positive) 
 then a sideways velocity, w, from right to left (looking in 
 the direction of flight) would cause an increase, wX w , in 
 the head resistance X, whereas a sideways velocity in the 
 opposite direction would cause a decrease. 
 
 As another instance, if Z r were different from zero and 
 positive, then if the machine were to dip down with angular 
 velocity r, it would experience a sideways resistance,. 
 rZ r) which would be reversed in direction if it were to tip 
 up, the angular velocity, r, being reversed. The reader 
 may be left to verify the other cases which present 
 themselves. 
 
26 FUNDAMENTAL PRINCIPLES CHAP. 
 
 In the small oscillations, moreover, 6 will differ from 
 by a small quantity e, and will be small, so that we shall 
 write : 
 
 sin 6 = sin 6 + e cos , cos 6 = cos 6 - sin 6 
 
 sin </> = < cos = 1 (10) 
 
 Finally we suppose the new propeller thrust to be 
 H + &H, h remaining constant. When all these substitu- 
 tions are made, U+u being of course written for u, the 
 equations (7) give 
 
 W-jfi = W(sm 6 + cos B ) + H + b H 
 
 - X - uX u - vX v - rX r ..- . . (lln) 
 
 F M - r, - rY r . . . (11 V ) 
 
 * 
 
 - tyZ w - ^ p - qZ q . . . (llw) 
 
 ~ L W - pL p -qL q . . (lip) 
 
 ~ F jlt 
 
 - N - uN. u - vN v - rN r . . . (llr) 
 
 We substitute from the equations of equilibrium and 
 rearrange the equations in two groups, the first group 
 containing those (the first, second, and sixth) involving 
 u, v, r, the second (consisting of the third, fourth, and 
 fifth) involving w, p } q. We thus obtain the groups 
 
 W^ = W cos + 8H - uX u - vX v - rX r (12t*) 
 
 sin0 - uY u - vY v - rY r (I2v) 
 
 - uN u - vN v - rN r (I2r) 
 
 and the group 
 
 (133) 
 
ii LONGITUDINAL STABILITY 27 
 
 The first group represent oscillations in the plane of x, y, 
 which we call longitudinal or symmetrical oscilla- 
 tions ; the second group represent rotations p, q about 
 the axes of x and y and motions w perpendicular to the 
 plane of x, y, which we should describe as lateral or 
 transverse oscillations. It will be noticed, however, 
 that while p represents a cant over sideways, q represents 
 a turning round of the direction of the machine, say a 
 directional rotation, and w determines a lateral motion of 
 translation. In view of the frequent attempts that have 
 been made to separate c; lateral" and "directional" 
 stability, and the fact that the motions are not all 
 rotations, it is better to distinguish the oscillations of the 
 second group by the term asymmetric oscillations. 
 Another alternative is to apply the term lateral to the 
 whole group of asymmetric oscillations, which are in 
 general mutually interdependent, and in this book it will 
 never be used except in this sense. 
 
 We further have, 
 
 < = 0, 6 = B + *,-= = >> t ^s = P cos e o - q sin (14) 
 
 as may easily be deduced from equations (4) or from 
 geometrical considerations. 
 
 Symmetric Oscillations. Condition of 
 Longitudinal Stability. 
 
 18. To investigate the symmetric oscillations we 
 assume u, v, r, and e to be proportional to e K \ so that 
 
 du dv dr de 
 
 ~dt = ^ dt=^> di = Xr > Tt = * 
 
 and we notice that the last equation gives r = \e ; but the 
 equations are of a more symmetrical form when rj\ is 
 substituted for e than they would be if \e were substituted 
 for r (this is, however, a mere matter of convenience). 
 
28 FUNDAMENTAL PRINCIPLES CHAP. 
 
 Equations (12 u, v, r) now become on transposing 
 
 Y u u + (w- + Y v }v + ( T sin 6 + W + Y r ] r = 
 \ 9 J v A 9 I 
 
 (15) 
 
 We consider at the outset the simplest case in which 
 the propeller thrust is independent of the velocity, so that 
 &H = ; in a later section we shall examine the necessary 
 modifications when this assumption does not hold good. 
 The right hand sides of the above equations vanish, and 
 on eliminating u, v, r, the result assumes the form of the 
 determinant 
 
 x ~w 
 
 9 "*" r ' 
 
 Y u , W- + Y n 2- sin 6 + W + Y r 
 
 9^9 
 
 9 
 
 (16) 
 
 Multiplying by X to remove X from the denominator 
 in the upper line and developing the determinant in 
 powers of X we get an equation of the fourth degree which 
 we write 
 
 gj o \4 + $g o x 3 + (E X 2 + 2> X + <g = . . . (17) 
 where 
 
 y - GW(X U + Y v ) H 
 
 || = C(X U Y V - X V Y U ) + W{(Y v N t . - Y r N v ) + (X u N r - X r N u )\ 
 
 ^ = X u (Y v N r - Y r N v ) + X v (Y r N u - Y u N r ) + X r (Y u N v - Y V N U ) 
 
 TT W 2 
 
 + W(X V N U - X U N V ) + (N u cos 6 - N v sin ) 
 
 9 9 
 
 @ W 
 
 4 = { - cos Q (YuNv - Y v Ny) - sin Q (X H N V - X V N U ] 
 
 (18) 
 
ii LONGITUDINAL STABILITY 29 
 
 These coefficients may be simplified with advantage by 
 the following notation : write A for the determinant 
 
 x u x v x r 
 
 Y u Y v Y r 
 
 N H N v N r 
 
 and write u x for the minor of X u , and similarly for other 
 minors, taken with their proper algebraic sign in such a 
 way that (e.g.) 
 
 X H u x + X v v x + X r r x = A = X u u x + Y u ii r + N u u Nt &c. 
 then we get 
 
 = GW(X U + Y v ) + W*N r 
 
 2) U W* 
 
 -? = A + Wr r + ~ 
 
 I ... (18a) 
 
 y v 
 
 19. The conditions of stability require that all the four 
 roots of the biquadratic equation for X shall have their real 
 part negative. This follows from the assumption that the 
 small disturbances u, v, r, etc. , are all proportional to e xt in 
 a typical oscillation. If \ 1? \ 2 > **> ^ are the roots, the 
 expressions for u, v, r, take the form 
 
 a ly a. 2 , a 3 , a being constant coefficients determined by the 
 initial conditions. 
 
 If any one of the roots Xj is real and positive a 
 disturbance of the form u = a 1 e* lt will increase indefinitely 
 with the time and steady motion will be unstable. 
 
 If on the other hand X x is real and negative and equal 
 say to K, the expressions u = ae~ Kt represent disturbances 
 which decrease with the time, the modulus of decay or 
 coefficient of subsidence being K. For such dis- 
 turbances the steady motion will be stable. 
 
30 FUNDAMENTAL PRINCIPLES CHAP. 
 
 If the biquadratic has a pair of complex roots 
 where i V 1, the corresponding disturbance takes the 
 form e at (a cos @t + b sin 8t), and if the real part a is positive 
 this represents an oscillation which increases with the time 
 and steady motion will be unstable. If on the other 
 hand the real part a is negative and equal to y, the 
 solution takes the form e~ yt (a cos ftt + b sin fit) and the 
 disturbance becomes a damped oscillation of which the 
 modulus of decay or coefficient of subsidence is equal 
 to <y. For such disturbances the system tends to revert to 
 its state of steady motion and is stable. 
 
 The condition that the roots of a biquadratic equation 
 shall all have their real part negative and thus indicate 
 stability of steady motion is given by Routh ("Advanced p, 
 Rigid Dynamics "). Supposing that 2( is positive (as is 
 the case above) this condition requires that 
 
 o, 33 , <o, 3> , <g oj and $ OT 
 
 where 
 
 - @oo 2 . . . (19) 
 
 shall all be positive. For a proof of these conditions 
 we refer to Routh's treatise, but a partial verification of 
 the last condition $ > may be given by examining what 
 happens if the real part of a pair of roots of the biqua- 
 dratic (17) 
 
 SU 4 + S3 \3 + CoX 8 + S) X + @ = 
 
 from being negative becomes zero. In such a case the 
 roots fit, 0i are roots of X 2 4- 2 = and by substitution 
 and equating the real and imaginary parts we find that 
 X 2 = $ 2 must be a solution of both equations 
 
 = 
 = 
 
 The second gives 
 
ii LATERAL STABILITY 31 
 
 which, substituted in the first, gives 
 
 or 
 
 which therefore represents the limiting case when the real 
 part of a pair of roots changes sign. The corresponding 
 inequality representing a condition for stability may also 
 be written 
 
 ~. ~. :.- ' < i9a > 
 
 and necessarily implies the condition 6 positive if all the 
 other four coefficients are positive. 
 
 Asymmetric Oscillations. Condition of Lateral 
 
 Stability. 
 
 20. For the asymmetric oscillations we take w, p, q, 
 and $ proportional to e K \ and notice that in this case we 
 
 have 
 
 d<f> 
 
 \q> cos Q -. cos 6 = p cos 6 - q sin 
 
 which we use to substitute for <f> in terms of p and q, 
 Our equations (13 w, p, q) now become 
 
 w + 
 
 M w w + ( - F- + M p \p + ( B - + M q \q = 
 
 (20) 
 
 giving on elimination 
 
 TX> W rrU W 
 
 W- + Z w , y cos 6 + Z p , - W - -r- sm + Z 
 9 A 9 A 
 
 ^ + 
 
 M w , -y--hM p , SP. + M q 
 
 9 
 
 + M q 
 
 (21) 
 
 = 
 
32 FUNDAMENTAL PRINCIPLES CHAP. 
 
 Multiplying by \ and expanding the determinant in 
 powers of \ in the form 
 
 ^X* + SBjX 3 + (^X 2 + 3>!X + @! = . . . (22) 
 
 we now have for the coefficients 
 
 i = Z W (AB - F*) + W{AM q + L P + JXAr + M p )} 
 ^ = Z w {AM q + BL P + F(L q + M p )} + W(L p M q - L q M p ) 
 
 - Z P (FM W + BL W ) - (Z q - W^(FL W + AM W ) 
 j = Z w (L p M q - L q M p ) + Z p (L q M w - M q L w ) 
 
 (L W M P - L P M W ) 
 
 + AM W ) sin 6 - (BL W + FM U ) cos 6 ] 
 - M q L w ) cos B - (L W M P - M W L P ) sin 6 \. (23) 
 
 The expression for (S^ may also be arranged thus : 
 
 = A(Z w M q - M w Z q ) + B(Z W L P - L W Z P ) 
 
 F{(Z W M P - M W Z P ) + (Z w L q - L w Z q }} + W(L p M q 
 W~(FL W + AM W ) 
 
 Writing as before 
 
 z w z p z q 
 
 L w L p L q 
 
 M w M p M q 
 
 and using as previously, e.g., w z to denote the minor of 
 Z w we get 
 
 %! = W(AB - F 2 ) 
 
 y 1 = Z W (AB - -F) + W{AM q + BL P + F(L q + M p )} 
 
 K 
 
 -k = Ap L + Bq M - F(q L + p M ) + Ww z 
 
 y 
 
 FL W ) 
 
 -y = Aj - Wq z + 
 
 (23a) 
 
ii RESISTANCE DERIVATIVES 33 
 
 and if, as before, 
 
 ft = 8A! - ^S, 2 - (S^ 2 . , . (24) 
 
 the conditions of asymmetric stability require that 
 
 2t 5 , $, 35, @ and 
 
 shall be positive ; 9l t being of necessity positive, as is 
 easily shown by the principles of elementary rigid 
 dynamics. 
 
 Observations on the Resistance Derivatives. 
 
 21. It will now be seen that if for a given aeroplane, 
 moving in a given manner, we knew the values of the 
 eighteen coefficients, X u . . . N r , or, as we may write them 
 for shortness ( X, Y, N) UI V) r and (Z, L, M) Wt p> 9 , as well as 
 the dynamical coefficients, namely, mass and moments of 
 inertia, we could, by substitution in the formulae obtained 
 above, ascertain if the motion in question were symmetric- 
 ally or asymmetrically stable ; also by numerical solution 
 of the biquadratic we could ascertain the character of 
 the small oscillations about steady motion, their periods 
 if periodic, their logarithmic decrements corresponding 
 to stability or logarithmic increments corresponding to 
 instability. For the coefficients (X, Y y N) U}Vtr and 
 (Z t L, M) Wtptqt I propose the name resistance derivatives, 
 as it is convenient to have some name for them. 
 
 Now assuming the air resistances to be proportional to 
 the square of the relative wind velocity and this is the one 
 assumption the validity of which (under normal conditions) 
 is generally admitted the six forces and couples X, Y, Z, 
 L, M, N will be quadratic functions of U+ u, v, tv, p, q, r, 
 and their derivatives X u . . . N will be proportional to U", 
 remembering that we are only going to a first approxima- 
 tion, and that squares and products of the small quantities 
 u,v t w,p, q, r are neglected in dealing with small oscillations. 
 The quotients X U /U, . . . N r /U will be constants for a 
 particular machine flying in a particular way. They 
 
34 FUNDAMENTAL PRINCIPLES CHAP. 
 
 will depend on the form, dimensions, and arrangement of 
 the aeroplanes and of the supporting frameworks (which of 
 course also encounter air resistance), and they will also 
 depend on the inclination of the aeroplane to the line of 
 flight. If they were determined experimentally for every 
 such inclination that might occur in practice, the problem 
 of stability would be reduced to arithmetical calculation. 
 At the same time, even granting the necessary experi- 
 mental data to be known, the calculations would be 
 exceedingly laborious, and in order to simplify them it 
 would still be necessary to search for approximate methods 
 of solution by developing the problem further on the 
 mathematical side as a first step. 
 
 In the next place, w r e observe that since in rectilinear 
 motion along the axis of x, we have X, Y, N propor- 
 tional to (U+uy, therefore 
 
 2X 2Y 2N 
 
 X u = -jj > YU = ~jj > N n =- ~^j . . .. (25) 
 
 and X 0) Y , N are given by the conditions of equilibrium, 
 
 = W sin 6 + H - X m = W cos 6 - Y , = - H h - N 
 
 and in particular if the propeller thrust is central so that 
 N = we have also N u = 0. 
 
 Remembering that the derivatives are proportional to Z7, 
 we find that the coefficients in the biquadratics in ^/g are 
 dependent on U in the following manner, the suffixes 
 being omitted from these coefficients as the conclusions 
 are the same for both the "symmetrical" and "asym- 
 metrical" biquadratic : 
 
 $ is independent of U 
 33 is proportional to U 
 
 V* 
 
 2) is of the form AU* + BU 
 
 G? is proportional to U 2 
 
 We may, however, write either biquadratic in the form 
 of an equation in \/g U, thus : 
 
 A 53 /A < / A 3) / A 
 
ii RESISTANCE DERIVATIVES 35 
 
 and the coefficients of powers of \/gU will then be as 
 follows : 
 
 21 constant 
 
 of form A + BU~* 
 proportional to ~U~' t 
 
 The discriminant of the equation in this form is 
 
 U 6 
 
 and is of the form P+QU^ + RU-*, where P, Q, R are 
 constants. The condition of stability would thus appear 
 to impose limitations on the value of U. 
 
 It must be remembered, however, that stability, at least 
 longitudinal, according to all accepted ideas, has no 
 meaning unless equilibrium exists, and when this is the 
 case we may put the conditions of stability into different 
 other forms by combining them with the conditions of 
 equilibrium in different ways. We thus have to dis- 
 tinguish between the primitive conditions of stability in 
 which no use has been made of the equations of equilibrium, 
 and modified conditions in which these equations have 
 been used to effect simplifications in the conditions of 
 stability. If, for example, we say that an aeroplane is 
 stable when its velocity exceeds a certain limit, it must be 
 remembered at the same time that this velocity is deter- 
 mined by the conditions of equilibrium, and that a modified 
 condition of stability may be substituted, in which the 
 velocity does not appear. 1 
 
 On the other hand, there may be cases -in which it is 
 desirable to consider stability apart from the question of 
 
 1 The discrepancies between certain results given in this book and those 
 given in Bryan and Williams' paper are thus accounted for. 
 
 D 2 
 
36 FUNDAMENTAL PRINCIPLES CHAP. 
 
 equilibrium. Thus it is important that when the motion 
 of an aeroplane is being accelerated in the vertical plane, 
 it should not tend to swing round sideways, in other 
 words, that it should possess asymmetric stability even 
 when the forces acting on it are not in longitudinal 
 equilibrium. Of course, in such a case the resistance 
 derivatives will really be functions of the time, so that the 
 investigation of the small oscillations will really be of a 
 complicated character. It may perhaps, however, be 
 sufficient to assume that the primitive conditions of 
 stability, determined on the hypothesis of steady motion, 
 hold good at every instant of the motion. 
 
 Stability of a dirigible. 
 
 22. If it be desired to investigate the stability of a 
 dirigible balloon by the methods of the present monograph 
 several modifications will have to be made. 
 
 In the first place account will have to be taken of the 
 inertia of the displaced air. In the case of a sphere 
 moving in a perfect liquid we know that this inertia is 
 equivalent to adding half the mass of the displaced liquid 
 to the mass of the sphere. For a balloon in equilibrium 
 the total weights of the airship and the displaced air are 
 equal, and hence the inertia of the air becomes of con- 
 siderable importance. The inertia of the contents of the 
 gas-bag must also be taken into account. If we suppose 
 the balloon moving in air to be represented by an ellipsoid 
 moving in an incompressible medium, we can find an 
 expression for the kinetic energy by hydrodynamical 
 methods. This expression will be, owing to the (assumed) 
 symmetry of the machine about a vertical plane, the sum 
 of two homogeneous quadratic functions, one of U+u, v, r, 
 and the other of w, p, q, and the equations of motion can 
 be written down by Hayward's method. 
 
IT DIRIGIBLES 37 
 
 In the second place the effects of gravity will be entirely 
 different, the weight of the airship being balanced by the 
 buoyancy of the air according to the principles of 
 Archimedes. In a state of equilibrium (at rest) the 
 centres of gravity and buoyancy are in the same vertical 
 line. If c is the distance between them, then when the 
 axis of x is been depressed from a horizontal position 
 through an angle 0, and the machine subsequently turned 
 through an angle < about this axis, couples are produced 
 about the axes of 
 
 x y z 
 
 of amount 
 
 - c W cos 6 sin $ - c W sin 6 
 
 and there are no component forces due to gravity along 
 the axes. Consequently the terms depending on the 
 action of gravity occur in the equations of rotation about 
 the axes of x and z instead of in the equations of trans- 
 lation, and the character of the oscillations is thus entirely 
 different. 
 
 The principal obstacle in the way of a satisfactory treat- 
 ment of the stability of a dirigible is the difficulty of 
 making suitable assumptions regarding air resistances due 
 to causes other than the inertia already mentioned. We 
 might, of course, assume that these effects were, as in the 
 case of aeroplanes, represented to the first order by terms 
 of the form 
 
 - X -uX u - vX v - rX r 
 
 where X is proportional to 7 2 and the rest are propor- 
 tional to U. But to suppose that it is legitimate merely 
 to add these terms to those dependent on the inertia 
 of the displaced air (which depend on the accelerations of 
 the system) is a very shaky assumption. All that could 
 be said is that it would lead to conclusions which might 
 be regarded as a basis of comparison with results of 
 experiment. 
 
CHAPTER III 
 
 GENERAL CONSIDERATIONS REGARDING SYMMETRICAL 
 DERIVATIVES 
 
 Expressions for Plane Areas. 
 
 23. We shall now show how in the case of plane 
 supporting surfaces (setting aside camber for the present) 
 the nine resistance derivatives (X, Y. N) u , v , r can be 
 found if the normal resistance and the position of the 
 centre of pressure are known functions of the angle of 
 attack and of the angular velocity r. In the first place 
 we assume, for simplicity, that there is no tangential 
 resistance ; such a resistance may be considered sub- 
 sequently. 
 
 FIG. 4. 
 
 Let ' S be a plane area perpendicular to the plane of 
 (x, y] and slanting upwards at an " angle of attack " a 
 
CH. in LONGITUDINAL DERIVATIVES 39 
 
 to the axis of x. Let C be a point whose coordinates are 
 x, y, centrally situated on S, which may be taken in the 
 first instance to be an origin from which the distances of 
 the centre of pressure and other points on the plane are 
 measured ; P the centre of pressure. Let R be the 
 resultant thrust, f the perpendicular distance of its line of 
 action from 0. Then the component forces and couple 
 due to resistance (taken positive when retarding motion) 
 are 
 
 X = R sin a Y = E .cos a N = R . . (26) 
 
 The angle of attack being a, we may write 
 
 R = KSU*f(a) 
 
 CP = a <f> (a) = s (suppose) . . . (27) 
 
 where K is the coefficient of resistance, 
 
 a is the semi breadth of the plane, or a quantity 
 determined by its linear dimensions, 
 
 /(a), (f>(a) are functions determined by experiment, 
 representing the relations connecting the resultant thrust, 
 and position of the centre of pressure with the angle 
 of attack. 
 
 We have further 
 
 = x cos a - y sin a + a <(a) = x cos a - y sin a + s . (28) 
 
 If now the additional velocity components u, v, r are 
 impressed, the velocity-components of C will be 
 
 U + u - yr and v + xr 
 
 and the angular velocity of the lamina about C will 
 be r. 
 
 To the first order of the small quantities u, v, r, the 
 effect of these added velocities is to increase the resultant 
 velocity of C to U -\-u-yr (as is evident by neglecting the 
 second term in the exact expression : 
 
 vel. 2 = (U + u - yrf + (v + xr) 2 ) 
 
 and to alter the direction of the velocity of (7, thus 
 
4 o SYMMETRICAL OR CHAP. 
 
 increasing the angle of attack relative to C by an 
 amount 
 
 + xr v + xr 
 
 tan- 1 
 
 U + u - yr U 
 
 radians to the first order of small quantities. 
 
 Hence, to the first order the new values of R and 
 the distance f, are, so far as they depend on the transla- 
 tional velocity- components of (7, given by 
 
 R = K8V*f(a) + 2KSU(u - yr)f(a) + KSU(v + xr)f'(a) . (29) 
 = x cos a y sin a + a (f) (a) + a - TJ <'( a ) . . (30) 
 
 where f'(a) and <f/(a) are the differential coefficients of 
 f(a) and </>(). 
 
 24. But the lamina possesses in addition an angular 
 velocity r about (7, and in consequence of this neither the 
 resultant thrust R, nor the coordinate of the centre 
 of pressure can be expected to be the same as if the 
 motion of the lamina were one of uniform translation. To 
 allow for this difference, it is necessary and sufficient 
 to assume that /(a) and </>(a) not only are functions of the 
 angle of attack a relative to the point (7, but also that they 
 depend on the angular velocity r of the lamina relative 
 to G. 
 
 Now, if the velocity U and the angular velocity r 
 are increased in any given ratio nil, it is clear that 
 the motion of the plane will be the same as previously, but 
 will take place n times as quickly. The resultant thrust 
 will be n 2 times as great, being proportional to U 2 , and the 
 position of the centre of pressure will be the same as 
 before. From this reasoning we see that f(a) and $(a) 
 must be regarded as functions of the ratio r/ U as well as 
 of -the angle of attack a relative to the point (7, and we 
 may, therefore, write their corresponding variations to the 
 first order 
 
in LONGITUDINAL DERIVATIVES 41 
 
 so that 
 
 where f r (a) and <j> r (a) are certain coefficients which we 
 may call the rotary derivatives of /(a) and $(a). The 
 physical assumption that is here made is merely the 
 generalisation of the law according to which the resistances 
 vary as the square of the velocity, so that if all the parts 
 of a system move n times as fast, the forces due to air 
 resistance are increased n 2 times without being changed in 
 direction or position. 
 
 25. Making the necessary substitutions we obtain the 
 following results, writing g = x cos ay 
 
 sn, _ = sn a, 
 
 ' 
 
 Y Y Y 
 
 ' = /(a) COS a, -=- = 2/(a) COS a, _^- =/'(a) COS a, 
 
 KSU KSU KSU 
 
 - 2yf(a) + / r ()}| + f(a){xa<t>'(a) 
 
 (31) 
 
 If tbere are two or more planes and the actions of the 
 air pressures on them are independent of each other, the 
 derivatives for the system are obtained by adding those 
 for the separate planes. This assumes that the air 
 encountered by one of the planes has not been disturbed 
 or set in motion by the other, in other words, that neither 
 plane comes into the " wash " caused by the other. The 
 effects of " wash " will be discussed briefly at the end of 
 the discussion on longitudinal stability, although a great 
 deal of experimental work would be necessary to attain 
 anything like finality in dealing with them. 
 
 26. We now notice the following results which greatly 
 simplify the subsequent calculations. These results, 
 
4 2 
 
 SYMMETRICAL OR 
 
 CHAP. 
 
 unfortunately, did not come to my notice until much 
 long and laborious algebra had been gone through, by 
 which they were proved in the first instance. 
 
 (i) If there is only a single plane surface, or even if 
 there are several parallel planes, the determinant A 
 vanishes and in addition the minors of its third line 
 U N> VN> T N vanish. 
 
 (ii) If there are two plane surfaces only, and if the 
 displacements of the centre of pressure, a$ f (a) ,a$ r (a) , be 
 neglected, then the determinant of the derivatives for 
 either plane has three rows and three columns proportional 
 to one another, i.e., using accented and unaccented letters 
 to refer to the two planes the determinants, 
 
 Y u 
 
 N u 
 
 X r 
 
 r, 
 
 N r 
 
 and 
 
 7', 
 
 N' 
 
 Y' r 
 
 N' 
 
 each have three rows and three columns proportional, and 
 the resulting determinant A , which is 
 
 X' 
 
 Y' u 
 
 N' 
 
 X v 
 
 X' v 
 Y' 
 
 X r + X' r 
 Y r + Y' r 
 
 N r + N' r 
 
 can be expressed as the sum of eight determinants, one 
 formed by taking the first determinant, three formed by 
 taking two rows or columns of the first and the remain- 
 ing row or column from the second, and so on. In such 
 cases again, each of these determinants vanishes since it 
 has at least two rows or columns proportional to each 
 other, and the result is that A again vanishes. 1 
 
 1 An alternative proof is given as follows. Writing " = xf'(n) + 2t//(a) +/ r (a) 
 and using suffixes 1, 2 to refer to the two planes, the determinant A is 
 immediately recognised as the product (written down by the ordinary rule) 
 of two determinants, namely : 
 
 sin ctj sin a 2 
 
 cos a x cos o 2 
 
 , 
 
 which is equal to zero. 
 
in LONGITUDINAL DERIVATIVES 43 
 
 (iii) If $'(a) and $ r (a) be not neglected, and there are two 
 plane surfaces, the only parts of A which do not vanish 
 will be those in which $'(ai). $'(o), $,( a i)> *(**) occur as 
 factors, a b a. 2 being the inclinations of the two planes, and 
 these will be multiplied into the minors V N , r N ; and it is 
 easy to see 
 
 T N = 2U*K 1 K 2 S l S 2 {f(a l )f(a 2 ) - f(aJfM sin (a, - a 2 ) . (32) 
 
 and that V N can be expressed in a somewhat similar form, 
 also containing sin (a-^ a 2 ) as a factor. Without looking 
 further at the present stage of the investigation, it will be 
 seen that the determinant A , so far from giving rise to 
 laborious calculations, either vanishes or reduces to some- 
 thing readily calculable and probably in most practical 
 applications negligible. An exception will be discussed in 
 due course, in which it will be shown that in some cases 
 stability may be increased by the use of three planes 
 especially arranged so as purposely to give a value of A 
 different from zero. 
 
 (iv) In the case of a single plane surface, using the 
 expression for ( , namely, 
 
 @ W 
 
 i = [r x cos 6 - r Y sin 6 \ 
 
 we obtain 
 
 cos (a - 6 Q ) . (33) 
 
 and it follows from the condition ( positive that a 
 single plane cannot be stable unless <f>'(a) is negative, 
 that is unless the centre of pressure moves forward 
 when the, angle of attack decreases. This is the case in 
 actual practice. Also we see that to secure stability, the 
 plane must be sufficiently broad for the shift of the 
 pressure to have an appreciable effect. In such cases, 
 however, it becomes necessary to take account of the 
 effects of the coefficients f r (a) and <j) r (a) in the other 
 conditions of stability. They do not occur in ^ . 
 
44 SYMMETRICAL OR CHAP. 
 
 Observations on the Rotary Derivatives. 
 
 27. In the foregoing work we have introduced two 
 derivatives, /(a) and r (a), such that f r (a)r/U and 
 <j) r (a)r/U, represent the changes in the values of /(a) and 
 (f>(a) when the lamina receives a small rotation r about 
 the point (x, y). It is important that the effects of 
 rotation on the resultant thrust and position of centre 
 of pressure should be investigated experimentally if broad 
 aeroplanes and aerocurves are to be used in aerial naviga- 
 tion. In the original results in Bryan and Williams' 
 paper, these rotary derivatives, as they may be called, 
 
 r\ 
 
 Vel. U of lamina ^>^ \ a Relative Vel. 
 
 U of wind 
 
 FIG. 5. 
 
 were neglected, failing the existence of such experimental 
 data, and the same thing has certainly been done in all 
 stability investigations till now. We may certainly get 
 rid of one of these by suitably choosing the point C (x, y). 
 For f r (a) only occurs in xf(a)- 2yf(a)+f r (a), and all that 
 is necessary is to write 
 
 xf(a) - 2yf(a) + f r (a) = x'f(a) - 2y'f(a) 
 
 this gives one relation determining the co-ordinates (x f , y f ) 
 of Ci, the new position of C, and we may further assume 
 that (7i is in the plane of the lamina itself. The point Ci 
 is characterised by the property that rotation about C l does 
 not affect the resultant thrust on the lamina. 
 
 Now let AB (Fig. 5) represent the lamina, C its 
 
in LONGITUDINAL DERIVATIVES 45 
 
 middle point. Then if the lamina receive an angular 
 velocity r about A it will move away from the wind, 
 and we naturally infer that the resultant thrust will be 
 decreased. If it rotate about B in the same direction 
 it will move towards the wind, and we infer that the 
 resultant thrust will be increased. We are thus led to 
 believe that Ci'lies between A and B. If the lamina 
 rotate about its centre (7, the thrust will be increased on 
 A C and decreased on EG, but as the thrust is greater on 
 the forward part AC than on the backward part BG, 
 it appears probable that the net result is to increase the 
 thrust. Hence we infer that the required point (7j is 
 probably in front of C and quite near the centre of 
 pressure. 
 
 A somewhat similar line of argument may be applied 
 in connection with <f> r (a). A positive rotation r about 
 a point on AB produced considerably beyond B will 
 certainly increase the angle of attack and cause the centre 
 of pressure to move backwards. A similar rotation about 
 a point on BA produced considerably beyond A will 
 decrease the angle of attack and shift the centre of 
 pressure forwards. But a rotation about C appears 
 likely to increase the pressure on the forward part of the 
 plane and decrease that on the backward part, shifting the 
 centre of pressure forwards, and hence it would appear 
 that the point C 2 about which the lamina must be rotated 
 in order that there may be no shifting of the centre of 
 pressure is probably behind (7, and does not coincide 
 therefore with C L . 
 
 The last argument is somewhat indefinite, as it is 
 impossible to bo quite sure that a current of air imping- 
 ing on a rotating lamina would behave in exactly the way 
 that appears most plausible. There are, however, two 
 methods of investigating the question experimentally ; a 
 much more satisfactory alternative. 
 
 28. The Whirling-table Method. Suppose the lamina 
 
SYMMETRICAL OR 
 
 CHAP. 
 
 attached to a rotating arm and whirled in such a way as 
 to give a forward velocity U at the point C. 
 
 If the lamina be rotated about where 
 OC=a, then we have r=U/a; if rotated 
 about a point O' on the opposite side we 
 should have r = U/a. If then f(a) and 
 (j>(a) be observed by determining experiment- 
 u ally the resultant thrust and position of the 
 centre of pressure when rotated first about 
 and secondly about O ', the difference of the 
 two values of f(a) multiplied by the length 
 a/2 will give f r (a), and the difference of the 
 two values of <j>(a) multiplied by a/2 will 
 similarly give r (a), the values being in either case 
 referred to the point C. 
 
 29. The Pendulum Method. According to the Blue 
 Book for 1909-1910, experiments in a wind tunnel which 
 produces a current of air that may be assumed uniform 
 give better results than whirling table experiments. To 
 adapt this method to the rotary derivatives, it is necessary 
 
 o 
 
 FIG. 6. 
 
 FIG. 7. 
 
 to attach the lamina under investigation to a pendulum 
 and observe the oscillations. To eliminate the effects 
 of air resistance on the pendulum the latter would have 
 
in LONGITUDINAL DERIVATIVES 47 
 
 to be made in the form of a disc or sphere, unequally 
 loaded, but having its axis in the axis of rotation (Fig. 7). 
 Then only tangential resistances would occur, the resultant 
 couple due to these would be proportional to the square 
 of the angular velocity (as may easily be shown), and its 
 effect on small oscillations might be neglected for an 
 approximate calculation. 
 
 Let AB be the lamina, O the centre of suspension, 
 6r the centre of gravity of the whole mass, 9 = inclination 
 of OG to vertical, a = inclination of AB to horizon, so 
 that 6 a = constant. If N is the moment of the resist- 
 ances about 0, then when the lamina is at rest we have as 
 in (31), 25, 
 
 N = KSU*f(a)(x COS a - y sin a + a </>()) . . (34) 
 
 The general equation of motion is 
 
 J2/3 
 
 I^ 2 =-W.OGsind + N 
 
 where /= moment of inertia of whole mass about 0. 
 For equilibrium we have 
 
 0= - W.OG sin0 + N 
 
 In a small oscillation if = + e and a = a + e, and the 
 angular velocity de/dt is equal to r, we have 
 
 sin 6 = sin 6 + e cos 6 
 
 and the equation of small oscillations becomes 
 
 (35) 
 
 whence the modulus of decay is gN r /2l, and if the period 
 is 2-TT/p, then 
 
 dN\ 
 
 '.OflfcoB*.-^-^ 
 
 This holds if the motion be oscillatory, a condition that 
 can be always secured by making OG sufficiently great. 
 
48 SYMMETRICAL OR CHAP. 
 
 The value of dN/da is got by directly differentiating N 
 with respect to a. 
 
 It follows that by observing the modulus of decay 
 N r can be found, or i/\N r be negative the logarithmic 
 increment of the oscillations ivill determine it. 
 
 Now using the notation of (31) 25 we see that if 
 N r is thus found for two different positions of the point C 
 (x, y) on the lamina, with the same angle of attack, a, the 
 formulae for N r in (31) will give sufficient data to find 
 both f r (a) arid ^ r (a). It may perhaps be safe to omit 
 further details of the method as the reader can probably 
 supply them. 
 
 This conclusion suggests, however, the possibility of 
 going further, and instead of using a single lamina, experi- 
 menting with a model aeroplane so as to find the values 
 of N r relative to different points in its plane of symmetry 
 as centres of rotation. The idea suggests itself, to use 
 the results to determine the resistance derivatives of the 
 aeroplane considered as a whole, instead of building them 
 up from a consideration of the separate surfaces. The 
 problem thus arises : Given the values of N r referred 
 to different origins in the plane of symmetry, to find 
 as much information as possible about the other resistance 
 derivatives referred to a single origin. This involves 
 consideration of the transformation-formulae for resistance 
 derivatives. 
 
 Formulae of Transformation. 
 
 30. Let U + u, v, r be the velocity components, X, Y, 
 N the resistance components referred to the origin (0) ; 
 let accented letters refer to parallel axes through the 
 point (x, y). We have 
 
 u = u' + yr, v = v f - xr, r' = r 
 
 X' = X, Y' = F, N' = N - xY + yX . . (36) 
 
in LONGITUDINAL DERIVATIVES 49 
 
 and the transformation formulae fall under the following 
 types : 
 
 (1) For (X, Y) m w v the type X'^ = X u 
 
 (2) For (X, Y\ the type 
 
 X'n = X r + yX u - xX v 
 
 (3) For N u , v the type 
 
 N' ul = N u + yX u - xY u 
 
 (4) For N r 'i the type 
 
 = N r - x(N v + Y r ) + y(N u + X r ) 
 
 v - xy(X v + Y u ) + y*X u . (37) 
 
 It will thus be seen that determinations of the resist- 
 ance-couple-derivative N r about different origins will give 
 the values of 
 
 X w T m N n N v + Y n N H + X r , Y u + X v 
 
 and as the values of X u , Y u , N u are equal to 2X /U, etc., 
 only one further datum is needed. If the condition 
 A = is satisfied this datum is available. 
 
 31. The formulae for change of axes, the origin being 
 constant, are as follows : using unaccented letters to refer 
 to old axes and accented letters to refer to axes making an 
 angle A, in the positive direction with them we have 
 
 N' - N = 
 
 u = u' cos A v' sin A , v = u' sin A + v' cos A 
 
 X' = X cos A + Y sin A Y' = - X sin A + Y cos A 
 
 d A A A- A d 
 
 (38) 
 
 whence the resistance derivatives referred to the new axes 
 can easily be expressed in terms of those referred to the 
 old axes, for example : 
 
 rJ. J\ 
 
 sin 
 
 sin 2 A 
 E 
 
 X' U ' = -y- r X' = i cos A -j + sin A -7- ) (X cos A + Y sL* ^ 
 du \ du dvj ^ 
 
 = X n cos 2 ^. + (X v + Y u ) sin A cos A + Y v sin 2 
 
 XT' 
 
5 o SYMMETRICAL OR CHAP. 
 
 Camber. 
 
 32. The chief difference between plane and curved 
 surfaces is that in the latter the direction of the 
 resultant pressure may vary with the angle of attack, 
 and we should have to use two letters, a, a, the first 
 denoting the inclination of, say, the chord of the aero- 
 curve to the axis of x, the other a denoting the 
 inclination of the resultant thrust to the axis of y. For 
 Turnbull's doubly curved surfaces experimental data would 
 alone be available, but if the section of a surface is a 
 circular arc of radius c, it is clear that the resultant 
 thrust will pass through the centre of curvature, and if the 
 centre of pressure be shifted forward through a distance 
 $s = a(//(a)Sa, the direction of R will change by an amount 
 a equal to Ss/c or Sa=Sa x a/c{ <//()} (for a plane 
 surface <'() is negative). The most important effect of 
 this when the angles are small will be to produce a 
 variation &X in the horizontal component equal to 
 R cosa Sa, and, when the necessary substitutions are made, 
 it is seen that the principal effect of curvature is to 
 introduce additional terms into the values of X v and X r . 
 The magnitude of these terms depends on a/c, that is, on 
 the angle which the arc subtends at its centre of curvature, 
 nob on the actual size of the aerocurve. They might 
 thus become important even with narrow aerocurves if this 
 angle is not small. The corrections thus introduced are 
 discussed in 69. Further corrections may be examined 
 at a future time if deemed advisable. 
 
 The principal difficulty arises from our want of 
 information regarding the rotatory coefficients </> r () 
 which also lead to variations in the angle a ; till these 
 variations have been studied the stability of aerocurves 
 must be to some extent uncertain, and the extension of 
 results calculated for plane areas must be regarded as a 
 
in LONGITUDINAL DERIVATIVES 51 
 
 working hypothesis, divergences between theory and 
 experiment being anticipated. 
 
 Narrow Planes at Small Angles. 
 
 33. Owing to the doubtful points raised in the previous 
 paragraphs, as well as for reasons of simplicity, the study 
 of aeroplane stability is best commenced by the considera- 
 tion of systems of narrow planes at small Angles. 
 
 By the term narrow planes applied to an aeroplane 
 with two or more plane surfaces, one behind the other, it 
 is to be implied that the breadth or " chord " of either 
 plane is small in comparison with their horizontal distance 
 apart. If the aeroplane receives a small angular velocity r, 
 the difference of velocity set up between the front and 
 rear edge of a plane will be small compared with the 
 difference in velocity set up between the two planes, and 
 for this reason, the rotary derivatives may be neglected 
 (in comparison with quantities actually taken into account). 
 Further it may be assumed in the first instance that the 
 shifting of the centres of pressure with varying angles of 
 attack [i.e., the quantities a$'(a)] are negligible owing 
 to the smallness of a. 
 
 We shall now assume that the point C (of 23) the 
 position of which has hitherto been left arbitrary is chosen 
 to coincide with the centre of pressure when the aeroplane 
 is flying steadily. That is, we now define x, y to be the 
 coordinates of the centre of pressure in steady flight, 
 so that <j)(a) = 0. The assumption that f r (a) and a^ r (a) 
 are negligible thus becomes equivalent to neglecting the 
 distances from the centre of pressure of the points C lt C 2 
 considered in 27, an assumption which, as we have seen, 
 is probably fulfilled more approximately in the case of Ci 
 than of (7 2 . 
 
 When the planes are gliding at small angles we imply 
 that the angles of attack a are small, and in such 
 
 E 2 
 
52 . SYMMETRICAL OR CHAP. 
 
 circumstances it will be assumed that the normal thrust 
 on a plane varies as the sine of the angle of attack. 
 This property is commonly expressed by the formula 
 
 P = 2P90 sin a 
 
 where P 90 is the pressure when the angle of attack 
 is 90 ; of course the formula fails when a is con- 
 siderable, for with a = 90 it would give P = 2P 9Q 
 instead of P = P 90 ; other formulae have been proposed 
 which get over the difficulty, such as Duchemin's 
 
 P _ op sina 
 
 ** 1 + sin 2 a 
 
 but for small values of a, the simpler one is usually 
 regarded as sufficient. We therefore write f(a) = sin a. 
 
 We further assume that sin a is small and that the 
 approximations sin a = a and cos a = 1 may be used if 
 desired, and that terms of the order a may be neglected 
 in comparison ivith finite quantities. Remembering, 
 however, that in horizontal flight 
 
 H = 2KSU 2 sin 2 a ..... (39) 
 
 it is evident that even sin 2 a cannot be neglected except 
 in comparison with quantities of lower order. 
 With these assumptions, and writing 
 
 = x cos a - y sin a, " = x cos a - 2y sin a 
 
 the table of derivatives in 25 reduces to 
 
 =2A^sin 2 a ^ = 2^KS sin 2 a, ^ = 2#smaeos a, -jj 
 
 Y Y Y Y 
 
 - a cos a, jj=2KScos*a, ^ 
 
 ina, ^ = 2O{' cos a, jf 
 
 (40) 
 
 The equations of equilibrium in horizontal flight 
 become 
 
 H = U*2KS sin 2 *, W = U*2,KS sin a cos a, - Hh = U*3K8' sin o (41) 
 
 34 The $', IJL notation. A further simplification which 
 will be found to be of great use, especially in dealing 
 
in LONGITUDINAL DERIVATIVES 53 
 
 with stability under complicated conditions, consists in 
 the substitutions S' = S cos 2 a and ^ = tan a. With this 
 substitution the foregoing formulae give : 
 
 
 (40a) 
 
 with 
 
 The Principle of Independence of Height. 
 
 35. Since the angles a are small, f' and f" are both ap- 
 proximately equal to x, and the resistance derivatives are 
 to this approximation independent of the y coordinates of 
 the planes. We thus see that if an aeroplane is stable 
 when the planes are on a level with the centre of gravity, 
 the stability is only very slightly affected by raising or 
 lowering the planes. 
 
 If instead of raising or lowering a plane vertically it be 
 displaced in a direction perpendicular to itself, the value 
 of f ' will be unaltered, while f " will be decreased by y sin a, 
 so that only this small difference between f' and f" will 
 affect the stability, and the conditions of equilibrium will 
 be unaffected. 
 
 In this case the line of action of the normal resultant 
 thrust on the plane is unaffected by the displacement, and 
 the only difference is that the tangential velocity of the 
 plane due to a rotation, r, of the aeroplane is altered, the 
 corresponding normal velocity being the same as before. 
 When the angles of attack are small, this change of 
 tangential velocity has only a small effect on the resultant 
 thrust of the air, and is far less important than a change 
 of normal velocity would be. 
 
54 
 
 SYMMETRICAL OR 
 
 CHAP. 
 
 The present deduction may be called the " Principle 
 of Independence of Height." This property seems at first 
 sight at variance with what one would naturally expect 
 from analogy with the case of a pendulum whose oscilla- 
 tions depend essentially on the height of the point of 
 suspension above the centre of gravity, or again from the 
 case of an aerostat. It must, however, be remembered 
 that the principle is based on the assumption that 
 tangential and head resistances are neglected, and the 
 difference will be made clear by taking moments in each 
 case about the centre of gravity, with the aid of the 
 following diagrams : 
 
 Resultant 
 
 In the pendulum (Fig. 8), the point of suspension (7 is 
 fixed so that the pendulum can only turn about C. In 
 this case rotation about G is set up by the moment of 
 the tangential force F acting at C at right angles to GO. 
 
 In the aerostat (Fig. 9), the hydrostatic thrust W, 
 equal to the weight of the air displaced, acts vertically 
 upwards although the centre of buoyancy C and its 
 moment about G tends to bring GO into the vertical 
 position by producing rotation about G. 
 
 In the plane parachute (Fig. 10) without tangential 
 
in LONGITUDINAL DERIVATIVES 55 
 
 resistance, if variations in the position of the centre of 
 pressure are neglected, the resultant thrust R always 
 passes through G, and has no moment tending to turn the 
 parachute about G. Of course the weight W has a 
 moment about (7, but all the forces pass through G and 
 their only effect is to produce an acceleration of the 
 parachute as a whole in the direction of the resultant force. 
 When the forces on a body have no moment about the 
 centre of gravity they have no tendency to produce 
 rotation. On the other hand, a tangential resistance would 
 tend to turn the parachute about G, bringing the axis GC 
 towards the vertical position. 
 
 Fig. 11 shows an aeroplane with two narrow planes, 
 A, B, and centre of gravity, G. If the normals to the two 
 
 FIG. 11. 
 
 planes intersect in /, the point / will present the nearest 
 analogy to the centre of suspension of the pendulum of 
 Fig. 8, or centre of buoyancy of the balloon in Fig. 9. 
 The resultant of the two normal thrusts at A and B wilJ 
 pass through /, and if it does not act along GI, its moment 
 about G will tend to rotate the aeroplane. But if A or 
 B are displaced perpendicular to themselves to A' and B f , 
 the normals to them will still meet in the same point /, 
 and the equation of moments about / will be the same as 
 before. The thrusts on A' and B' will also be the same 
 as on A,B when the aeroplane has no rotation about G, 
 
56 LONGITUDINAL DERIVATIVES CH.III 
 
 and when it rotates the normal velocities of A', B' will be 
 the same as of A, B, while the difference of tangential 
 velocity between A and A only has a small influence 
 on the resultant thrusts. 
 
 In actual practice it is certain that, owing to head resist- 
 ances and tangential forces, stability will be affected to a 
 sensible extent by raising or lowering the planes, and 
 also, if the planes are too high, a sudden gust of wind 
 will cause the aeroplane to heel over. 
 
 The inference, then, is that this effect must be in large 
 measure attributed to tangential forces or head resistances. 
 
 On the other hand, the principle receives confirmation 
 from the fact that successful flights have been made with 
 machines in which the centre of gravity is above the 
 aeroplanes. 
 
 It must not be forgotten, too, that we are now only 
 considering systems of planes, not curved surfaces. 
 
 An important corollary is that there is no practical 
 difference between monoplane and biplane machines in 
 respect of stability. It is the horizontal, not the vertical, 
 disposition of the planes on which stability mainly 
 depends. The nomenclature " monoplane " and " biplane " 
 does not cover the necessary distinctions. The differences 
 with which we are concerned are between the single-lifting 
 system in which the whole of the weight is supported 
 either by one plane or by two or more vertically super- 
 posed planes, the remaining planes taking the form of a 
 tail or rudder only, and double- or multiple-lifting systems 
 in which the weight is supported by two or more planes 
 (or sets of superposed planes) disposed in a fore and aft 
 direction. 
 
CHAPTER IV. 
 
 GRAPHIC STATICS OF LONGITUDINAL EQUILIBRIUM. 
 
 Single-Lifting Systems. 
 
 36. Before considering stability further it is useful to 
 discuss the problem of longitudinal equilibrium with 
 special reference to steering in a vertical plane. This can 
 best be done by graphic methods. 
 
 In the following discussions we shall in general suppose 
 the aeroplane to be longitudinally stable. In such cases a 
 variation in the direction of the rudder plane or in the 
 propeller thrust will cause the aeroplane soon to assume 
 the new position of equilibrium determined by the altered 
 conditions. If the aeroplane is not stable we can still 
 examine the effects of a change of the kind on the con- 
 ditions of equilibrium, although the initial effect of the 
 change may be the reverse of that needed to bring the 
 aeroplane into equilibrium and the new equilibrium 
 position may only be attainable by careful manoeuvring. 
 
 Case 1. Single-Lifting System with a Neutral Tail or 
 Rudder. We speak of the tail or rudder as neutral when 
 its plane is parallel to the direction of the relative wind 
 so as to encounter no normal thrust. This is sometimes 
 referred to as " grazing incidence." In this case the 
 following three forces have to be in equilibrium : 
 
 1. The weight W acting vertically through the centre 
 of gravity G. 
 
 2. The resultant thrust R on the supporting plane. 
 
 3. The thrust H of the propeller. 
 
 57 
 
58 GRAPHIC STATICS CHAP. 
 
 The first condition of equilibrium is that these three 
 forces must pass through one point. 
 
 Suppose H passes through the centre of gravity G, 
 then R will also pass through G (Fig. 12). 
 
 The other condition of equilibrium is that the three 
 forces W, R, H are parallel and proportional to the three 
 sides of a triangle. Also W is constant while the angle 
 
 FIG. I. A CHANUTE GLIDER. 
 
 The two superposed planes constitute a " single lifting system," the tail plane 
 being in general " neutral " ( 36). Being used as a glider, the longitudinal 
 stability would be greater than in a similar aeroplane propelled hori- 
 zontally ( 55). The photograph was given to the author by the late 
 Mr. Octave Chanute. 
 
 between H and the main plane is constant and we will 
 denote this angle by &>. We then have the following 
 geometrical construction for the triangle of forces : 
 
 Draw DE vertical and of length W units. On DE 
 describe a segment of a circle containing an angle 90 o>, 
 and in it place a line DF containing H units of length. 
 
IV 
 
 LONGITUDINAL EQUILIBRIUM 
 
 59 
 
 Then FD will be parallel to the force H and EF will be 
 parallel and proportional to the normal force R (Fig. 13). 
 
 The aeroplane will therefore be in equilibrium when 
 the axis of the propeller is parallel to FD and the main 
 plane is perpendicular to EF. In order that FD and 
 therefore H may be horizontal we must have H = W tan CD. 
 
 The direction of the aeroplane relative to the horizon is 
 thus determined by the magnitude of the thrust H. If 
 H be increased F will be lowered relative to D and the 
 aeroplane will point more upwards. 
 
 The actual direction of flight is in the present instance 
 
 din plane 
 
 H 
 
 W 
 FIG. 12. 
 
 parallel to the tail or rudder plane ; if this be altered in 
 direction for steering purposes the direction of flight will 
 alter accordingly. This conclusion depends essentially on 
 the hypothesis, made in dealing with narrow planes, that 
 we neglect the shift of the centre of pressure when the 
 angle of attack is varied. In this case if the three forces 
 W, R, H ever pass through the* common point O they 
 will always do so, hence the conditions of equilibrium 
 require that the moment about G of the thrust on the 
 rudder vanishes, therefore the thrust itself must vanish 
 and the rudder must always be neutral. 
 
 Finally, if the angle of attack is a, the velocity of the 
 aeroplane is determined from' the equation 
 
 R = KSU 2 sin a 
 
60 GRAPHIC STATICS CHAP. 
 
 or indeed from the assumed law of resistance, supposing the 
 sine law be not assumed to hold good. Thus in conclusion, 
 when there is equilibrium 
 
 The inclination of the machine to the horizon depends 
 exclusively on the propeller thrust. Increasing this 
 causes the machine to tip up in front, decreasing it 
 causes the machine to tip down. 
 
 The direction of flight is always parallel to the 
 plane of the tail. If this be turned in direction, a 
 corresponding change will take place in the direction 
 of flight. 
 
 The magnitude of the velocity U depends on both 
 causes, being determined by the law of resistance. 
 
 37. If the propeller thrust H does not pass through G, 
 the forces H and R will intersect in a point /, which is 
 fixed relative to the machine. In order for the tail to 
 be neutral the conditions of equilibrium between the forces 
 W, R, H require that GI shall be vertical. If the machine 
 be tilted forwards or backwards this will cease to be the case, 
 and the rudder plane will necessarily cease to be neutral 
 as is evident from the equation of moments. And since 
 the inclination to the horizon depends on the value of H, 
 there will be only one value of H for which the pressure 
 on the tail plane vanishes. 
 
 If the engines are stopped and the machine glides 
 steadily under gravity, as it should do with the rudder 
 neutral, then (according to our assumptions) the only 
 forces which maintain -equilibrium are R and W 9 and it 
 follows that R must pass through G. If the engines 
 are re- started and exert a thrust H, the rudder cannot still 
 remain neutral unless the thrust H also passes through G. 
 
 Unless, therefore, an aeroplane is provided with sub- 
 stantial fore and aft planes capable of sustaining a 
 considerable pressure, it is important that the propeller 
 thrust should pass as nearly through the centre of gravity 
 as is possible under practical conditions. Otherwise the 
 
iv LONGITUDINAL EQUILIBRIUM 61 
 
 machine must be regarded as a doubly-lifting one for all 
 practical purposes. It is very important that in case of a 
 stoppage of the engines the machine should glide " on an 
 even keel " without the rudder being subjected to excessive 
 strains. 
 
 A. second cause of pressure on the rudder plane is the 
 shifting of the centre of pressure when the angle of attack 
 varies, the main plane being no longer assumed to be 
 " narrow," according to our signification. In this case the 
 rudder, if neutral, remains so, as long as the angle of 
 attack remains constant. To secure this, when it is 
 required to rise or descend at an angle the propeller 
 thrust must be increased or decreased so as to tilt the 
 machine up or down through the same angle 6. The 
 angle of attack being constant, steering in a vertical 
 plane will have to be effected exclusively by varying the 
 propeller thrust, and not by turning the rudder plane 
 relative to the machine. 
 
 If, on the other hand, the angle of attack is decreased, 
 the centre of pressure will move forward and the air will 
 have to exert an upward thrust on the rudder plane if it 
 is fixed posteriorly. When the angle of attack is increased, 
 the centre of pressure in general will move backwards and 
 the opposite will be the case. 
 
 Double-Lifting Systems. 
 
 38. In the general case of a double-lifting aeroplane, 
 where neither the front nor the rear planes are neutral, 
 there are four forces in equilibrium when the machine is 
 moving uniformly. These are the weight W at 6r, the 
 propeller thrust H and the resultant thrusts R and R 2 on 
 the two surfaces S t and S 2 at their centres of pressure A 
 and B. As before we shall suppose in the first instance 
 that H passes through the point G. Let the normals to 
 the surfaces at their centres of pressure meet in M 9 
 
62 
 
 GRAPHIC STATICS 
 
 CHAP. 
 
 so that for narrow planes M is regarded as a fixed point 
 (Fig. 14). For equilibrium the resultant of R l and R 2 , 
 which acts through M, must be equal, and opposite to the 
 resultant of W and H acting at G. Calling this resultant 
 R its line of action is along GM, and the planes Si, S 2 are 
 equivalent to a single plane of suitable area, having its 
 centre of pressure in the line GM and its plane perpen- 
 dicular to GM. If co is the inclination to GH of this 
 equivalent plane, the angle between ffand R is 90 + o>. 
 
 As before the three forces at G are represented by the 
 sides of the triangle DEF in Fig. 13, the angle at F 
 being constant and equal to 90 o>, and if H be increased, 
 
 O r*- B 
 
 Direction 
 
 of 
 flight 
 
 /A 
 
 FIG. 13. 
 
 FIG. 14. 
 
 it is obvious that F will be lowered relative to D, and the 
 machine will be tilted upwards ; conversely., if H be 
 decreased, the machine will be tilted downwards. If the 
 engines are stopped without moving the planes, the 
 aeroplane will be in equilibrium when GM is vertical. 
 
 It thus appears that the inclination of the machine to 
 the horizon depends entirely on the propeller thrust. In 
 fact, we may say that, so far as the equilibrium is con- 
 cerned, the machine behaves exactly like a glider, the 
 resultant of propeller thrust and gravity taking the place 
 of gravity alone. 
 
 It remains to determine the motion. Knowing R (the 
 
iv LONGITUDINAL EQUILIBRIUM 63 
 
 values of H and W being given) the components ^ and 
 jR 2 are known, and if aj and a 2 are the angles of attack, 
 we have 
 
 E l = K^U'^in a lt R 2 = K 2 S 2 U* sin a 2 . (42) 
 
 Now from the conditions of equilibrium we have 
 
 E! sin GMA = E 2 sin GMB 
 
 Hence 
 
 li^ sin a x sin GMA = ^^ sin a 2 sin M . . (43) 
 
 which determines the ratio sin a^ : sin a 2 . Also a x a 2 > the 
 angle between the directions of the planes is known, being 
 equal to A MB, and a t and a 2 may be found. When 
 this has been done, Z7 2 may be determined from either of 
 the above equations (42) or otherwise. 
 
 39. If one or other of the planes S l9 S 2 is used for steering 
 purposes so that its inclination to the line AB can be 
 varied, it is necessary to have recourse to a geometrical 
 construction in order to ascertain the effects of steering on 
 the position of equilibrium. 
 
 Suppose A, G, B to be in a straight line, and let MO 
 be drawn through M perpendicular to the direction of 
 flight to meet AB in 0. Then ai = AMO, a 2 = BMO, and 
 (43) becomes 
 
 K^ sin AMO sin AMG =K 2 S 2 sin BMO sin BMG . (44) 
 
 But 
 
 AO AGr 
 
 sin AMO sin 0AM ; sin AMG = ^ sin GAM 
 
 T>rv TZC* 
 
 sin BMO = sin OEM ; sin BMG = sm GBM 
 
 whence (44) reduces to 
 
 K&AO.AG sin 2 0AM = K 2 S 2 BO.BG sin 2 OEM . (45) 
 
 again if i^, i 2 are the angles which the planes S^ S 2 make 
 with the line AB, OAM^Vtf i^ OBM = $Q-i 2 , 
 giving 
 
 K&AO.AG cos 2 ^ = K 2 S 2 BO.BG cos 2 i 2 . . (46) 
 
 This determines the ratio AO'.BO, and hence O is 
 
64 GRAPHIC STATICS CHAP. 
 
 found. If i lt 4 are small angles, as is generally the case, 
 we may substitute the approximate relation 
 
 K&AO.AG = K 2 S 2 BO.BG . . . . (46a) 
 
 based on the approximate assumption that cos 2 4 = 
 cos 2 4 = 1. To this degree of approximation the position 
 of O is unaltered by varying the inclinations of the planes 
 to AB. 
 
 40. If the front plane be tilted up by increasing i lt then 
 M will be displaced to M ' along BM, and the direction of 
 
 N M 
 
 FIG. 15. 
 
 flight will be perpendicular to M'O instead of MO (Fig. 15), 
 The effect is to increase the angles of attack, which now 
 become OM'A, OM'B (for, by Euclid, these are greater 
 than OMA) OMB respectively), and the direction of 
 R will be changed from MG to M'G. The inclination of 
 the machine to the horizon will be altered in consequence, 
 and to estimate the net effect on the direction of flight 
 relative to the horizon, the figure must be combined with 
 the triangle of forces at the point G, say GDE. If, then, 
 
iv LONGITUDINAL EQUILIBRIUM 65 
 
 GD is measured containing H units of length, and with 
 centre D and radius W units, we describe a circle cutting 
 MG,M'G in E and E', DE and DE' will be the directions 
 of the vertical relative to the machine. The result of 
 the change will be that the machine will be tilted over 
 forwards through an angle EDE', and since the direction 
 of the line of flight relative to the machine is altered 
 by the amount MOM', the net result of tilting up the 
 front plane is to depress the direction of the flight path 
 through an angle MOM' + EDE'. In other words, raising 
 the front plane causes the machine to descend, and con- 
 versely depressing the front plane causes it to rise. 
 
 If the machine was originally flying horizontally, OM 
 will be parallel to DE, and if with centre and radius 
 OM we draw a circle cutting GM' in N, the triangles 
 GOM, GON, will be similar to GDE, GDE', so that ON 
 will be parallel to DE'. In this case the angle NOM' 
 represents the inclination to the horizon at which the 
 machine begins to descend. 
 
 41. The effect of tilting up the rear plane through an 
 angle MBM f is to alter the relative direction of the flight 
 path through MOM f (Fig. 16). The angles of attack 
 are altered from OMA, OMB, to OM'A and OM'B 
 respectively. It is obvious that OM'A < OMA, so that 
 the angle of attack on S l is decreased. That OM'B< 
 OMB is pretty evident from Figure 16. A formal demon- 
 stration, however, consists in observing that if OMB, 
 OM' B were equal, the points M, M', O, B would lie on a 
 circle, so that we should have MM'B^MOB-, but in 
 the cases which occur in practice MM'B is small, being 
 the angle between the front and rear planes, whereas 
 MOB is usually somewhere near a right angle, and, at 
 any rate, not a small angle. 
 
 The direction of R relative to the machine will be 
 turned through an angle MGM, and the direction of the 
 vertical through EDE' where GDE, GDE' are the two 
 
 F 
 
66 GRAPHIC STATICS CHAP. 
 
 positions of the force- triangle at G. The net change in 
 the direction of the flight-path relative to the horizon is 
 represented by the difference of angle of MOM'EDE'. 
 
 If the machine was originally flying horizontally, and 
 we draw a circle centre O and radius OM cutting GM in 
 
 O 
 
 FIG. 16. 
 
 N we see that angle MON=EDE r and the net result of 
 tilting up the rear plane is to cause the machine to rise 
 at an angle NOM 7 . Thus elevating the rear plane 
 causes the machine to rise, and conversely depressing it 
 causes the machine to descend. 
 
 42. Corrections. The above investigation will probably 
 be sufficiently accurate for most practical purposes. It 
 remains, however, to be shown that where the underlying 
 hypotheses are only approximately valid, more accurate 
 results might be obtained by introducing modifications in 
 the construction. 
 
 If cos 2 ij and cos 2 i 2 are not taken to be unity, the 
 point will no longer be fixed when the values of i x and 
 i 2 are varied. 
 
 If ij be increased the ratio AO : BO will be increased, 
 
iv LONGITUDINAL EQUILIBRIUM 67 
 
 and O will approach A moving to a position 0', and O'M' 
 will cut OM in a point below AB. 
 
 The perpendicular to the new flight path relative to the 
 machine will be M'O instead of MO, and the depression 
 of the path relative to the horizon will be slightly reduced 
 by the angle OM'O'. 
 
 If on the other hand i 2 be increased A O : SO will be 
 decreased and will move outwards to a position 0', so 
 that the elevation of the line of flight due to the change 
 will be reduced by an amount OMO '. In either case the 
 result is slightly to lessen the steering effect due to 
 turning the plane. 
 
 In either case the effect could be more fully studied by 
 constructing the envelope of the line MO, first with ^ 
 constant and secondly with i 2 constant. 
 
 43. If G does not lie in the line AS, and we wish 
 to study the effect of varying the inclination of the front 
 plane S 1 relative to the aeroplane, we must join AG 
 
 FIG. 17. 
 
 meeting BM produced in B', and the point must be 
 constructed on the line AGB f produced. In fact, since 
 the plane S 2 is kept fixed the construction is just the 
 same as if this plane were shifted parallel to itself to the 
 point B' . subject always to the proviso that the inclina- 
 tions of the planes to the line GA are small (Fig. 17). 
 
 F 2 
 
68 GRAPHIC STATICS CHAP. 
 
 If GB be produced to meet OM in O ', the point O' 
 will similarly represent the turning point of the line MO 
 when the inclination of S 2 is varied. 
 
 44. Effect of Shifting of Centre of Pressure If, owing 
 to a change in the angles of attack, the centre of pressure 
 of one or both planes is displaced, this will cause a further 
 change in the direction of flight and in the inclination 
 of the aeroplane to the horizon. 
 
 In Fig. 18 if A A', BB f represent the displacements 
 
 Ml 
 
 FIG. 18. 
 
 of the centres of pressure, MM' will represent the dis- 
 placement of M, and the constructions of 39 will give 
 the angle EDE' through which the direction of the 
 aeroplane is turned relative to the vertical. If the 
 centres of pressure are displaced forwards, as occurs when 
 the angles of attack are decreased, the figure shows 
 that the aeroplane will tilt upwards in front through the 
 angle EDE'. 
 
 To ascertain the effect on the direction of the flight- 
 
iv LONGITUDINAL EQUILIBRIUM 69 
 
 path relative to the aeroplane it is easiest to imagine 
 the angle A'ME' moved backwards and brought into 
 coincidence with A MB, in which case G must be dis- 
 placed through a distance GG f equal and parallel to M 1 M 
 (Fig. 19). Here MG' is nearer to MB than MG, and 
 it readily follows from the relation AO.AG-+ BO.BG = 
 constant, that is shifted to a point O f on the side of 
 remote from B. This depresses the direction of flight 
 relative to the aeroplane through an angle OMO ', and 
 as the aeroplane itself has tilted upwards, the net effect 
 is to change the inclination of the flight-path downwards 
 
 M 
 
 through an angle equal to OMO EDE f , the two causes 
 to a certain extent counterbalancing each other. If GG f 
 be small, and MG f meets AB in G", it may be shown, by 
 differentiating the relation A O.A G+ BO.BG = constant, 
 that 
 
 00' GG-" 
 
 AG.BG 
 
 (466) 
 
 and, further, that if the aeroplane was initially flying 
 horizontally the effect of the change is to depress or 
 elevate the flight path according as OO' is greater or 
 less than GG". It is scarcely worth while, however, to 
 reproduce here the discussion of these points in fuller 
 detail, 
 
yo GRAPHIC STATICS CHAP. 
 
 45. Prof. Marcel Brillouin's Metacentric Curves. A 
 more complete theory of the effects due to displacement 
 of the centre of pressure has been formulated by Prof. 
 Marcel Brillouin. 1 So long as these displacements are 
 neglected, the point M, which is the intersection of the 
 normals to the two planes, may be regarded as the 
 metacentre of the machine, being the point through 
 which the resultant thrust, R, always passes. If the 
 planes are broad, and if the line of action of the resultant 
 thrust is constructed for different directions of the relative 
 wind velocity, through the whole range of 360, these lines 
 of action when marked relative to the planes will envelop 
 a curve, called the metacentric curve. It will be noticed 
 that this curve is not the locus of the various correspond- 
 ing positions of M; for the direction of the resultant R 
 depends on the areas S lt S 2 of the planes, and is thus 
 independent of the direction of the tangent to the locus 
 of M. 
 
 M. Brillouin has determined the metacentric curves for 
 certain pairs of planes, assuming Lord Kayleigh's formula 
 for the position of the centre of pressure ; and similar 
 methods are obviously applicable, based on Joessel's or 
 any other alternative ]aw. The metacentric curves are 
 in general more or less star-shaped, having eight cusps, 
 four of which correspond to grazing incidence on one 
 or other of the two planes, and the intersection of the 
 normals to the two planes at their geometrical centres 
 is a centre of symmetry of the curve. The existence 
 of four other cusps is most easily shown in the case 
 of two equal plane areas. 
 
 If the metacentric curve be constructed, and if the 
 propeller- thrust passes through the centre of gravity 6r, 
 the positions of equilibrium are obtained by drawing 
 tangents to the curve through G, and then determining 
 the direction of the vertical by the force-triangle GDE. 
 
 1 Revue de Mecaniyue, 1909. 
 
LONGITUDINAL EQUILIBRIUM 
 
 7 1 
 
 46. Case where the propeller-thrust does not pass 
 through the centre of gravity. In Fig. 20 the three 
 forces H, W and R will pass through a point Z, which is 
 the intersection of the vertical through G with the line of 
 action of H. Draw GL parallel to the line of action of H, 
 and let it meet the line of action of R in L. Then ZGL 
 is a triangle of forces for the three forces at Z, and we 
 
 have therefore 
 
 H W R 
 GL == ZG == LZ' 
 
 Suppose that W and H are kept constant. 
 
 W 
 FIG. 20. 
 
 If the direction of H is nearly horizontal, so that GZ 
 is nearly perpendicular to ZH, then for small angular dis- 
 placements the length GZ is stationary, and L may be 
 regarded as a fixed point through which R always passes. 
 The point L thus takes the place of G in the previous 
 investigations. If the position of M undergoes a small 
 displacement to M f owing to a change in the inclination of 
 one or other of the planes, the new resultant R will act 
 along LM ', cutting ZH in Z f , and the new direction of 
 the vertical will be GZ' . 
 
72 GRAPHIC STATICS CHAP. 
 
 If ZH be not nearly horizontal, the relation GL : GZ ' = 
 H : W shows that ZL is a tangent to a conic whose focus 
 is G, and directrix is ZH, and the eccentricity (e) of which 
 is equal to the ratio H: W. 1 The point of contact P is 
 obtained by making ZGP a right angle, and this is the 
 point about which the line of action of R turns in the 
 case of a small displacement. 
 
 Hence, given the position of the metacentre M and the 
 magnitude q/*H. the position of equilibrium is obtained by 
 drawing a tangent from M to the conic ; and if this cuts 
 ZH in Z, then GZ is the position of the vertical. 
 
 The direction of the flight -normal MO can be con- 
 structed by the same method as before, the point of 
 
 FIG. 21. 
 
 contact P (or L if H is horizontal) taking the place of G 
 in the construction for 0. 
 
 Since in an aeroplane H is small compared with W, 
 the conic is an ellipse and two tangents can be drawn 
 to it from the point M, one on either side of G. If, 
 however, we construct the force-triangle for the second 
 tangent, we shall find that it determines a position of 
 equilibrium in which the aeroplane is inverted. 
 
 47. If the engine be stopped, then, in the new T position 
 of equilibrium MG will be the direction of the resultant 
 
 1 The intercept of a tangent to a conic on the latus rectum is e times the 
 distance from the focus of the point at which the tangent meets the directrix. 
 This easily follows from the normal property SG = eSP, and the property 
 that PSZ is a right angle, the letter S being substituted for G above and G 
 now being the foot of the normal. 
 
IV 
 
 LONGITUDINAL EQUILIBRIUM 
 
 73 
 
 R, and will be vertical. If G is above the line of action 
 of H (instead of being below, as in Fig. 21), the effects 
 of the change will be illustrated by Fig. 22. One 
 effect will be to chancre the relative direction of the 
 vertical from GZ to GZ' (in MG produced) thus causing 
 the aeroplane to dip downwards in front through an angle 
 ZGZ'. But as R moves forward from MZ to MZ', the 
 flight normal will also move forward from MO to a new 
 position MO', so that the angles of attack of both planes 
 will be decreased by an amount OMO'. The net result 
 will be to depress the direction of the flight-path relative 
 
 to the horizon by an amount ZGZ' OMO' . If instead 
 of stopping the engine the propeller-thrust H be reduced, 
 the eccentricity of the conic to which MZ is a tangent 
 will be reduced in the same proportion and in Fig. 22 MZ 
 will therefore be displaced towards MGZ', so that the 
 resulting changes in the position of equilibrium will be of 
 the same kind as before, but less in amount. For an 
 increase in the propeller-thrust the effects will be reversed, 
 and the aeroplane will begin to rise. 
 
 If it were possible to make the angles ZGZ' and 
 OMO' equal, the direction of the flight-path would be 
 unaffected by stopping the engines, and under closely 
 
74 GRAPHIC STATICS CH. iv 
 
 analogous circumstances the direction of the flight-path 
 might be made independent of the propeller-thrust. 
 This arrangement would be advantageous, but it is not 
 in general practicable, as Brillouin has shown in the 
 paper above referred to. If, as is usually the case, the 
 angles of attack are small, MG will be large compared 
 with the dimensions of the aeroplane, and it follows that the 
 angle ZGZ' is large compared with ZMZ' unless GZ', and 
 therefore the distance of the propeller-axis below 6r, are 
 also large compared with the other dimensions of the 
 aeroplane, a condition impracticable for other fairly 
 obvious reasons. The only possible alternative would 
 be to make OMO' large compared with ZMZ'. Even 
 if this were mathematically possible (which appears not 
 to be the case), it would have the result that a very 
 small displacement of the centre of gravity would produce 
 a very large change in the direction of the flight-path, 
 a condition possessing obvious grave objections. 
 
 The reader will have no difficulty in drawing the 
 corresponding figure, with the aid of Figs. 20 and 21, 
 for the case where the propeller-axis is below the centre 
 of gravity. In this case, when the engine is stopped, or 
 the propeller-thrust is reduced, the angular displacements 
 of the lines GZ and MO will be in opposite senses so 
 that if GZ' and MO denote their new positions the 
 direction of the flight-path will be depressed through an 
 angle ZGZ' + OMO . At the same time the angles of 
 attack on both planes will be increased by OMO ', so that 
 the descent will be less violent than it would otherwise 
 be. Conversely, if the propeller-thrust be increased, the 
 direction of the flight-path will be correspondingly 
 elevated. 
 
CHAPTER V 
 
 LONGITUDINAL STABILITY OF SINGLE-LIFTING SYSTEMS 
 
 The Simplest Case. 
 
 Single Lifting Plane Propelled Horizontally by a Central 
 Thrust Lanchester's Condition. 
 
 48. In order to avoid algebraic complications at the 
 outset, we first consider a system specified as follows : 
 Two surfaces (either of which may be replaced by a 
 
 pair of superposed surfaces), S i9 S 2 , of which the front 
 surface S l supports the whole weight of the aeroplane, 
 being inclined to the line of flight at an angle of attack a, 
 while the rear surface S 2 acts as a tail or rudder or 
 auxiliary plane, being placed in a neutral direction so 
 that 2 = 0. Distance between centres of pressure == /. 
 
 The line of action of the propeller- thrust passes through 
 the centre of gravity (and this fact will in the future be 
 represented by the statement that the thrust is central), the 
 direction of the thrust being along the line of flight, 
 which in this section is taken to be horizontal. In these 
 circumstances the resultant thrust of the air on Si also 
 
 75 
 
76 SINGLE LIFTING SYSTEMS CHAP. 
 
 passes through the centre of gravity; thus the centre of 
 gravity either coincides with the centre of pressure or is 
 on the normal to S l through the centre of pressure. In the 
 former case we have for S 1 both ' and f //= =0, x = and 
 y ; in the latter f ' = exactly and f " = approxi- 
 mately, while for S 2 we have f ' = f " = I. 
 The conditions of equilibrium give : 
 
 H = KSiU 2 sin 2 a 
 
 W = KS^sm a cos a 
 
 = KSi U 2 g sin a, or ' = as above . . . (47) 
 
 The nine derivatives are 
 
 X u = 2#Sj U sin 2 a, X v = KS 1 U sin a cos a, X r = 
 
 Y u = 2KS 1 U sin a cos a, Y v = KS^ cos 2 a + K8 2 U, Y r = ~KS 2 Ul 
 
 N u = 0, N n = -KS Z UI, N r = KS 2 Ul 2 (48) 
 
 We obtain 
 
 (49a) 
 . . . (496) 
 
 (49c) 
 
 W 2 
 sin2 + W^^iWl + sin 2 a) + - 
 
 It will be noticed that the conditions T) O and ( positive 
 require I to be positive, that is, the auxiliary plane must 
 be placed behind the main plane, as a tail, not in front. 
 
 In virtue of the condition of equilibrium we have also 
 
 g+ ff1t . 
 
 = S^S 2 l sin 2 a cos 2 a 
 
 = SfSJ sin 2 a (1 - sin 2 a) . . (49e 2 ) 
 
 J 
 
 When expressed in this form we see that D^/'ZT* and 
 ^ jU^ contain the factor sin 2 a, while the three previous 
 coefficients remain finite in the limit when a = 0. In the 
 biquadratic for \U, the last two coefficients are therefore 
 small in comparison with the first three, and approximate 
 
v LONGITUDINAL STABILITY 77 
 
 methods of solution may be used. In particular we 
 notice that in the expression S 3 6 ^) o aj 2 2UD 2 , the 
 first two terms contain sin 2 a as a factor, while ^T),, 2 con- 
 tains sin 4 a ; the latter may therefore be neglected in a 
 first approximation and the condition of stability then 
 reduces on division by 33 to 
 
 ( 2> - @ 8 >0 . .. ' .' . .. (50) 
 Now neglecting sin 2 a we have 
 
 m*S. 2 \ .... (516) 
 } . . . ; . (51c) 
 
 and using the first form of ( /</ 4 7 4 we obtain from (50) 
 the approximate primitive condition of stability 
 
 cW -\ 
 
 I + KSJ, \SJtn tan a - {(^ + S^G + WFS 2 \ > . (52) 
 
 Making use, however, of the equations of equilibrium 
 (47) we have 
 
 WPS t = KSfifU* sin a cos 
 
 tan a (1 - sin 2 a) 
 
 to a first approximation. Hence the terms underlined 
 cancel to this order of approximation, leaving 
 
 W 
 
 . . . (53) 
 
 whence if C= Wk 2 , so that k is the radius of gyration 
 about the axis of z, 
 
 ^ S^+S, _F_ 
 g > S 2 2 tan a 
 
 This is the condition ^of stability obtained by Lanchester 
 by an entirely independent method. As use has been 
 made of the conditions of equilibrium in cancelling the 
 parts underlined in (52) we ought to regard it as a 
 modified form (to this extent) of the conditions of 
 stability. 
 
78 SINGLE LIFTING SYSTEMS CHAP. 
 
 Another modified condition of stability can be obtained 
 by expressing U 2 in terms of W by means of the condition 
 of equilibrium, again neglecting the difference between 
 tana and sin a cos a. We thus obtain 
 
 This form is probably better than the previous one, as it 
 only involves quantities depending on the weight and 
 measurements of the aeroplane, in addition to the co- 
 efficient of resistance K. If the propeller-thrust, velocity 
 of propulsion U and angle of attack a are varied in a way 
 consistent with the conditions of equilibrium the stability 
 determined by (54a) is independent of their variations. 
 
 49. If it is desired to proceed to a higher degree of 
 approximation in which cos a is not taken to be equal to 
 unity, the best plan is to write, as in 34, S\ = Si cos 2 a 
 and fju = tan a in the equations of equilibrium and the 
 coefficients, and we thus obtain, 
 
 H = KS\UY W = KSfJPp .... (55) 
 
 (56a) 
 
 } . . . (566) 
 
 * + fcVJ . . (56c) 
 3) 2W 
 
 . . . (56d) 
 
 + V . (57) 
 
 ^ 2 ) + 2^ sin 2 a{^ 2 (? 2 + /; 
 
 W _ SfiJM sin 2 a 
 2 
 
 cos Sa^Sfj cos 2 a + ^ 2 ) + 2^ sin 2 a{^ 2 (? 2 + /; 2 ) - ^fc 2 cos 2 a} 
 
 sin 2 a) 
 
 and the condition for stability requires that this expres- 
 
v LONGITUDINAL STABILITY 79 
 
 sion shall be positive. Putting cos a= 1, sin a = 0, this 
 reduces to the condition previously obtained. 
 
 The foregoing expressions give the exact conditions of 
 stability whether the angle a be large or small, provided 
 that the resistance follows the sine law, and that the 
 shift of the centre of pressure is negligible when the angle 
 of attack is varied. In view, however, of these assump- 
 tions not being exactly fulfilled in actual practice it is 
 probably better to be satisfied with the approximate 
 solutions obtained by neglecting sin 2 a, leading to 
 
 W k*g 8, cos 2 a + S 2 
 
 KS.co^a^ 1 'I- ^ST~ 
 
 or the even simpler forms (54), (54a) obtained by substi- 
 tuting unity for cos 2 a in these. In the S\, //, notation, the 
 above conditions may be written 
 
 and these we shall regard as the standard forms of the 
 condition of stability as it will appear later on in our work, 
 that the " S\, p, notation " considerably simplifies the 
 algebra. 
 
 Separation of the long and short oscillations. 
 
 50. In the biquadratic equation for X, the last two 
 coefficients (49c, e 2 ) contain the factor sin 2 a, so that if a 
 is small these coefficients are small quantities of the second 
 order in comparison with the others. This fact enables 
 us to obtain approximate solutions of the biquadratic, and 
 to separate the different oscillations of the system. 
 
 Let us write the biquadratic for X 
 
 2IX 4 + 33X 3 + \ 2 + 2>"X + (T = 0, 
 
 the suffixes being dropped as they were only used to 
 distinguish the coefficients for symmetric and asymmetric 
 
80 LONG AND SHORT OSCILLATIONS CHAP. 
 
 oscillations, and the accents in $)" and (" being used to 
 show that these are small quantities of the second order. 
 
 If X is finite, then for a first approximation we may 
 neglect )" and (,", and the equation on dividing through 
 by X 2 becomes 
 
 21X 2 + 23X + ( = 
 
 The roots of this equation determine oscillations whose 
 times remain finite, however small be the angles of attack, 
 and we shall call these the short oscillations. If, on the 
 other hand, X is small, ^"x may be neglected in com- 
 parison with (!' and 2(x 4 , and 93x 3 in comparison with 
 (5x 2 . Hence to the lowest order of approximation we get 
 
 or 
 
 This solution is of the first order of small quantities, 
 and the values of X being imaginary the motion is 
 oscillatory. To find a closer approximation we substitute 
 the value of X 2 given above in 3Jx 3 (still neglecting 5lx 4 ) 
 and write the equation 
 
 /TC'/CO^ 
 
 + @" = . . . (60) 
 
 showing that to this order the real part of the values 
 of X is equal to 
 
 The small values of X thus determine sloiv or long 
 oscillations whose period approximates to 27r*/(G/6*) and 
 modulus of decay is equal to 
 
 and is positive if the approximate condition for stability is 
 satisfied, namely, GS)* S"SB< 0. 
 
v LONGITUDINAL STABILITY 81 
 
 To examine the character of the oscillations we must go 
 back to the equations of motion, which are 
 
 (w\ \ w 
 
 2S* sin 2 a + - ) u + S, sin a cos a . v rr ^ r = (61u) 
 
 KUg) KUK 
 
 / W\\ / W\ 
 
 2$! sin a cos a u + ( S x cos 2 a + ^ 2 + vjf~ ) v + ( ~ '-V + ~K~ } r ~ " W^ 1 ') 
 
 s 
 
 ^ 2 + 1 )r = 0(61r) 
 
 51. Short oscillations. If we neglect sm a altogether 
 the equations become 
 
 u - r = . . (62*0 
 
 and these are applicable to the short oscillations. The 
 equation for X is in this case obtained by eliminating 
 v and r from the last two equations, and this gives 
 
 8 ^ ~ ' (63) 
 
 This equation is identical with the equation 9t\ 2 -f 
 = if we take the values of 5( o , 33 o , & given 
 in 48, and it is thus seen that the motions here considered 
 are really the short oscillations of the system. 
 
 The ratio v : r is given by either of these equations, say 
 the second, which gives 
 
 r ~ KUgSJ 
 
 or, if preferred, 
 
 v - Ir _ WV\ 
 
 ~7 -- KUgSj, . -: l ' ' (64) 
 
 where v - Ir is obviously the velocity of the tail plane 
 perpendicular to the axis of x. This equation is of course 
 merely the equation of moments, being equivalent to 
 
 - = KUS 2 (v - lr)l. 
 
 The velocity u is found in terms of r from the equation 
 
 G 
 
82 SHORT OSCILLATIONS CHAP. 
 
 *?u/g r = 0, which is derived from the equation of 
 motion 
 
 W = W sin 6 where = T . . . (65) 
 
 gdt dt 
 
 the assumption being in this case that the effects of the air 
 pressures on the planes may be neglected in considering 
 the motion along the line of flight. 
 
 Since, however, u does not enter into the two equations 
 from which the quadratic for A. is found, it follows that the 
 short oscillations are independent of the variations of 
 velocity in the line of flight and are determined entirely 
 by the normal and rotational motions, v and r. 
 
 In the next place we observe that A, only enters into the 
 equation (63) in the form \/U, and that the values of \/U 
 given by this equation are independent of the velocity U. 
 The rates of oscillation are thus proportional to the 
 velocity U. If, instead of taking the time as independent 
 variable, we take the distance flown, s, where s= Ut, we 
 see that the actual path described by the machine ivhen 
 oscillating is independent of U, the velocity of flight, 1 
 so that if the velocity of flight is changed the machine 
 will still perform the same number of oscillations, and 
 these oscillations will subside to the same fraction of their 
 initial amplitude when the machine travels over the same 
 distance. 
 
 The condition of stability requires that in the equation 
 for \ or \JU, all the coefficients shall be positive and 
 therefore 
 
 - S 2 ) + 1 2 S 2 > . . . . (66a) 
 
 W 
 
 4- S 2 l^> .... (666) 
 
 The first expression is essentially positive. The second 
 will be positive if / is positive, or if I is negative and 
 numerically greater than W[KS^g. 
 
 In the former case the auxiliary plane is behind the 
 
 1 Assuming, of course, that u is neglected. 
 
v LONGITUDINAL STABILITY 83 
 
 main planes. The corresponding condition in the case of 
 a doubly lifting machine is that the front plane must have 
 a larger angle of attack than the rear one, as will be 
 shown later. 
 
 The negative solution is interesting in connection with 
 aeroplanes having the rudder planes in front (e.g. the 
 Wright machine). The conclusion is that by placing the 
 rudder either behind or sufficiently far forward in 
 front of the main planes it is possible to secure stability 
 so far as the short oscillations are concerned. 
 
 The motion will be either really oscillatory or subsiding 
 in character, according to whether the values of x/ U are 
 
 Line of steady flight 
 
 FIG. 24. 
 
 complex or real; the conditions being 33 2 491 (S negative 
 or positive, respectively, that is 
 
 or > 
 
 This condition may be transformed into 
 
 or > . (67) 
 
 and it will thus be seen that if I is negative, or if I is 
 positive and less than a certain limit, the disturbance is 
 subsiding in character instead of being periodic. 
 
 Taking now, as an example, the case where X is real and 
 negative so that I is positive, equation (64) shows that 
 v Ir and r are of opposite signs, and hence the angular 
 displacement of the aeroplane is in the opposite direction 
 to the displacement of the tail plane. The motion of the 
 tail plane is represented diagrammatically in Fig. 24 for 
 
 G 2 
 
84 LONG OSCILLATIONS CHAP. 
 
 two different displacements, one above and the other 
 below the line of flight. Other cases may be similarly 
 discussed. 
 
 Long Oscillations. 
 
 52. For the second solution of the biquadratic, the 
 values of \ are to the lowest sufficient order of 
 approximation 
 
 the imaginary part being of the first order in a and the 
 real part, determining the modulus of decay, being of the 
 second order. To obtain a general idea of the character 
 of the oscillations we shall neglect the latter part, and we 
 thus assume the oscillations to be periodic. 
 
 Writing S\ for fii cos 2 a and p for tan a, equations (61) 
 give 
 
 = Q . . (69.) 
 - S 2 lv + ^ 2 + r = 0. . (69r) 
 
 With fju small, the only way of obtaining a solution 
 differing essentially from that obtained for the short 
 oscillations is by assuming that u is large compared with 
 v and r so that u sin a or u //, is comparable with v and r. 
 Moreover, in view of the fact that the terms containing X, 
 in the second and third equations (61) were previously 
 comparable with the others, it follows that for the long 
 oscillations these terms may be neglected for a first 
 approximation. 
 
 If we neglect X in the third equation we obtain 
 
 - v + Ir = ..... (70) 
 
 or the normal velocity of the tail is zero. More generally 
 the equation, which may be written 
 
v LONGITUDINAL STABILITY 85 
 
 shows that the normal velocity of the tail is in every case 
 small. 
 
 Again we may eliminate S\ and S 2 from the three 
 equations by multiplying the first by 1//-1, the second 
 by 1, and the third by I /I and adding, and we get on 
 removing superfluous factors 
 
 and this, since it does not involve S t and S 2 , must be 
 independent of the resistances and must represent the 
 equation of moments about the metacentre. Observing 
 that it is assumed large compared with v, the terms of 
 lowest order are those underlined, leading to 
 
 u \* = 9 r . . . . :. . (7ia) 
 
 On the other hand, if we neglect X in the second 
 and third equations, we have, on eliminating $ 2 , 
 
 W 
 
 2S\nu + S\v + -g-r = - . . .. - . (72) 
 
 or substituting v = IT again, we obtain 
 
 '* + ^) r = - W 
 From (73) and (71a): 
 
 X2 - - ~*W- ... . (74) 
 
 which agrees with the first approximation 'for the long 
 oscillations given by 
 
 o _ , 
 (S ~ * g{WK*S\SJ? + 
 
 taking the values of the coefficients in 49, and neglecting 
 terms containing p? in the value of (5 . 
 
 53. To trace the undulating path described by the 
 centre of gravity, we suppose that the co-ordinates of 
 this point at a time t are f + Ut and 97, the axis of x 
 
86 LONG OSCILLATIONS CHAP. 
 
 coinciding with the direction of steady motion. Then 
 if 6 be the angular co-ordinate 
 
 ^| + U = (U + u) costf - v sin 6 . . . (75a) 
 = U + u to a first approximation 
 
 -^ = ( U -\- w) sin B + v cos 
 
 = UB + u to a first approximation . (756) 
 
 and since f , rj, 6 all vary as e M , 
 
 or ? =. 
 
 Assuming the approximate expressions r = X 2 u/</, 
 we have finally 
 
 or writing X = \'i where i = *J 1 , 
 
 where x' is real. 
 
 Put f = ^4(cosX^ + i sinX^) and equate the real parts, 
 then we get the solution 
 
 = A cos \'t 1 
 
 17 = - \U\' sin X'* + l\' 2 cos X'*} J * 
 
 If we write x = f + C7^, i/ = ?;, then a, y will be the 
 co-ordinates of the centre of gravity referred to a fixed 
 origin, so that the equations of the actual path will be 
 given by 
 
 x = Ut + A cos X' 
 
 y = - {U\' sin \'t + l\' 2 cos \'t} I 
 
 and to trace the path it is only necessary to add the 
 ordinates of the two curves determined, one by the 
 equations 
 
 x = Ut + A cos \'t y = - - X' sin X' . . (78) 
 
LONGITUDINAL STABILITY 
 
 and the other by the equations 
 
 x = Ut + A cos \'t 
 
 cos \'t 
 
 (79) 
 
 of which the first represents waves having the crests more 
 pointed than the troughs (upper curve, Fig. 26), while 
 the second represents waves the descending parts of which 
 are steeper than the ascending ones (lower curve). The 
 actual path, which is a combination of the two, should be 
 familiar to anyone who has thrown paper gliders. 
 
 In regard to the assumption r = \ 2 u/g, in connection 
 with the fact that we have retained terms containing A/ 2 in 
 the expression for 17, it should be noticed that the terms 
 
 7T 3 / 2 1T 27T 
 
 FIG. 25. 
 
 neglected in (7 la) would only affect the final result by 
 quantities of a higher order than the second. 
 
 The path of the centre of pressure of the tail plane is 
 very approximately the first curve of Fig. 25, the equation 
 of motion being 
 
 Xn = ue + (o - ir) 
 
 where v Ir is a negligible quantity. 
 
 Effect of inclination of propeller axis. 
 
 54. If the axis of the propeller is not along the direction 
 of motion, the primitive conditions of stability will, 
 
88 INCLINATION OF FLIGHT PATH CHAP. 
 
 of course, be unchanged, but as the conditions of equi- 
 librium now give 
 
 W + H sin r) = KS\U*ii .... (80) 
 
 the terms previously cancelled in modifying the conditions 
 of stability will no longer cancel. The approximate con- 
 dition of stability corresponding to (54) 
 
 -> -^- -cota . . (81) 
 
 while corresponding to (54a) we shall have 
 
 g " W + H sin r/ 
 
 According to (8 la), if t] is positive so that the propeller 
 thrust tends to depress the machine, stability is increased, 
 and the reverse is the case if the propeller thrust tends to 
 elevate the machine. 
 
 Effect of inclination of the flight path. 
 
 55. That the stability, both symmetric and asymmetric, 
 of an aeroplane depends very largely on the angle which 
 the direction of flight makes with the horizon was first 
 pointed out to me by Mr. E. H. Harper, to whom the 
 following investigations are largely due. In this section 
 and the next we assume the propeller thrust to be central 
 and acting along the direction of motion, so that the tail 
 plane is neutral and the shift of the centre of pressure 
 is neglected. 
 
 The first three coefficients of the biquadratic are as given 
 in (49a e), but we have 
 
 sin 6 . . 
 
 2W 
 
 JJT K 2 S l S 9 l{sin a cos o cos 6 + sin' 2 a sin 6} . (82e) 
 
 In virtue of the equation of equilibrium 
 
 W cos 6 = KS l U* sin a cos a . . . (83) 
 
LONGITUDINAL STABILITY 89 
 
 these reduce to 
 
 25 W 
 
 sin 2 a -f sin a cos a tan 0) 
 
 t/ 
 
 W 
 
 sin a cos a (2 tan a + tan 0) . 
 
 = KtSiSJ. sin a cos (B - a) . . . . (840) 
 
 Hence $) vanishes when the machine is ascending at 
 an angle given by tan 6= 2 tan a. In this case the 
 discriminant^ becomes equal to d 53 2 , and is negative, 
 so that the flight is distinctly unstable. The limit for the 
 angle of elevation of the flight path at which stability 
 ceases is thus less than the value given by tan 6 = 2 tan a, 
 showing that, even other things being the same, a machine 
 rapidly loses its stability when it begins to rise in the 
 air. 
 
 It should be noticed that o vanishes when 6 a = 90, 
 but this condition is not likely to occur in practice. 
 
 On the other hand, a positive value of is favourable 
 for stability, so that if an aeroplane is symmetrically 
 stable ivhen flying horizontally it will be more stable when 
 descending. 
 
 An important consequence is that it is not safe to draw 
 inferences regarding the stability of motor-driven 
 machines from experiments with gliders. 
 
 56. For purposes of approximation and simplification 
 we shall suppose and a both small so that cos (0 a) may 
 be taken to be equal to unity ; in such cases the value of 
 5) alone is affected by the inclination of the flight path. 
 Corresponding to the approximate condition (52) we 
 now have 
 
 jy + Jf^SV J 2 I7 2 Z(tan a + i tan 6) - {(S\ + S 2 )C + WPSj > (85) 
 
 Making use of the conditions of equilibrium, we may 
 write the modified condition in the form corresponding 
 to (53) 
 
 C7_ 2 (S\ + S 2 )k* - S 2 l* tan 6/2 tan a 
 + | tan ff) 
 
90 SINGLE LIFTING SYSTEMS CHAP. 
 
 this being the modification required to adapt Lanchester's 
 condition to flight at an inclination 6 to the horizon. For 
 the particular case where the machine is descending as 
 a glider, so that = a, this condition gives 
 
 w 2 (S\ + S 2 )k* 
 
 g 3 S 2 l tan a 
 
 so that the minimum velocity-height ( U 2 /2y) required for 
 stability is less than two-thirds that required by Lan- 
 chester's condition. 
 
 On the other hand, if the machine is ascending at an 
 angle equal to the angle of attack, so that = - a, the 
 condition becomes 
 
 , , 
 ~JT SJ tan a 
 
 requiring a value of U' 2 /g, more than double that required 
 for stability in horizontal flight. 
 
 From the fact that %,, 33 , 6 are independent of 0, and 
 d only contains 6 in the form cos (9 a), which is nearly 
 equal to unity, it follows that to our assumed degree of 
 approximation, the short oscillations are independent 
 of the inclination of the flight path, as is also the period 
 of the long oscillations ; on the contrary, thv modulus of 
 decay of the lat ter depends greatly on the inclination. 
 
 Effect (a) of head resistances and (b) of 
 variable propeller thrust. 
 
 * # * In what follows we shall omit the suffixes in 2t 23 ( @ , the 
 context showing that we are dealing with longitudinal stability. 
 
 57. (a) Under this heading we include all resistances 
 not covered by the previous investigation, namely the 
 resistances encountered by the body of the machine and 
 engine and the aeronaut, also tangential forces due to skin 
 friction and so forth. 
 
 The most general supposition consistent with the 
 
v LONGITUDINAL STABILITY 91 
 
 assumption that the pressure of the air on a moving body 
 varies as the square of the relative velocity, consists in 
 admitting that the effect of these resistances is represented 
 by additional terms in X , Y , N proportional to 7 2 , and 
 in the nine coefficients X u . . .N r proportional to U. We 
 will for the present use accented letters X f . . . N f r to 
 represent these added terms, which, moreover, must 
 satisfy the relations X' U = 2X' /U, F tt = 2F /^, and 
 
 Now in the first place if Y f be different from zero, this 
 represents an additional lift due to the air resistances, 
 not accounted for by the action of the aeroplanes them- 
 selves ; similarly, N' represents a couple due to this cause. 
 It will, I think, be generally admitted that these quanti- 
 ties may be assumed to vanish in an ordinary machine, 
 leaving only the resistance X r to be taken into account. 
 At the same time, the following method shows that in 
 normal circumstances they would have little effect on the 
 stability. 
 
 In the first place, it is easy to show (cf. Lanchester, 
 vol. I., chap. vii. 164) that for an aeroplane of given 
 weight the propeller thrust is a minimum when the head 
 resistance is equal to the drift, while the horse-power is a 
 minimum when the head resistance is one-third of the 
 drift. In either case the head resistance X' is comparable 
 ivith the drift KSxU 2 sin 2 a, and is not large compared 
 with it. If it be necessary to consider flights performed 
 at speeds largely in excess of the most economical speed, 
 in which the head resistance is large compared with the 
 drift, a fresh examination of the conditions of stability 
 will be required. Except, however, at these excessive 
 speeds, the head resistance X f will not be large compared 
 with KS^U 2 sin 2 a, arid the part X' u will not be large 
 compared with the portion ( < 2KS l U sin 2 a) of X u due to 
 the planes. In fact X' u will be a small quantity of order 
 a (at least). 
 
92 SINGLE LIFTING SYSTEMS CHAP. 
 
 Of the remaining coefficients of the group (X\ Y',N f } u 
 those which do not vanish will not in general be large 
 compared with X u , and, therefore, 1 they will also be at 
 least of order a 2 . 
 
 Bearing this fact in mind, and neglecting quantities of 
 higher order where these are added to quantities of lower 
 order, we obtain the approximate results, 
 
 = W*C ............ (89a) 
 
 , + S 2 )C + WPSJ ....... (896) 
 
 j + } as in (51c) ...... (89c) 
 
 W \ 
 cos 2 a Sf + JX8J) 
 
 2W W 2 
 
 + - KiSjSJ sin 2 a . + ^ KSJ sin ^ .... (89d) 
 
 ! sin a cos a cos 6 + fs, sin 2 a + sin ^ (89e) 
 
 - 
 
 the terms involving the other coefficients JSr' M . . . Y' u . N f r 
 being of higher order than those retained and therefore 
 negligible. It is thus seen that the coefficient X' u alone 
 enters into the lowest approximations, and thus has an 
 important effect on the stability. 
 
 Now let the ratio of the head resistance X f to the 
 drift KSiU^am 2 a, be denoted by u, then we shall have 
 X f u = v. 2,KS l U sin 2 a, and by making use of the condition 
 of equilibrium (83) the expression for $) simplifies down to 
 
 J~L = ^K'S^l sin a cos a J2(l + v) tan a + tan 6 + 2 -^jL cos 0} (90) 
 
 while (, is to the lowest approximation given by 
 
 (g 2W 
 fU* = Wg S?e W sin a 
 
 (as in horizontal flight). 
 
 It will thus be seen that the main effect of head resist- 
 ance is to increase the coefficient ^), thus increasing the 
 
 1 Any structure which gives rise to resistances not satisfying this condition 
 would practically be of the nature of a "plane" and should be included 
 among these. 
 
v LONGITUDINAL STABILITY 93 
 
 stability by increasing the modulus of decay of the long 
 oscillation. This result, which is best obtained in the first 
 instance by algebra as is done above, is entirely in accord- 
 ance with what we should expect from general con- 
 siderations regarding the nature of the long and short 
 oscillations as discussed in 50 53. 
 
 Head resistance thus enables the machine to rise at a 
 steeper angle to the horizon than would be possible if 
 such resistance did not exist. For example, the effects of 
 head resistance and inclination of flight path will cancel 
 one another if 
 
 2v tan a + tan 6 + j cos 6 = . . . (91) 
 
 or putting cos 6 = 1 , 
 
 tan 6 = - 2v(tan a + ^) =-- - 2,(tan a + 2 . veloc / ty height ) (91a) 
 
 If the machine is ascending at an angle given by this 
 formula, it will be as stable as it would when moving 
 horizontally in the absence of head resistance, and 
 generally the increased stability due to head resistance 
 enables the tangent of the inclination of the flight path 
 to be increased by the amount approximately given above. 
 
 (b) In all the previous work the propeller thrust has 
 been assumed constant, but if either the horse-power or 
 the rate of revolution of the propeller is maintained 
 constant the thrust will decrease when the velocity 
 increases. In either case the effect of these variations 
 can be represented by the addition of a positive term to 
 X w and will be precisely similar to that of head resistance. 
 It will be expressible in the form of an increase in the 
 value of v defined above, and will increase the elevation of 
 the flight path for given stability. 
 
 The effect of such variations when the propeller thrust 
 does not pass through the centre of gravity can be 
 investigated if the necessity arises, but the formulae will 
 necessarily become somewhat complicated. 
 
94 SINGLE LIFTING SYSTEMS CH. v 
 
 58. Mr. E. H. Harper has expressed the condition of 
 symmetric stability in terms of the forces, taking account 
 of inclination of the flight path and also of head resistance. 
 This condition may, of course, be easily modified to take 
 account of variable propeller thrust, though in the follow- 
 ing formula the thrust P is assumed constant : 
 
 Let 
 
 B = head resistance 
 
 E 1 = KU*S! sin 2 a = component of drift 
 
 P propeller thrust 
 
 then by the conditions of equilibrium we have 
 
 W sin e = R + R 1 - P . .' . . . . (92) 
 
 and Mr. Harper finds that the approximate condition of 
 stability 63) (23 > reduces to 
 
 For the case of horizontal flight without head resistance 
 M = Q, R l = P, and the condition reduces at once to (54) 
 or (54a). 
 
CHAPTER VI 
 
 LONGITUDINAL STABILITY OF DOUBLE-LIFTING SYSTEMS. 
 EXTENSION OF RESULTS TO SYSTEMS OTHER THAN 
 NARROW PLANES MOVING AT SMALL ANGLES 
 
 The principle of equivalent systems. 
 
 59. We have hitherto only considered cases where the 
 weight of the machine is entirely supported by the front 
 plane or planes, the rear plane acting only as a tail, and 
 being placed in a neutral direction. 
 
 We shall now show how any results proved for such a 
 
 S 2 
 
 system can be extended to a system of two supporting 
 planes placed tandem, and this we call a double lifting 
 system. We assume the planes to be so near the axis of x 
 that the principle of independence of height holds good ; at 
 the same time that principle allows the front and rear 
 planes to be " stepped," or placed at a somewhat different 
 level, so that the results may be less modified by the 
 " wash " which the front planes may produce on the 
 back ones. 
 
 Let S lt S 2 be the areas of the front and rear planes, a 1} 
 
 95 
 
DOUBLE LIFTING SYSTEMS 
 
 CHAP. 
 
 Photo.] [The Sport and General Illustrations Co. 
 
 FIG. II. THE LATE M. E. LEFEBVRE FLYING AT RHEIMS. 
 
 The earlier type of Wright Biplane with auxiliary planes only in front of the 
 main planes. Partial longitudinal stability dependent on the short oscilla- 
 tions is obtainable by placing the auxiliary planes sufficiently in advance of 
 the main planes ( 51). To secure longitudinal stability for the long oscilla- 
 tions as well, the front planes must be tilted up at a greater inclination than 
 those at the rear ; the aeroplane thus becomes a "double lifting system," as 
 discussed in 60. 
 
vi LONGITUDINAL STABILITY 97 
 
 2 their angles of attack, x l} x. 2 the co-ordinates of their 
 centres of pressure. Using the " S', /* " notation, 
 
 S\ = Si cos 2 a, $' 2 = $ 2 cos 2 a 2 . ^ = tan a lt /x 2 = tan a 2 , 
 
 we see that the resistance derivatives can be written in 
 the form 
 
 XJKU = 22S> 2 X v jKU = 
 
 = ss> 
 
 NJKU* 
 
 (94) 
 
 These involve the six sums, 
 
 i, sflf'/i*, sfif*, 
 
 the values of which depend on the values of the six 
 quantities, S' l9 ' 2 , ^ ^ x^ x 2 . 
 
 The six sums are, however, not independent in virtue of 
 the determinantal relation 
 
 which has been proved for systems with two planes in 
 26. 
 
 If follows, therefore, that if any two systems have^i'e of 
 the six sums equal, the remaining sums will be equal, and 
 the conditions of equilibrium and stability of the two 
 systems will be identical. Such systems will be described 
 as equivalent. 
 
 A tandem system of two planes can thus be replaced by 
 an equivalent system satisfying one additional arbitrary 
 condition. If we assume as this condition ^ = 0, the 
 equivalent system will have its rear plane in the neutral 
 position, and the connections between the two may be 
 given in the following table : 
 
 H 
 
9 8 
 
 DOUBLE LIFTING SYSTEMS 
 
 CHAP 
 
 Double Lifting System 
 
 Coefficients 
 
 Equivalent 
 System 
 
 (i) 
 
 S 1 sin 2 oj + $ 2 sin 2 02 
 
 ^ and 2^ 
 
 S l sin 2 a 
 
 (nj 
 
 &! sin Oj Cos ttj + j sin a 2 cos o 2 
 
 F K JC 
 
 ATT 2 ' 2^C7' CT' 
 
 >S'j sin a cos a 
 
 (iii) 8 
 
 !#! sin a x cos Oj + /S^j sin a 2 cos Og 
 
 JVo ^ M X r 
 
 (Sfj^! sin a cos a 
 
 AC/*-' 2/TC7' KU 
 
 (iv) 
 
 /8j cos 2 a : + >S 2 cos 2 03 
 
 KU 
 
 Sfj cos 2 a + S 2 
 
 (v) 
 
 $!#! COS 2 a x + ^^ COS 2 O 2 
 
 N v Y r 
 KU' KU 
 
 . S^ cos 2 a + $2a: 2 
 
 (vi) 
 
 Aja?! 2 cos 2 a x + 2 a: 2 2 cos 2 a 2 
 
 N r 
 KU 
 
 <S' 1 a; 1 2 cos 2 a -i- S^x^ 
 
 -. : ,, (95) 
 
 "We have supposed for the sake of greater generality 
 that the propeller thrust does not necessarily pass through 
 the centre of gravity. If it does, N = 0, x 1 = 0, # 2 = / 
 where / is the distance of the tail plane behind the centre 
 of pressure of the main plane, and each member of (iii) 
 vanishes. 
 
 It will be noticed, by adding (i) and (iv), that the 
 total superficial area /Si + S 2 of the surfaces is the same in 
 the two equivalent systems. In the $', p notation the 
 
 area is ^(1+^ 2 )- 
 
 The principle of equivalence thus established can now 
 
 be stated as follows : 
 
 To every double lifting system of two narrow planes 
 there corresponds an equivalent single lifting system 
 having the same conditions of equilibrium and stability, 
 and the same total superficial area of its planes. 
 
 The conditions of stability of a double lifting system 
 are usually considerably simplified by replacing it by the 
 equivalent single lifting system. Without using this 
 artifice the algebra is very long and cumbersome. 
 
 Example. Let ^ = 20, a 2 =10, fii = 2(), S 2 = 10 units 
 of area, and let x l x 2 , the distance between the centres of 
 pressure, be equal to 10 units of length. Let the propeller 
 
vi LONGITUDINAL STABILITY 99 
 
 thrust be central ; then the equation of moments 
 
 S. 2 a. 2 x 2 = gives 4iX 1 + x 2 = 0, whence x l = 2, x 2 = 8 in the 
 
 original system. 
 
 For the equivalent system we have x 1 = 0, # 2 = ~^ ^ 
 where Z is the distance of the tail plane, and the remaining 
 equations give, remembering that the angles are small, 
 
 Sp 2 = 20. 20 2 + 10. 10 3 = 9000 (a) 
 
 S ia = 20.20 + 10.10 . = 500 (6) 
 
 S l + S 2 = 20 +10 = 30 (c) 
 
 - 8J = 20.2 - 10.8 = - 40 (d) 
 
 S 2 l* = 20. 2 2 + 10. 8 2 = 720 (e) 
 
 From (a) and (b) we find a =18, /Sj = 27-J, hence by 
 (c) Sz = 2| = - 2 9-, and by substitution in (c?) 7= 18, which 
 agrees with the result obtained by dividing (e) by (d). 
 
 Thus, in the equivalent system, the areas of the sup- 
 porting and tail planes are 27^ and 2 f units ; the 
 inclination of the former is 18, and the distance between 
 the planes is 1 8 units of length. 
 
 Invariants of equivalent systems. 
 
 60. In deducing the conditions of stability of a tandem 
 or double lifting system from those of its equivalent 
 single system a great deal of time may still be wasted 
 in algebraical complications, and this was done in 
 Mr. Harper's and my early solutions. A considerable 
 simplification may be made by first making a list of the 
 invariants of equivalent systems as follows : 
 
 (96) 
 
 then c,/*and R are invariants. Of these d represents the 
 distance behind the origin of the centre of gravity of 
 S\ and $' 2 , f the distance of the centre of pressure of 
 the planes themselves, R the radius of gyration of S'i,S' 2 
 
 H 2 
 
ioo DOUBLE LIFTING SYSTEMS CHAP. 
 
 about the origin. The inclinations a l9 a 2 being small, d 
 and R may to a sufficient degree of approximation be 
 assumed to refer to the areas S lt S. 2 instead of S\ and S' 2 . 
 (2) The minors of the determinant in A are also 
 invariants, and these are 
 
 Minor of 
 a?! - ac 2 ) 
 
 ^20*1*2 - 
 
 (97) 
 
 so that the ratios /*! - /* 2 : #1 - # 2 - A*i#2 A^i are also 
 invariants. It will be noticed that x l x 2 is the distance 
 between the front and rear planes, which may be denoted 
 in every case by Z, and that S f i$ f 2(1*1 fa)* = S^ sin 2 (a x a 2 ) 
 where a a 2 is the angle between the directions of the 
 front and rear planes. This angle we shall denote by a, 
 and we shall further write ^ ^ = v so that v = a approxi- 
 mately, the angles being small. 
 
 61. To extend the conditions of stability of a single 
 lifting system to a double lifting one, the other conditions 
 of the problem being the same, it is only necessary to 
 express the inequalities representing them in terms of 
 invariants. 
 
 Taking the " simplest case " of 48, where the direction 
 of flight is horizontal, the propeller thrust is also in the 
 same direction and passes through the centre of gravity, 
 we may write (58), 
 
 l/fjbis an invariant and =l/v or (x l x 2 )/(fjL l fjL 2 ) for the 
 compound system. 
 
 S 2 1 2 /(S', + S 2 ) is also the invariant 2S'x*/2S' or IP as 
 defined above, and the condition becomes 
 
 - > F r 
 
 g ' k & r 
 
VI 
 
 LONGITUDINAL STABILftt ' '' J ib' 
 
 Taking the other form, we may write it 
 
 The invariants here are 
 
 S\S. 2 l 2 = S' 1 S' 2 (x l - xjf or again 
 S\ + S 2 = S\ + flT, 
 
 (S\ 
 
 and we may write the condition 
 
 or in a number of different other ways ; for example, we 
 may write 
 
 K ^ 
 
 Kg > > 
 
 which reduces by invariants to 
 
 W s asr 
 
 ' or ' ' (100) 
 
 It is further to be pointed out that the condition vjl 
 positive, which is itself in invariant form, shows that the 
 front plane must have the larger angle of attack. It is 
 easy to waste time in turning and twisting the conditions 
 of stability into various other forms, and much of this 
 danger is saved by a clear understanding of the 
 invariants. 
 
 The propeller thrust does not pass through the 
 centre of gravity. 
 
 62. If the propeller thrust does not pass through the 
 centre of gravity it is obvious by moments or other- 
 wise that if the tail plane is in a neutral direction the 
 centre of pressure of the front plane cannot coincide with 
 the centre of gravity. Taking, however, a double lifting 
 system, supposed descending at an angle with the 
 horizon and assuming H the propeller thrust, h its distance 
 
DOUBLE LIFTING SYSTEMS CHAP. 
 
 from the origin (Fig. 27), 77 its inclination to the line 
 of flight, the conditions of equilibrium become, with 
 the usual notation 
 
 = W sin 6 4- H cos r, 
 = W cos 6 + Hs'mr) 
 = - Hh ...... (101) 
 
 In practice the angles 6 and 77 will be small, and from 
 the first equation it is evident that H is then small com- 
 pared with W, and still more is H sin 77 compared with 
 W cos 6. The term underlined will therefore be neglected 
 in what follows, and we will further assume that cos 77 may 
 
 FIG. 27. 
 
 be put equal to unity. This is equivalent to assuming 
 either that 17=0, or that even if 77 is not zero, the 
 corrections discussed in 54 are negligible. 
 
 The third equation shows, as stated above, that the 
 system cannot be replaced by an equivalent system with 
 both x l = and /* 2 = 0. All we can do to simplify the 
 algebra is to assume an equivalent system where either 
 Xi = or fj. 2 = 0. Which is the better choice ? Algebraically 
 the simplest plan would be to assume the rear plane to be 
 neutral or /* 2 = 0, the third condition then giving 
 KS'JJ^x^ = Hh. For the purpose of applications 
 the reverse assumption x 1 = seems on the whole most 
 desirable, giving KS' 2 U 2 fi 2 x 2 = Hh. For if H is variable, 
 or, indeed, H becomes zero through stopping the engines, 
 the conditions of equilibrium are modified, and it is clear 
 that the easiest way of adjusting equilibrium is by turning 
 the tail plane so as to alter /u 2 . On the other hand, if the 
 adjustment were made by varying x 1} this would involve 
 shifting the centre of gravity of the machine, thereby 
 
vi LONGITUDINAL STABILITY 103 
 
 probably affecting the moments of inertia, and certainly 
 effecting changes in the equivalent system that would be 
 less easy to take into proper account. 
 
 Assuming, therefore, for the equivalent system . x l = 
 and x 2 = I, the determinantal equation in X becomes 
 
 W cos 
 
 = . . (102) 
 
 The coefficients of the biquadratic are given by 
 
 = fc 2 - . : . - : . > . - '-' . (103a) 
 
 (1036) 
 
 S' 2 K~ 2/i 2 cos + sin 0) . . .. ., . (103^) 
 
 ^ (cos ' + ^ sin ^ ' ' (103ei) 
 
 where the terms underlined are small (being of order rf if 
 is comparable with /*) in comparison with those retained, 
 and they may therefore be neglected. 
 
 We have, further, by the approximate conditions of 
 equilibrium (101) 
 
 w cos e = 
 whence 
 
 3) / W \s 2W 
 
 W\KUg) = "^ " ^' ~ + " * tan 
 
io 4 DOUBLE LIFTING SYSTEMS CHAP. 
 
 When S is expressed in this form, it will be seen that 
 $Vi + S' 2 [JL 2 occurs as a factor, not only of ($, but also of the 
 whole of <) except the last term, which vanishes when 
 fi 2 = 0, as occurs in the case of h = 0. Writing /* x /* 2 "> 
 we now find that the condition 63) (93 > 0, when 
 reduced to its simplest form, becomes 
 
 w w fcan 6 
 
 >0 . (104) 
 
 When we transform this inequality by means of 
 invariants to cover the case of a system where x l is 
 different from zero, we may use the conditions of 
 equilibrium, 
 
 Til. T-T}l 
 
 and the invariants of 60, including the notation, 
 
 S = S i + S 2 , 
 
 and we may express the result in a variety of forms 
 such as 
 
 ^ Sd - k*S\S' 2 -jp + ( SFfif + fy Sd J^{~2~~ W cos ^^ ^/' 7 
 
 >0 . (105) 
 
 63. Mr. Harper has verified this condition by an 
 independent method, starting with a system in which 
 the tail is neutral, but the front plane has its centre of 
 pressure not coinciding with the centre of gravity. In 
 this case, ^ 2 = 0, but x 1 is different from ; his 
 condition is 
 w 
 
 ^(S'fx* - ^V) - S\8' 2 (x, - x 2 }(S\ 4- S'JW + Xl x 2 ) 
 + ^Vi^i + ^"jj^{JSr^ + S' 2 x 2 ) - S\S 2 ( Xl - x^] >0 (106) 
 
 which reduces to the above as he has verified. 
 
vi LONGITUDINAL STABILITY 105 
 
 He has also obtained the following form for this case : 
 
 dor) 
 
 where, as in 60, /is the distance of the centre of pressure 
 behind the centre of gravity, S\ is equal to the invariant 
 V 2 , and 
 
 + S' 2 IP + f* - 2fd 
 
 It has been necessary also to work through a number of 
 other formulae which are not of sufficient interest to justify 
 their publication at the present time. 
 
 As H is small compared with W, it easily follows that 
 unless h is large compared with I the correction for non- 
 central propeller thrust will be small. In an actual 
 flying machine h would certainly be small in comparison 
 with I. 
 
 The above formulae seem to suggest that making h 
 positive, or lowering the propeller thrust, tends to decrease 
 the stability, but since any change of this kind also affects 
 the conditions of equilibrium, the question is not a definite 
 one until it is clearly specified what other changes in the 
 system are effected in order to balance the moment of the 
 thrust. If we suppose the balance to be restored by 
 moving the centre of gravity forward, or, what is the 
 same thing, shifting the planes both backwards through 
 a small distance, we may show from (106) that in the case 
 of horizontal flight, stability is increased by lowering the 
 propeller axis avid decreased by raising it if 
 
 \ 2 /ino , 
 
 ) . (109) 
 
 but the corresponding converse when (109) is not satisfied 
 is less simple. In this case, lowering the propeller axis 
 
io6 BROAD PLANES CHAP. 
 
 decreases the stability if this stability is small and 
 increases it if already considerable. 
 
 A case of more frequent occurrence in flying machines 
 is that in which the propeller axis and the planes are 
 atout the same level and the centre of gravity is below 
 them. This is equivalent to raising the propeller axis 
 and the planes simultaneously above the centre of gravity, 
 so that the moments of the drift and propeller thrust 
 continue to balance each other. It will be seen that in 
 this case the ''principle of independence of height" musb 
 no longer be assumed, the corrections to be applied to 
 the formulae depending essentially on the small deviations 
 from that property. 
 
 Broad Planes. 
 Effect of shifting of centre of pressure. 
 
 64. We shall now suppose that when the angle of 
 attack a is increased by da, the co-ordinate of the centre of 
 pressure of a surface S increases by a$\a)da as in 23 ; 
 this will in general be negative ; thus taking Joessel 
 and Avanzini's formula, the distance of the centre of 
 pressure from the centre of figure = 0*6a(l sin a) 
 = 0*3(1 -sin a) x (whole breadth 2a), and thus 
 
 a<j)'(a) = -0-6 a cos a . 7 . . (110) 
 
 It will be convenient to write 
 
 a(f)'(a) = e cos a . . . . (HI) 
 
 and e will then represent, according to this formula, 
 the distance of the centre of pressure in front of the 
 centre of area at " grazing incidence " when the angle 
 of attack just vanishes. If any other formula for the 
 centre of pressure be assumed, we need only assume e 
 to be a function of a defined by a$'(a) = e cos a. We 
 
vi LONGITUDINAL STABILITY 107 
 
 do not in this section take account of the " rotary co- 
 efficients " of 23. 
 
 The effect of the shifting is to alter the coefficient 
 N v to KU2&(x-ep) and N f to KU2S'(x-e^x. 
 
 In this section we shall assume that the propeller thrust 
 acts along the direction of steady flight and passes through 
 the centre of gravity, i.e. we neglect the corrections of 
 54, 62. 
 
 Now in a single lifting system with neutral tail x 
 vanishes for the front plane and p for the rear plane, so that 
 ^S'epx = 0, that is, the part added to N r vanishes. This 
 is not necessarily the case for a doubly lifting or tandem 
 system. For this reason such a system does not in general 
 possess a corresponding singly lifting system which is 
 equivalent to it in every respect both algebraically and 
 physically. An exception occurs when e is the same for 
 the front and rear planes, so that HS'eux = e^S^x = by 
 the equation of moments. We may, however, still simplify 
 the algebra very greatly by replacing the doubly lifting 
 system by a singly lifting one which is equivalent to it in 
 every respect but the presence of the added term in N r , 
 that is, we employ the substitutions of 59, with the addi- 
 tional substitution of S\ep for ZS'e/j, (this determines the 
 value of e), but with the retention of the additional term 
 SS'e/xx in the third column, bottom row,, of the 
 determinant. 
 
 [I have worked out the solution without the use of an 
 equivalent system, but the algebra is very much longer, 
 and it is therefore desirable to use this short cut.] 
 
 The determinantal equation in X is now 
 
 = 
 
 (112) 
 
 a. * 
 
 + KUg> *> KU\ 
 
 ."*, + 
 
 0, - 8J 
 
 KUg 
 
 Neglecting terms of order /u, 2 in the first three coefficients 
 
io8 BROAD PLANES CHAP. 
 
 and making use of the equilibrium condition W cos = 
 we get 
 
 
 W \ 4 2TI 7 
 
 + tan 
 (113d) 
 
 In the approximate condition 6 (^X) the pro- 
 ducts of the terms once underlined cancel ; and in (, we 
 may neglect JJL tan compared with unity. For horizontal 
 flight the condition reduces to 
 
 W 
 
 " W ~ S^ + 
 
 >0 . . . (114) 
 
 65. For a double lifting system this expression is easily 
 transformed into the correct formula by means of in- 
 variants. In this case 
 
 S 2 l + S\e[j, becomes 2$'(e/i x) 
 
 while to transform S\efjLl-\-^S f efix we observe that the 
 term twice underlined in ^D arises from the determinant 
 
 which becomes 
 
 0, 
 
 22/Sf/i, 
 0, - 
 
 Now if the bottom term in the first column had been 
 2^S f e/j, 2 instead of zero the determinant would have 
 vanished, and hence we find that 
 
vi LONGITUDINAL STABILITY 109 
 
 
 
 = 22SV 2 2SV2S'.x (with SS'fjiX = 0) or generally 
 
 = - 2zS'en 2 S\S' 2 vl with v = ^ - /x 2 and I = x l - x. 2 . (115) 
 
 It is not worth while to reproduce all the algebra that 
 has been gone through in our preliminary investigations, 
 as in all cases likely to be of interest considerable simpli- 
 fications occur. For any system with two planes en will 
 be small compared with I, and only first powers of e need 
 be retained. It is only in the case of a single plane with- 
 out a tail of any kind that this might become inadmissible, 
 and then the algebra becomes very simple indeed. We 
 consider only the following cases : 
 
 66. CASE I. Single lifting system with neutral tail. 
 In this case ^S'epx = 0, and in the case of horizontal 
 flight the condition reduces to 
 
 g- (S 2 l + S\en) - k 2 S\(S\ + S 2 ) - 28' \S 2 enl - - -^ S\ 2 S 2 e^l z >0 . (116) 
 
 and the condition when higher powers of e/j, are retained 
 is not much more complicated. 
 
 For flight at an angle downwards with the horizon, if 
 fj> tan 6 be neglected in ( we get 
 
 + t flr l i} 
 Putting W=KU z S lt j. sec 0, we get 
 
 >o . . (117) 
 
 a quadratic in 
 
no 
 
 BROAD PLANES 
 
 CHAP. 
 
 67. CASE II. Single plane without tail or rudder. 
 This case may be deduced from the preceding one by 
 putting S 2 and I = 0. But where the stability is entirely 
 dependent on the shifting of the centre of pressure, it 
 is really easier to start afresh and use the complete 
 
 Pfioto.] IThe Sport and General Illustrations Co. 
 
 FIG. III. EDWARD MINES BIPLANE AT DONCASTEK. 
 
 An aeroplane having no auxiliary tail planes or vertical fins. Longitudinal 
 stability could only be attained by means of the shift of the centre of pressure 
 with varying angle of attack ( 67), and the stability condition would thus 
 impose a superior limit to the speed. There is no lateral stability, all the 
 coefficients in the biquadratic except the first two vanishing. 
 
 discriminant >0 instead of the approximate substitute 
 65>-<g93>0. We have 
 
 3>/ W V 2 
 
 W 
 Kg 
 
 tan 
 
 W 
 
vi LONGITUDINAL STABILITY in 
 
 whence 
 
 and with this substitution the condition which may be 
 written 
 
 reduces to 
 
 --e/ti(l - /x tan 0)(1 + '- : ^ : ) - k 2 S'(l + 2/x 2 ) 2 (l + p. tan 0)>0 . (120) 
 
 that is 
 
 V k 2 (l + 2/i 2 ) 2 cos 0(1 + /A tan 
 
 or, going back to the substitutions S' = $cos 2 a, /-t^tan a. 
 
 fr 2 k 2 cos ^(1 + 2 tan 2 a) 2 (l '+ tan a tan #) / 
 
 ~g~ > .e tan (1 - tan a tan 0)(tan a .+ ^ tsin-0) ' ( 12 l a ) 
 
 For horizontal flight (0 = 0) this becomes 
 
 f/ 2 fc 2 (l + 2tan 2 a) 2 
 
 > ~r 9 i or 
 
 g e tan z a 
 
 or again 
 
 + sin 2 a) 2 
 
 sin a cos a 
 
 For a glider = a giving 
 
 + tan 2 a) 2 fc 2 (l + 2 tan 2 q) 2 
 J a) r 3 e sin ^a(l - tar 
 
 (123) , 
 
 2 2 fc 2 cos a (1 + 2 tan 2 a) 2 (l + tan 2 a) 2 fc 2 (l + 2 tan 2 a) 2 cos 
 (/ > 3 etan 2 a(l - tan 2 a) or 3 e sin *a(l - tan 2 a) 
 
 or approximately 
 
 ^T >3 
 
 2 k 2 
 
 thus the minimum value of C/' 2 /^ required for the stability 
 of a glider is only about two -thirds of that required for 
 the same plane when driven horizontally through the air. 
 Stability becomes impossible if the plane is rising at a 
 greater angle than tan' 1 (2 tan a), 
 
U2 BROAD PLANES CHAP. 
 
 68. Now taking the exact condition for the case of 
 horizontal flight 
 
 W WS(1 + sin 2 a) 2 
 Kg > e sin a cos a ' ' ' ' (122a ' 
 
 and supposing W and S are kept constant, there is an 
 inferior limit to sin a, and consequently a superior limit 
 to the velocity 7 consistent with stability ; that is to say, 
 it is impossible to increase the speed of propulsion of a 
 single plane above a certain limit without the plane 
 becoming unstable. There will also be a maximum value 
 of a consistent with (I22a) corresponding to a minimum 
 velocity. The left hand side of (I22a) will be found to 
 be a minimum when a = 28 roughly. 
 
 On the other hand, in the case of a system of narrow 
 planes the condition of stability when expressed in the 
 form (54a) of ( 48) gives a limiting value for W which is 
 practically independent of the angles of attack of the 
 planes and depends only on their areas and their distances 
 in front of and behind the centre of gravity. This 
 condition is thus independent of the speed. 
 
 For these reasons it is highly undesirable that the 
 stability of a flying machine should be made to depend 
 on the shifting of the centre of pressure. The machine 
 should be constructed to satisfy the conditions of stability 
 based on the theory of narrow planes discussed in this 
 treatise. The effect of the shift of the centre of pressure 
 is certainly to increase the stability in certain cases, but 
 the higher the velocity the less is the advantage, as may 
 easily be seen for the case of a single lifting machine 
 from equation (118). 
 
 Furthermore, the fact that no account has been taken of 
 "rotary derivatives"- the effects of which may be com- 
 parable with those of the shifting of the centre of 
 pressure renders it still more necessary to rely for 
 stability on the separation and difference of inclination 
 of the front and tail planes. 
 
vi LONGITUDINAL STABILITY 113 
 
 It is important to point out that these conclusions are 
 not based on the assumption that the rate of shift of the 
 centre of pressure &<j) r (a) becomes small when a or p is 
 small, for according to our assumption it tends to the 
 limiting value e. 
 
 For large values of a the investigation will require 
 corrections owing to the deviation of the pressure on the 
 plane from the simple " sine law." 
 
 Effects of friction Raised planes. 
 Deviations from the " sine " law. 
 
 69. Suppose that, owing to frictional resistance, the 
 resultant thrust on a plane makes a small angle e with the 
 normal. Then a difference will exist between the angle of 
 attack a and the angle a or a + e which the resultant 
 thrust makes with the axis of y. The case of e variable 
 is practically covered by the corrections of 70 below. 
 We shall write in this case S' = S cos a cos a instead of 
 $ cos 2 a, and /* = tan a, // = tan a. 
 
 Further, instead of assuming the principle of independ- 
 ence of height we may take account of the height of the 
 planes by writing x' = x yn f , x" = x 2yp ; x' will then be 
 the abscissa of the point where the resultant thrust meets 
 the axis of x. If we then write 2/* // = //', so that 
 x" = x' yp!', the determinantal equation takes the form 
 
 
 = . (123) 
 
 If y is the same for both planes it is obvious that the 
 system may be replaced by an equivalent single lifting 
 system with neutral tail raised at the same height y above 
 
 I 
 
ii4 CAMBER CHAP. 
 
 the centre of gravity, and the calculations might thus be 
 used to investigate the effects on longitudinal stability of 
 a low centre of gravity. The formation of the coefficients 
 of the biquadratic and substitution in the approximate 
 form of the discriminantal condition of stability presents 
 no special difficulty even in the most general case, but it 
 scarcely appeared desirable to work out the formulae in 
 this memoir. If the results are wanted, this can always 
 be done subsequently. 
 
 70. If the " sine " law of resistance be not assumed, we 
 may still write 
 
 E = KSU* sin a, 
 
 where K is now a function of a, say K(a), and according 
 to experimental evidence 
 
 In this case the resistance derivatives only involve the 
 values of R and dR/da, and we have 
 
 ^ = SU*(K cos a + ^ sin a \ . . . (124) 
 
 If we give K the proper value for the value of a under 
 consideration, the whole of the previous work holds good 
 subject to the assumption that dK/da sin a is negligible 
 in comparison with K cos a. If not, corrections may be 
 applied. For the single lifting system, these corrections 
 will only occur in X v , Y w where S\ will be replaced by 
 
 Principal effect of curvature of main surfaces 
 
 (Camber). 
 
 71. From 32 it easily follows that for a concave surface 
 with neutral tail the variations in the direction of R con- 
 tribute to X v an additional term, which is equal to 
 
 KS^Ua/c j>'(a) sin a cos a'. 
 
vi LONGITUDINAL STABILITY 115 
 
 If we examine the correction due to this cause alone, 
 assuming the other derivatives to have the same values as 
 in 48, we find that the correction does not affect 91, 93, 
 or (, ; it introduces only terms of order sin 2 a into 6, 
 while for <) we now find 
 
 a 
 - sin a sin a' + ZIPS^ sin 2 a cos 2 a' - <'() (125) 
 
 If, as is usual with plane surfaces, the centre of pres- 
 sure shifts forward as the angle of attack decreases, <'( a ) 
 is negative, and the effect of concavity is to decrease 
 ), and thus to decrease stability. If, for very small angles 
 of attack, however, the centre of pressure recedes as the 
 angle of attack decreases, the effect of concavity then 
 increases stability. In any case, the effect depends on 
 the term X u Y v N r in 1) (assuming N U = Q). In view of 
 the difference in the resultant pressure on plane and 
 curved surfaces, it will be more exact if we assume S 1 to 
 be the area of the equivalent plane surface. 
 
 Effects of "wash." 
 
 72. Another case in which the coefficients are modified 
 in a similar way is where the wash produced by the front 
 plane alters the direction in which the air impinges on 
 the rear plane. The angle of attack on the rear plane 
 will be less than the angle which that plane makes with 
 the line of flight, so that the resistance derivatives of the 
 rear plane will again involve two angles a, a. 
 
 In Lanchester's investigation, allowance is further made 
 for the fact that oscillations of the front plane may produce 
 fluctuations in the direction of the wash which affect the 
 pressure on the tail plane. In connection with the present 
 investigation, the method of applying the latter correction 
 for a single lifting system would be as follows : Let the 
 machine receive a small velocity v perpendicular to the 
 
 I 2 
 
n6 WASH CHAP. 
 
 line of flight. This will produce an additional component 
 velocity of the " wash " current in the same direction 
 which will everywhere be proportional to v when v is 
 small. If the value of this component at the tail be 
 (1 e)y, the effective change in the corresponding relative 
 velocity of the wind on the tail will be ev instead of v, 
 consequently the portions of X v , Y V9 N v due to the tail will 
 have to be multiplied by e. 
 
 In this case the correction is very easily applied. Con- 
 sidering the " simplest case," the determinantal biquadratic, 
 after its last line has been multiplied by 1/6, becomes 
 
 , , W\ , , W JFsintf 
 
 S 1+ S 2 < + _, r- SJ + ^ + 
 
 0, - S A 8 4- 
 
 KUg 
 
 = 
 
 (126) 
 
 This equation is deducible from the original one by 
 substituting S. 2 e for $ 2 , l/e for I, and k 2 /e for k 2 , and we 
 only have to make the corresponding substitutions in the 
 final result, which for horizontal flight now gives 
 
 agreeing with Lanchester's result. It should be observed 
 that in this case we assume the tail plane to be placed 
 neutrally with respect to the wind. It will therefore make 
 a small angle a 2 with the line of flight, so that the first 
 line of the above determinant ought to read 
 w\ w 
 
 2$' 1 /z 2 + ^^ , S'iH + $ 2 e sin a 2 - j^j^ cos 6 - S 2 l sin a 2 , 
 
 even if cos a 2 is taken to be unity in the second and third 
 lines. This correction only introduces a small additional 
 term, 
 
 (E / W 
 sin a 2 in 
 
 and this is usually negligible. 
 
vi 
 
 LONGITUDINAL STABILITY 
 
 117 
 
 Effect of " lag" The fluctuations in wash probably do 
 not affect the tail plane at exactly the instant when they 
 are set up by the main plane, but it would be rather 
 premature and unnecessary to investigate the correction 
 for the " lag " thus produced in the present state of our 
 knowledge. It is obvious that the greater the speed of 
 the aeroplane, the less will this correction be. Its existence 
 should, however, be admitted. 
 
 Three plane systems. 
 
 73. We have seen that the loss of stability which 
 occurs when an aeroplane rises at more than a certain 
 angle is due to S decreasing and changing sign. Now 
 3)/<7 3 contains the determinant A o of the resistance 
 coefficients, which for a two plane machine has been 
 proved to be zero. If, however, for two we substitute 
 three planes or sets of superposed planes at different 
 
 y 
 
 FIG. 28. 
 
 horizontal distances from the centre of gravity, A o may 
 have a value different from zero, and this fact may be 
 used to increase the stability. For example, the machine 
 may be provided with a tail plane in the rear, and a 
 rudder plane or elevator in front of the main-supporting 
 surfaces, as in the Farman and Curtiss biplanes (Fig. 28). 
 
 Let S i9 S. 2 , S 3 be the areas, x^ x. 2 , x s the abscissae or 
 horizontal co-ordinates, a 1? o 2 , a 3 the angles of attack of the 
 three planes, then with the usual notation S\ = S l cos 2 a l9 
 a 1} etc., we have 
 
n8 
 
 THREE PLANES 
 
 CHAP. 
 
 By the rule for multiplication of two determinants this 
 is equal to 
 
 "It $ 21 8 
 
 1, 1, 1 
 
 1, 1, 1 
 
 asj, x 2 , X 3 
 
 * 3 )} 2 (128) 
 
 and is essentially positive, unless it be zero. 
 
 Photo.] 
 
 [The Sport and General Illustrations Co. 
 
 FIG. IV. M. HENRY FARMAN FLYING AT BLACKPOOL. 
 
 An aeroplane with auxiliary surfaces fore and aft constituting a "three plane 1 ' 
 system as in 73, each pair of superposed planes counting as a single plane 
 for the purposes of longitudinal stability. The conditions of lateral stability 
 are not all satisfied. The vertical rudder planes in the rear are equivalent to 
 a single vertical fin behind and not much above the centre of gravity, so that 
 the coefficient @j is negative (indicating instability), though JDj and ^ are 
 positive. 
 
 The effect of this term is thus to increase, never to 
 decrease, symmetric stability. 
 
 It remains to examine the effect on the other co- 
 efficients in the biquadratic, and for this purpose we 
 
VI 
 
 LONGITUDINAL STABILITY 
 
 119 
 
 shall again introduce the notion of an equivalent system 
 of planes. 
 
 We can, as in 59, choose a single lifting surface with 
 neutral tail having the values of 2/Sy, 2 ^V, %&, SS'sc/*, 
 Sfifa, the same as for the given three plane system, but 
 in view of the fact that A o is not zero, the value of 
 S/SW will no longer be the same, that is, it will 110 
 longer be equal to SJ? in the second system. 
 
 To get over this difficulty we imagine the neutral plane 
 divided into two parts, say equal parts for simplicity, and 
 
 FIG. 29. 
 
 these parts separated by a distance 2c, the middle point 
 between them still being at a distance I behind the 
 centre of gravity. The new system is thus specified 
 as follows (Fig. 29) : 
 
 Areas of planes 
 
 Inclinations 
 
 Co-ordinates 
 
 
 
 - I + c 
 
 The value of SSV will then be equal to S 2 (/ 2 + c 2 ), the 
 relations between the original and the new system being 
 as follows : 
 
 Original System. 
 
 Equivalent System. 
 
 28 sin 2 a 
 2$ sin a cos a 
 2S cos 2 a 
 
 S 1 sin 2 a 
 Sj_ sin a cos a 
 S l cos 2 a + 2 
 
 
 2Sx sin a cos a ( = 0) 
 
 2$x cos 2 a 
 
 
 
 
 2&K. 2 cos 2 a 
 
 + S 2 (l* + c 2 ) 
 
 (129) 
 
120 THREE PLANES CHAP 
 
 For the equivalent system we easily find 
 
 n 2 a .... (130) 
 
 and since A o for the original system has been shown to be 
 essentially positive, it follows that c 2 is positive, so that 
 the distance of separation of the two parts of the neutral 
 plane is real ; in other words, the equivalent system is 
 a possible one. 
 
 74. We now form the coefficients of the biquadratic for 
 the equivalent system, taking into account inclination 
 of flight path but not head resistance, always supposing 
 the propeller thrust to act along the axis of x and to 
 be constant. To the usual first approximation 
 
 = -- S r l *S 2 p?l to first order ...... (131e) 
 
 If we suppose ^ = in the first instance and form 
 the approximate condition 6^) (93 > 0, we shall find 
 that the only parts containing c 2 which do not cancel 
 are those arising from the product of S into the term 
 underlined in 5), which arises from the determinant A o , and 
 the condition of stability becomes 
 
 (132) 
 
 The approximate condition for stability is the same as it 
 would be in horizontal flight with c = 0, when the direction 
 
vi LONGITUDINAL STABILITY 121 
 
 of flight makes an upward angle with the horizon 
 given by 
 
 , . (133) 
 
 and this represents the increase in the angle of elevation 
 consistent with stability obtainable by separating the 
 halves of the neutral plane through a distance 2c. 
 
 To sum up, then, the values of S3 and 6 are increased 
 by the substitution of I 2 + c 2 for I 2 . The value J) is 
 increased by the additional term due to the deter- 
 minant A , though for ascending flight it is decreased 
 by the term involving tan 0, which is then negative. The 
 net result is an increase in the approximate discriminant 
 SS) - $93, this increase being equal to the increase in 5) 
 multiplied by the increased value of 6. For the original 
 system the increased elevation obtainable by this means is 
 given approximately by the relation 
 
 A - W-/g N v sin 6 = (N v negative, 6 negative) . (133a) 
 
 as is evident from equations (18a). 
 
 The question of the hitherto neglected rotary 
 coefficients of broad planes. 
 
 75. The transformations and methods of the last article 
 help us to obtain some estimate of the general effects 
 of the rotary coefficients/,, (a) and <f> r (a), if only as deductions 
 from assumed hypotheses concerning them. 
 
 A plausible assumption is that when a plane rotates 
 about its centre of pressure, the resultant thrust is not 
 affected by the rotation to any important extent possibly 
 not at all, but that the moment of the pressures so set up 
 tends to retard rotation. If this be assumed, the effect is 
 represented by an additional positive term in N r , and can 
 be made equivalent to the effect, considered in the last 
 article, of changing SjP into S 2 (l* + c 2 ) by a suitable choice 
 
122 ROTARY DERIVATIVES CH. vi 
 
 of c. The effect is thus to increase stability and in 
 particular to increase the angle of elevation of the flight 
 path consistent with stability, as indeed is almost evident 
 from general considerations. 
 
 If rotation about the centre decreases the moment of the 
 resistances, the effect will be to decrease stability. This 
 alternative hypothesis seems much less plausible than the 
 first, but questions of the kind cannot be decided by rule- 
 of-thumb reasoning. In this connection, it is a significant 
 fact that a balanced plane which is free to rotate on its 
 axis will rotate continuously when placed in a wind, but 
 the decisive test is obtainable ( 29) by attaching the plane 
 to a pendulum placed in an air current, the axis of 
 suspension passing through the centre of pressure, and 
 observing whether the oscillations subside or increase. 
 
CHAPTER VII. 
 
 ASYMMETRIC OR : LATERAL STABILITY STRAIGHT PLANES 
 AND VERTICAL FINS. 
 
 Further hypotheses regarding narrow planes. 
 
 76. In dealing with the longitudinal or symmetric 
 oscillations, we assumed as a definition of narrow planes, 
 that the chord or distance from the front to the back edge 
 of a plane was so small that the unequal velocities of the 
 two edges due to rotation had a negligible effect on the 
 distribution of pressure. On the other hand, in studying 
 
 FIG. 30. 
 
 the asymmetric oscillations we have to take account of the 
 fact that the span of the planes is often considerable, so 
 that rotation may produce considerable differences in the 
 velocity at different distances from the axis of x 9 or rather 
 from the plane of symmetry. 
 
 Let the plane be divided into strips such as dSin the 
 accompanying figure. Then the new assumption which 
 we have to make, and which we agree to regard as implied 
 in the definition of a narrow plane, is that the pressure on 
 the element dS is independent of the motion of parts 
 
 123 
 
I2 4 
 
 NARROW PLANES 
 
 CHAP. 
 
 to the right and left of the element, and depends only on 
 the velocity of the element itself. A further assumption 
 which we regard as implied in the statement that the angle 
 of attack is small is that the pressure on dS follows the 
 well-known sine law, or, what is equivalent, that the 
 resultant thrust on dS = KdS x resultant velocity x 
 normal velocity relative to the air. 
 
 A correction is probably necessary, at least near the 
 extremities of the plane, but it is no use trying to take all 
 such corrections into account at the outset of the discussion 
 (see 93). 
 
 To make the argument more general, and applicable in 
 particular to surfaces which are bent up or down towards 
 
 y 
 
 FIG. 31. 
 
 their extremities, let us suppose that x, y, z are the 
 co-ordinates of the element dS, and that /, m, n are the 
 direction cosines of the normal to this element (Fig. 31). 
 The resultant velocity of dS is 
 
 U + u - yr + zq, 
 
 neglecting terms of the second order, while the normal 
 velocity is 
 
 l(U + u - yr + zq) + m(v - zp + xr) + n(w - xq + yp), 
 
 so that, again to the first order, we have for the resultant 
 thrust dR on the element 
 
 dE = KdS{lU* + 2lU(u - yr + zq) 
 
 + mU(v - zp -f xr) + nU(w - xq + yp)} (134) 
 
vii LATERAL STABILITY 125 
 
 and the corresponding force and couple components, when 
 integrated over all the elements, become 
 
 X = fldR, Y = [mdR, Z = [ndR, 
 
 L = f(ny - mz}dE, M <= f(lz - nx)dR, N = f(mx - ly)dR. (135) 
 
 If the aeroplane is symmetrical about the plane of x, y, 
 such integrals as contain only odd powers of z or of 
 n (when dR is expressed in terms of dS by means of 134) 
 will vanish, and this will secure the conditions that the 
 resistance derivatives (X, Y, N) w , P , q and (Z, L, M) U)V)V 
 vanish, i.e. the independence of the symmetric and 
 asymmetric oscillations. We obtain 
 
 X = L^lKl^dS + 2Wit KPdS + Uvf/ClmdS 
 
 + UrfKl(mx-2ly)dS . . (136*) 
 
 Y = U 2 i KlmdS + ZUufKlmdS + Uv f Km*dS 
 
 + Ur[Km(mx-2ly)dS . . (136y) 
 
 N = U 2 fKl(mx-ly)dS -h 2Uu f Kl(mx - ly)dS + Uv ( Km(mx - ly)dS 
 
 + UrfK(mx-ly)(mx-2ly)dS (136n) 
 
 Z =r- UwJKtfdS + Upfznfa-mttfS 
 
 + Uql Kn(2lz-nx)dS . (136z) 
 L = Uiv[Kn(ny- mz)dS + Up ( K(ny - mz) 2 dS 
 
 + UqfK(ny-mz)(2h-nx)dS (1361) 
 M = Uivj Kn(lz - nx)dS + &pJK(*y - mz)(lz - nx}dS 
 
 + Uq(K(lz-nx)(2lz-nx)dS (136w) 
 
 The first three refer to longitudinal stability. By 
 putting I = sin a, m = cos a cos /3, n = cos a sin they 
 might be applied to the case of an aeroplane whose surfaces 
 slope upwards at an angle ft towards their extremities. 
 This case reduces to the problems considered in Chapters 
 V, VI by putting cos ft=l. In most cases of immediate 
 interest, this approximation will be amply sufficient, and 
 
126 STRAIGHT PLANES CHAP. 
 
 therefore it is considered undesirable to reproduce a fuller 
 discussion here. In lateral stability the difference is 
 important and we start with the simplest case. 
 
 Straight planes. 
 
 77. By this we mean planes perpendicular to the plane 
 of x, y, in contradistinction to the bent up planes just 
 mentioned. For these we put I = sin a, m = cos a, n = 0, 
 and obtain 
 
 Z m Z p , Z q , L m M w each = 
 
 and writing 
 
 e 
 
 I=lz 2 dS= moment of inertia of area of plane with 
 respect to the plane of x, y, we have 
 
 L p = KUIcos? a L q = - ZKUfsm a cos a 
 
 M p = - KUI sin a cos a, M q = + 2KUI sin 2 a . . (137) 
 
 Substituting in the expressions for the coefficients in 
 20, we find 2) and (g (omitting the suffixes) both zero, 
 also for a single plane 6 = 0. If there are two planes 
 whose angles of attack are a l3 2 > and moments of inertia 
 ./! and 7 2 it will be found that 
 
 ^2y = 2W 1 I 2 sin2(a 1 - a 2 ) ..... (138) 
 
 which also vanishes if ^ = a 2 or the planes are parallel, 
 likewise if /i or J 2 = or one of the planes is a rudder 
 plane of negligible span. 
 
 In every case two, if not three, roots of the biquadratic 
 vanish. This indicates lack of stability, the effects of 
 which can be studied by writing down the equations of 
 motion 
 
 t 9 ) 
 
 - - F = KUI( - p cos 2 a + 2q sin a cos a) . (139p) 
 ^ " F = KUI + Sin " C S a " 2q Shl2 a) ' 
 
vii LATERAL STABILITY 127 
 
 The last two show that there is no tendency for the 
 machine to right itself if it heels on one side, for there is 
 no term in them depending on the action of gravity, arid 
 therefore no tendency to assume a definite position relative 
 to the vertical. The first tells us that if the machine has 
 heeled over through an angle $, gravity will tend to 
 make it slide down sideways, and there is nothing to 
 check this tendency. The maintenance of lateral balance 
 must therefore be effected by furnishing the machine with 
 warping devices to be controlled by the operator, or else 
 other planes, such as vertical fins or rudders, must be 
 added. 
 
 The explanation of the third vanishing root of the 
 biquadratic in the case of a single plane may 'be seen by 
 multiplying the second and third equations above (139 p, q) 
 by sin a and cos a, and adding ; this eliminates the right 
 hand side, and the resulting equation expresses the fact that 
 the couple about a normal to the plane is zero, so that 
 there is nothing tending to check rotation about this 
 normal, i.e. there is no directional stability. 
 
 The following points should be noticed. 
 
 (1) The effect of a straight narrow plane on the lateral 
 stability depends on the moment of inertia of its area 
 about its plane of symmetry, and does not depend on the 
 position of the plane. For x and y do not enter into 
 the coefficients. 
 
 (2) A horizontal rudder plane for steering in a vertical 
 plane does not affect the stability unless its span is 
 considerable. 
 
 (3) We may write I=SR 2 where S is the area and R 
 the radius of gyration of the plane. We may, following 
 Lanchester, extend the method to cases other than those 
 of narrow straight planes by suitable conventions as to 
 the value of R in the various terms, these becoming his 
 " aerodynamic " and " aerodromic radii." 
 
128 STRAIGHT PLANES AND FINS CHAP. 
 
 Effect of a single vertical fin. 
 
 78. If we suppose a vertical fin or plane of area T 
 placed in or parallel to the plane of x, y with its centre of 
 pressure (for grazing incidence) at the point (x, y, 0) or 
 (x, y, z), the resistance derivatives due to this fin are 
 given by 
 
 Z w = K'TU Z p = K'TUy Z q = - K'TUx 
 
 L w = K'TUy L p = K'TUy* L q = - K'TUxy 
 
 M w = - K'TUx M p = - K'TUxy M q = K'TUx* . (140) 
 
 while for a system of several such fins we only have to 
 prefix the sign of summation to expressions of these 
 forms. These expressions are, of course, to be added to 
 the corresponding derivatives due to the main planes. In 
 the case of a single main plane and a single fin we 
 obtain 
 
 2) W 
 
 1J3-3 = KK'IT ~(y sin a - x cos a) cos a 
 
 - ^ K'T{(By - Fx) cos 6 + (Ax - Fy) sin 6} . (Uld) 
 
 (S W 
 
 -f-4 = -j KK'TI (2 sin a cos 6 - cos a sin &) (x cos a - y sin a) (141e) 
 
 Consider the case of horizontal flight (0 = 0), and, to 
 simplify matters, suppose the axis of x is a principal axis 
 of inertia, so that F=Q, an assumption we shall frequently 
 find it convenient to make. In this case, the second term 
 in the expression giving <) reduces to the part containing 
 By. 
 
 Now the only way of securing the stability condition 
 ( positive is by making x cos a y sin a positive, that is, 
 placing the fin or rudder in front of a line through the 
 centre of gravity perpendicular to the main plane ; i.e. 
 briefly speaking, placing the rudder in front. But this 
 necessarily makes the first term of <D negative, a result 
 which can only be counteracted by making y negative and 
 sufficiently large in the second term of 2), that is, by 
 
vii LATERAL STABILITY 129 
 
 raising the fin considerably above the centre of gravity. 
 Mr. Harper and I have investigated the matter at some 
 length, and find that even when stability could theoretic- 
 ally be secured by this means the height of the fin would 
 have to be far too large, in proportion to its horizontal 
 co-ordinate x, to enable the method to be applied in the 
 construction of a convenient flying machine, and further 
 that the difficulty increases with the velocity U. There 
 is, of course, a possibility of improving matters by 
 making the product of inertia F positive, but this again 
 does not appear very promising. 
 
 In a flying machine of the now prevalent types, y is as 
 a rule not very different from zero, and the choice lies 
 between putting the vertical rudder behind and making 
 the machine unstable through (g negative or putting it in 
 front and making it unstable through 2) negative. For 
 a machine, the dimensions of which were given, it would 
 be easy to obtain in either case numerical approximate 
 solutions of the biquadratic and to decide which was the 
 lesser of the two evils, but a more interesting line of 
 investigation is to examine how two fins are better than 
 one in securing stability. 
 
 General case of a number of fins. 
 
 79. It will be observed that (1) the resistance deriv- 
 atives due to vertical fins parallel to the plane of x, y are 
 independent of their z co-ordinates ; thus, for example, two 
 vertical planes symmetrically placed at opposite sides of 
 the plane of x, y are equivalent to a single plane of 
 double the area in the plane of x, y midway between 
 them. 
 
 (2) For a number of fins the resistance derivatives 
 contain the sums of first and second powers of their 
 co-ordinates, and are thus transformable by the rules 
 
 K 
 
1 3 o STRAIGHT PLANES AND FINS CHAP. 
 
 applicable to centres of gravity and moments of inertia 
 (compare Lanchester's " Fin Resolution"). 
 
 Writing T for the sum of the areas of the fins, x, y 
 for the co-ordinates of their centre of mean position (or 
 centre of pressure), and M lt M 2 , P for the moments and 
 product of inertia of the areas of the fins with respect to 
 axes parallel to the co-ordinate axes through the latter 
 
 [The Sport and General Illustrations Co- 
 
 FIG. V. GLEN CURTISS BIPLANE AT RHEIMS. 
 
 An aeroplane with two vertical fins, namely a triangular fin in front and a 
 rectangular rudder plane behind. This arrangement is favourable to lateral 
 stability if the centre of pressure of the two fins is slightly above or in front 
 of the centre of gravity and certain other conditions are satisfied ( 84-86). 
 
 point, the expressions Z w , Z p , Z q , L w , M w of the last 
 article hold good, but we have 
 
 L p = K'U(Ty* + MJ, L q = M p = - K'U(Txy + P), M q =K'U(Tx* + M 2 ) 
 
 (142) 
 
 To shorten the algebra assume provisionally that 
 K f = K. If the fins have a different resistance-coefficient 
 K' from the main planes we can subsequently restore the 
 difference by putting a factor K' IK into the terms 
 depending on the areas and moments of the fins. 
 
 The determinantal equation is most conveniently now 
 put in the form of an equation in \/KUg, and becomes 
 
vii 
 
 LATERAL STABILITY 
 
 131 
 
 W 
 
 KUg 
 2V, 
 
 W 
 
 "' " ~ 
 
 + /cos 2 a, 
 
 (Txy+P) 
 I sin o cos a, 
 
 - 21 sin a cos a 
 
 + (Tx* + M 
 -f2Isin 2 a 
 
 (143) 
 
 At this stage, it will be found convenient to write I' for 
 / cos 2 a and ^ for tan a, consistently with the practice 
 previously adopted in dealing with longitudinal stability. 
 
 When the determinant is developed into the biquadratic 
 form, the following expressions represent the coefficients if 
 \jKUy is taken as variable, 5(, $3, 6, etc., representing 
 the coefficients if X is taken as variable : 
 
 
 KUg 
 
 W(AB - 
 T(AB - 
 
 ay 
 
 - P(3I> + 2Txy + 2P) + B(T + 
 - 2P) 4 
 - 3(Txy + 1 
 
 . (144a) 
 
 . (1446) 
 
 WT 
 
 (Ax - Fy) 
 
 . (144c) 
 
 - P 2 ) 
 
 WT 
 
 - sin ^(M^ - Py)\ 
 
 {cos 
 
 These expressions become much simplified when (1) the 
 angle of attack a of the main planes is small so that ft is 
 small, (2) the fins are so small that we need only retain 
 the lowest significant terms involving Tor the correspond- 
 ing moments, and in particular M^ M 2 , P ar p small com- 
 
 K 2 
 
132 STRAIGHT PLANES AND FINS CHAP 
 
 pared with /, (3) the product of inertia F vanishes. (4) 
 = or is small. In these cases, we have to a sufficient 
 approximation 
 
 WI'B ....... (I45a,6) 
 
 80. From these forms, we notice the following 
 points : 
 
 (1) The moment of inertia M l of the fins does not 
 enter into the approximate values except in the part of 
 6 which is of order T 2 , although the height y of their 
 centre does enter. There is thus no advantage in making 
 -Mi different from zero, that is, placing two fins one above 
 the other. 
 
 (2) It is easy to see that M 1 M 2 P 2 is zero for two fins, 
 and positive for any number greater than two unless 
 collinear. As this expression occurs among the neglected 
 terms, there is no advantage in using more than two fins. 
 
 General character of the oscillations Ap- 
 proximate resolution of the biquadratic 
 Resistance to rolling. 
 
 81. The conclusions obtained in connection with the 
 separation of long and short oscillations in the case of 
 longitudinal stability do not hold good in the case of 
 lateral or asymmetric oscillations, and in the present case 
 the separation of the roots of the biquadratic assumes a 
 different form. 
 
 In the first place, if there are no fins, only the first two 
 terms of the biquadratic remain finite, and these give rise 
 
vii LATERAL STABILITY 133 
 
 to one finite value of X, say X 1} which = 93/21. With 
 the approximate values of (145 a, b) we have 
 
 A- -2- ' ' ' ' (146) 
 
 whence X x is essentially negative as required for stability. 
 This solution determines the rate of damping of an 
 angular velocity p about the axis of x, as is evident from 
 first principles. It is, in fact, equivalent to the dynamical 
 equation, 
 
 A jdt - - KUI 'P ' - '. < 146a ) 
 
 While this damping tends to check the rolling motion 
 about the line of flight, it indicates no tendency to restore 
 equilibrium when the machine has heeled over through a 
 definite angle about this line. 
 
 Even when F is not zero and a not small, X x will in 
 general be negative. For AB F 2 is essentially positive 
 by the theory of moments of inertia, so that 21 is positive, 
 and writing A = 2,m(y 2 + z 2 ), etc., as in elementary Rigid 
 Dynamics, we find (with T, M l} M 2 and P each zero) 
 
 \ 
 
 (x cos a y sin a)(x cos a 2y sin a) + 32mz 2 / . (147) 
 
 j^jj = 
 
 an expression which could only be negative in the case of 
 a preponderance of the mass of the flying machine between 
 the planes x cos a y sin a = and x cos a 2y sin a = 0. 
 A machine the mass of which was so distributed might be 
 expected to exhibit anomalous behaviour, but would not 
 occur ordinarily in practice. 
 
 If the fin area T instead of being zero is small, the 
 remaining roots X 2 , X 3 , X 4 , of the biquadratic will be small, 
 compared with X lt They will, therefore, all make 2lx 4 
 small compared with 33x 3 , and therefore will be given to 
 a first approximation by the cubic 
 
 m 3 + (SX 2 + 2>X + <g = . , . (148) 
 
134 STRAIGHT PLANES AND FINS CHAP. 
 
 the condition of stability assuming the simpler approxi- 
 mate form 
 
 2) - (23 > 0, 
 
 which greatly simplifies some of the calculations, and 
 will be used hereafter when the full condition is too 
 complicated. 
 
 82. In certain circumstances the separation can be 
 carried further. We notice that g/^Z/y and 
 contain terms of the first order in T, while in 
 one term contains a factor of order Ta, and the other is of 
 the second order in. T. As there are two independent 
 small quantities a and T, it is not possible to lay down a 
 universal rule, but we may generally expect the biquadratic 
 to take the form 
 
 2U 4 + 23X 3 + ('X 2 + 3>'A + <T = 0, 
 
 the first two coefficients 21, 35 being finite 6', 5)' of the 
 first order, @" of the second order. We have discussed 
 the finite root \j = 33/21. Another root, X 4 , is given 
 approximately by \ 4 = (g"/5/ and is of the first order. 
 The middle pair of roots, X 2 and \ 3 , are of order J, 
 the first approximation to their values being given by 
 
 8A 2 + 3)' = 0, 
 and using this value we can substitute 
 
 2U* = - -^'x 2 , r = -- ^ A*, 
 leading to the second approximation given by 
 
 + --0 . . (149) 
 
 Omitting the accents, we see that the modulus of decay of 
 the oscillations determined by the middle pair of roots 
 \ 2 and \ 3 is 
 
 (150) 
 
vii LATERAL STABILITY 135 
 
 thus being directly related to the discriminant . The 
 method will fail and all the roots will be of the same 
 order, if 6/5) be of the order of smallness of (5>/)* that 
 is if T be of order a 2 . If the method is applicable, then 
 the various roots correspond to : 
 
 (1) A rapid subsidence determined by Xj (damping of 
 the angular velocity of rolling). 
 
 (2) A slow subsidence determined by X 4 . 
 
 (3) An oscillation of medium period determined by\ 2 
 and \ 3 , but the rate of subsidence of which may be 
 comparable with that corresponding to \ 4 . 
 
 83. An important point is carefully to distinguish the 
 terms depending directly on the action of gravity. These 
 are the terms containing cos 6 and sin 9 in the full 
 expressions for ^jK^IPg^ and (,/K*U*g\ In the abbrevi- 
 ated or approximate expressions, they are distinguished 
 by having U' 2 in the denominator. These are the only 
 terms which determine any tendency on the part of the 
 machine to assume or return to a definite position relative 
 to the vertical. No method which ignores these terms will 
 give inherent stability such as will enable a machine to 
 right itself when displaced. The presence of U' 2 in the 
 denominator is in agreement with the fact that while the air- 
 pressure on a moving plane increases with the square of the 
 velocity, the effects of gravity remain constant, and, there- 
 fore, their relative influence decreases. 
 
 The occurrence of g elsewhere is due to our having 
 measured resistances in pounds instead of poundals, and 
 our use of the homogeneous form of the equations of 
 dynamics. 
 
 Case I. Systems with two raised fins at the 
 same height. 
 
 84. We have already pointed out in dealing with one fin 
 that x cos a y sin a occurs with a positive coefficient 
 
i 3 6 
 
 STRAIGHT PLANES AND FINS CHAP. 
 
 in (, and with a negative coefficient in <). The simplest 
 way out of this difficulty is to make x cos a y sin a = 0, 
 that is, to put the centre of pressure of the fins in a 
 line through the centre of gravity perpendicular to the 
 main plane. Since a is small, it comes practically to the 
 same thing if x = Q, i.e. if the centre of pressure is 
 vertically above the centre of gravity. This arrangement 
 is mentioned by Lan Chester, who refers to its advantages 
 in securing stability. A further simplification consists in 
 assuming the two fins to be on the same level, so that y is 
 
 Centre of 
 Pressure 
 
 .c, 
 
 T, 
 
 y 
 
 FIG. 32. 
 
 the same for both ; in this case M l and P will vanish, and 
 if T lt T 2 (Fig. 32) are the areas, x lt x 2 the co-ordinates of 
 the fins, we shall have 
 T^ 2 
 
 T T 
 
 = -m , x square of distance between fins (151) 
 
 The condition (5; positive now reduces to y negative, that 
 is, for stability the fins must be raised above the centre of 
 gravity. 
 
 The expression for 2) reduces to 
 
 TW ( \ 
 
 = TM 2 r - ^u* \ (By - Fx "> cos e + (Ax ~ Fy "> sin *} (152) 
 
 Remembering that x is small, the condition y negative 
 will certainly make <) positive if F=0 } and 6 will also be 
 necessarily positive under the same conditions. If F 
 
vii LATERAL STABILITY 137 
 
 is different from zero, there will still be wide limits con- 
 sistent with the stability conditions <) and 6 positive. 
 
 85. The final condition or $6>-2lT) 2 - (B 2 > 
 necessarily becomes complicated, and it is useful to obtain 
 some simpler condition which will suffice for the purpose. 
 Now, after a good deal of laborious substitution, I find that 
 ivhen the tangent of the inclination //, is small, the con- 
 dition 
 
 substituted in makes that expression essentially 
 positive ; this condition is therefore sufficient for stability. 
 
 In verifying this conclusion, it will be found best, in the 
 first instance, to make the substitution in (5<3) (93, and 
 with the value so obtained to substitute in 
 
 (63)- (8)93 -2l> 2 . 
 The work is rather laborious. 
 
 For the case where higher powers of /-i are retained, I 
 have tried the same assumed condition, which, of course, 
 may be written 
 
 but this substituted in leaves certain negative terms in 
 it. These are probably much smaller than the positive 
 parts in cases likely to occur in applications ; at the same 
 time their presence shows that the above condition can 
 only be relied on as sufficient provided that a further con- 
 dition is satisfied. It might perhaps be worth while to 
 examine this particular case at some future time. 
 
 The above condition may be put into various modified 
 forms by combining it with the condition of equilibrium 
 W= KSU' 2 sin a. If we put A = Wr, 2 and /= Sp*, then r, 
 and p will be radii of gyration of the mass of the machine 
 and of the area of its main plane respectively, and we find, 
 on substituting 
 
 ( - y)2H sin* < * , if H = . . , (1536) 
 
138 STRAIGHT PLANES AND FINS CHAP. 
 
 H being the height to which the velocity U is 
 due. 
 
 Even the above formula is not required if the total area 
 of the fins is small compared with that of the main planes. 
 It will, in fact, be found that the terms of , which are of 
 the lowest order in T and M 2 , are essentially positive. 
 
 It will thus be seen that the present arrangement 
 possesses a wide range of stability. Moreover, this stability 
 is not much affected by the angle which the flight path 
 makes with the horizon, (g, and the important parts of 2) 
 containing cos 9 and not sin 6. A further obvious property 
 is that, should such a machine be turned upside down, with 
 the fins below, it would become unstable and tend to right 
 itself. 
 
 CASE II. Both fins at the level of the centre 
 
 of gravity. 
 
 86. If we put y = Q and M 1 = P = Q, the condition of 
 stability ( > reduces when 6 = to x cos ay sin a > 0, 
 that is, the centre of pressure of the two fins must be in 
 
 T 2 T, 
 
 
 G^Xf 
 
 
 ^^ C.P. 
 
 s 
 
 FIG. 33. 
 
 
 
 front of the centre of gravity (Fig. 33). To make 2) 
 positive in this case, it will be necessary to have 
 
 VPt 
 
 ff<*i , . . (154) 
 
 The stability, as dependent on (S, will in this case be 
 limited by the condition 
 
 tan B < 2 tan a. 
 
VII 
 
 LATERAL STABILITY 
 
 139 
 
 Now it may be true that this condition would usually be 
 satisfied in uniform flight, since even if a plane were 
 gliding down uniformly under gravity, 6 would only be 
 equal to a, and any value of 6 greater than a could only 
 be consistent with steady motion if the machine were 
 retarded by reversing the engines or by a sufficiently 
 great head resistance. At the same time, we have to con- 
 sider the possibility of a machine diving down at a steep 
 angle either in order to recover lost speed or as the result 
 of a sudden gust of wind, and we have further to 
 remember that the conditions of lateral stability are 
 
 FIG. VI. DIAGRAMMATIC VIEW OF A WRIGHT BIPLANE USED AT PAU. 
 
 An aeroplane having vertical auxiliary fins situated fore and aft of its centre of 
 gravity. From the figure it appears probable that the centre of pressure of 
 the fins in this case was rather behind and not much above the centre 
 of gravity, and that the arrangement failed to secure lateral stability for 
 this reason. 
 
 independent of those for longitudinal equilibrium. The 
 conclusion is that, if for any reason such a machine should 
 dive downwards at an angle greater than that consistent 
 with the relation tan < 2 tan a, it would lose its lateral 
 stability, and would probably turn round sideways. This 
 might result in an accident, either through the machine 
 becoming uncontrollable or through breakage caused by 
 undue strains. The erroneous impression might at least 
 be produced that the machine had suddenly been struck 
 by a side gust of wind, causing it to overturn sideways. 
 A knowledge of the risk and its cause might enable an 
 aviator to save the situation. 
 
1 40 STRAIGHT PLANES AND FINS CHAP. 
 
 87. Using the approximate forms of 80 (which, be it 
 remembered, assume ^=0), we find 
 
 - ir = TI>B 
 
 and the stability condition >0 reduces to 
 
 Wx r WE cos 6 (2 tan a - tan 6) n 
 
 x r cos an a - an n 
 
 Kg L + KU*[TB + W(Tx* + MJ\ J 
 
 This condition can be put in various forms or replaced 
 by a simpler condition, which is sufficient, although not 
 necessary for stability, by omitting some of the terms in 
 the denominator of the last fraction. For example, 
 omitting Tx 2 + M 2 , we see that stability will be assured by 
 taking 
 
 which from the equations of equilibrium gives 
 
 (2p " tan ^ ' ' (1566) 
 
 Or, again, we may write 
 
 M 2 = T^ 2 , B = 
 
 where f is the radius of gyration of the areas of the fins 
 about their centroid, and k 2 is the radius of gyration of the 
 mass of the machine about the axis of y. In this case, the 
 condition reduces to 
 
 T * " > TT g + SJ f ^ ~ ten * *.' +^ + f ] < 156c > 
 
 In any case, stability is secured by making M 2 large, and 
 x small, that is, by making the distance between the fore 
 and aft fins sufficiently large, and placing the centre of 
 pressure not too much in advance of the centre of gravity. 
 At the same time, x must not be too small, otherwise ( 
 will become too small. 
 
 Further, if tan 6 is negative, or the aeroplane is rising, 
 the condition of stability > becomes more difficult to 
 
vii LATERAL STABILITY 141 
 
 satisfy as the angle of rising increases, while, as we have 
 already shown, stability ceases when the angle of descent 
 becomes greater than that given by tan 6 = 2 tan a. 
 
 The existence of two limits is a direct consequence of 
 the fact that ( occurs with a negative sign in $. The 
 present system is one in which a numerical calculation of 
 the actual oscillations obtained by solving the biquadratic 
 for a machine of given dimensions would be of con- 
 siderable interest. This suggests a subject for future 
 investigation. 
 
 CASE III. One fin above and one behind of 
 the centre of gravity. 
 
 88. The following investigation of this case is due to 
 Mr. Harper. We suppose the fins to be of areas J\ at the 
 point (0, y) and T 2 at (x, 0) (Fig. 34), and we clearly have 
 
 = -T 2 x 
 
 
 The approximate coefficients in the biquadratic where //- 
 is small will be given by 
 
 21 = W{AB - F*\ . . . .... . . (158a) 
 
 = WET + W(By^T l + Ax*T 2 ) + (2\ + T 2 ){AB - -F*} (1586) 
 
 (AxT 2 - FyT,) (158c) 
 
 W 
 
 (x cos 6 + y sin 6)xyT T 2 ..... (158e) 
 
I 4 2 
 
 STRAIGHT PLANES AND FINS CHAP 
 
 We suppose F = 0. Also if the fins are above and 
 behind the centre of gravity, x and y will both be negative. 
 In these circumstances, all the coefficients except (S 
 are essentially positive. Putting 6 = 0, we see that the 
 condition ( positive is secured by making 
 
 xyTi > 21' p or 21 sin a cos a . . . (159) 
 
 which is equivalent to the condition 
 
 yL w + L q positive . (159) 
 
 It will thus be seen that for stability the distance of the 
 tail Jin behind the centre of gravity must not be less than 
 a certain inferior limit. It will be noticed, moreover, 
 
 -y 
 
 T 2 
 
 FIG. 34. 
 
 that the coefficient of sin 6 is positive in (, consequently 
 stability from this cause is increased when flying down- 
 wards, though a limit for stability occurs in flying 
 upwards at too steep an angle. 
 
 89. Coming next to the approximate condition 
 g<> _ gsg > o, we notice that for 6 = the only positive 
 part of ( is the term underlined, and if the fin area is 
 small, the term underlined in 93 need alone be retained for 
 a first approximation. The product of these terms cancels 
 out the product of the terms underlined in (52), and hence 
 in (S$)-(S9[3, all the terms which are not small in com- 
 parison with other terms are positive. The approximate 
 discriminantal condition of stability is thus satisfied 
 subject to the assumptions that have been made. 
 
 Mr. Harper further points out that the stability condi- 
 
VII 
 
 LATERAL STABILITY 
 
 tion xL w + L q positive represents the condition that 
 if the machine receives a small angular velocity of rota- 
 tion about a vertical axis passing through the tail plane 
 (x, 0), the moment of the air resistances that is set up 
 tends to cause the machine to heel over about the axis of 
 
 Photo.] 
 
 [The Sport and General Illustrations Co. 
 
 FIG. VII. ROUGIER FLYING AT BLACKPOOL. 
 
 The Voisin type with its box-like arrangement of vertical fins is equivalent, so 
 far as lateral stability is concerned, to a system with one fin above and one 
 fin behind the centre of gravity, the conditions of stability of which are 
 discussed in 88. 
 
 x towards the outside of the curve that it ivould describe 
 in virtue of the rotation. 
 
 Effect of two or more transverse planes. 
 
 90. We have already referred to the case where, instead 
 of a single main plane, there are two planes the moments of 
 inertia of which are I l9 / 2 , and the angles of attack of which 
 are a ly 2 . The differences in this case (and the argument 
 applies to any number of planes) may be conveniently 
 exhibited by the following artifice : 
 
 Construct the vector sum of two vectors (I l9 2aj) and 
 
144 STRAIGHT PLANES AND FINS CHAP. 
 
 (/ 2 , 2a 2 ) and let it be (/, 2a). Then in the formulae for the 
 resistance derivatives we have 
 
 Jj cos 2 a : + I 2 cos 2 a 2 = ^(I I + I 2 - I) + I cos 2 a 1 
 
 /! sin 2 ai + I 2 sin 2 a 2 = i(I x + 7 2 - I) + I sin 2 a V . (160) 
 
 I I sin aj cos H! + J 2 sul a 2 cos a 2 == ^ g i n a cos a - 
 
 It follows that the values of L q and M p are Ae same as 
 for the single plane (I, a), wA&7e Ae values o/L p a^cZ M q 
 are increased by the same amount as they ivould be if the 
 moments of inertia M l5 M 2 of the fins were each increased 
 
 The effect will in general be to increase the stability 
 particularly in such arrangements as those of Cases I and 
 II above. Of course, if aj a 2 is small, I I + / 2 / will be 
 small, and generally of the second order in a x a. 2 . The 
 increase of stability will depend mainly on the equivalent 
 increase in M 2 , in view of the fact that the value of M l 
 has little effect on the stability. 
 
 Effects of structural and head resistances 
 and twin screws. 
 
 91. As in 57, the most general effect of air resistances 
 acting on the framework, tangential forces such as friction 
 on the aeroplanes, and other resistances, will be to introduce 
 additional terms into the resistance derivatives which we 
 shall denote by accented letters (Z f , L', M f ) W)p>q . In an 
 ordinary flying machine some of these terms will probably 
 be small or unimportant, and it will probably be only 
 possible to make rough and ready estimates of their 
 value. In this connection, experiments on models afford 
 the most desirable tests. 
 
 When the machine is turning about the axis of y with 
 angular velocity q, the parts of the framework which are 
 moving with the greatest velocity encounter the greatest 
 resistance, and this circumstance, as well as the resistances 
 on the fuselage and framework supporting the tail, give 
 
vri LATERAL STABILITY 
 
 J 4S 
 
 rise to the positive coefficient M v This coefficient divided 
 by KU must be added to the determinant (143) of 79 
 in the same term as the moment of inertia M 2 of the fins. 
 The effect of the moment of the resistances is thus 
 equivalent to an increase in M 2 , and from what has 
 already been shown this increase generally increases the 
 stability. 
 
 A similar effect occurs in a machine furnished with 
 twin screws (like the Wright machine), arising from the 
 fact that in turning round, the outer propeller moves 
 through the air faster than the inner one. If the one 
 that is moving quickest exerts the least thrust, as 
 assumed in 57 (6), the effect will be to produce a further 
 couple tending against the rotation q and further 
 increasing the value of M' q . 
 
 From elementary considerations, it should be pretty 
 obvious (without further discussion) that if the speed U 
 of the machine is not largely in excess of that required 
 for maximum efficiency, so that the head resistance is 
 not large compared with the drift (see 57 a), the 
 value of M\ will not be large with ZKUI&m* a or 
 ZKUI'jj?. In a reasonably designed machine, L f p will 
 be of the same order of magnitude, and will thus be 
 small compared with KUIcos 2 a or KUI r . This property 
 is equivalent to the obvious statement that if the machine 
 receives a small rotation p about the axis of x, the 
 moment of the air resistances on the framework is small 
 compared with the moment of the air resistances on the 
 main planes. It follows that the coefficient L f p has 
 generally a negligible effect on the stability. The same 
 thing is evident if we notice that the presence of this 
 added term in the middle term of the determinant (143) 
 is equivalent to an increase in the moment of inertia M l9 
 which has but little effect on the stability. 
 
 As regards the terms M' p and L' q , these will probably 
 either vanish or be very small even in comparison with 
 
 L 
 
146 STRAIGHT PLANES AND FINS CHAP. 
 
 L' p and M' q in the case of a well-designed flying machine. 
 At the most, they will be comparable with ^KUI^ and 
 small compared with the corresponding terms KUIp 
 dependent on the main planes. If they are equal to 
 one another their effect will be equivalent to a change in 
 the product of inertia P of the fins. In any case, there 
 would be no difficulty in taking account of them should 
 the necessity arise, but this appears hardly probable. 
 
 It will not do rashly to neglect effects of resistance 
 represented by Z' w , Z f p , Z f q , L f w M' w . These include the 
 effects of lateral resistance of the fuselage and framework. 
 For a machine with a low centre of gravity, Z' p and 
 L' w will evidently be important ; for a machine with a 
 long tail, Z f q and M' w will be important. The terms must 
 not be neglected, because they occur in the small terms of 
 the determinantal equation for X, and they are only 
 associated with terms depending on the fins which are 
 also in general small. If the new coefficients satisfy the 
 reciprocal relations Z' P = U W and 'Z' q = M f w , their effects 
 can be represented by means of vertical fins. For 
 instance, Z' p and L' w are then equivalent to an increase 
 Z f p jKU in the value of ' Tx, and, if positive, have thus 
 the same effect as shifting the centre of pressure of the 
 fins forward, and so on. This interpretation may perhaps 
 be more useful than a direct calculation of the effects 
 of these resistances. At the same time, when the deter- 
 minantal equation is developed and the coefficients 
 are expanded, it is sufficient for a first approximation to 
 retain only the terms of lowest significant order in the 
 resistance derivatives in question, as done in the case 
 of fins in (145). 
 
 Probably the reciprocal relations Z' p = L' w and Z ' q = M' w 
 are satisfied in almost every case, but though further 
 discussion thus becomes rather uninteresting, it may be 
 pointed out that if these relations are not satisfied it 
 is still possible to represent Z' w , L ' w , M f w by suitable 
 
vii LATERAL STABILITY 147 
 
 increases in T, Ty, Tx, and to allow for the difference 
 by writing T(y + vi) and T(x + g) for Ty and Tx in the 
 second and third columns of the first row of the deter- 
 minant (143). The only important additional term 
 introduced into the coefficients of the biquadratic is a 
 term Tx% I' in the value of ^/K B U^g 3 , other changes 
 being negligible for a first approximation. 
 
 There is one other way in which the propeller may 
 affect stability apart from gyrostatic action. The torque 
 produced by the propeller affects the lateral equilibrium, 
 and this, of course, indirectly influences stability, but it 
 is further possible that when the aeroplane possesses a 
 small angular velocity p about the axis of x, the propeller 
 torque is slightly altered, being increased when the 
 aeroplane rotates in the same direction as the propeller, 
 decreased when it rotates in the opposite direction. This 
 effect would be represented by an additional positive term 
 added to L p . We have seen, however, that such an 
 addition would in general have little effect on the 
 stability. . 
 
 Effects of friction and curvature of main 
 surfaces. 
 
 92. Making use of the property of narrow planes 
 assumed in 76, we may easily deduce the corrections 
 analogous to those of 69 applicable to lateral stability. 
 In the first place, we observe that if the angle of attack a 
 (defined by R = KSU* sin a) is not the same as the angle 
 d (defined by the property that R makes an angle 90 a 
 with the axis of x), then supposing d to remain constant. 
 we write I f I cos a cos a', /*, = tan a, // tan a, and the 
 resistance derivatives become 
 
 L p = KUI f , L q = - ZKUI'n, M, v = - KUI'p.', M Q = 
 
 (161) 
 L 2 
 
148 STRAIGHT PLANES AND FINS CHAP. 
 
 The effects of concavity of the main surface may be 
 investigated by applying the considerations of 71 to a 
 strip dS of the main plane, the legitimacy of this step 
 being implied in the definition of narrow planes. If the 
 machine receives an angular velocity p, about the axis of 
 x, and we consider an element dS, distant z from that 
 axis, the change in the angle of attack is da = zpj U, the 
 centre of pressure of dS shifts forward through a distance 
 a<j)'(a)da, and c being the radius of curvature, the direc- 
 tion of R on the element changes by an amount 
 da! a<f)'(a)da/c, the resolved part of R along the axis of x 
 changes by an amount R cos a da, and its moment about 
 Oy by Rz cos a da. The final result is to add to M p a 
 term 
 
 = / KUa/c(f>'(a) sin a cos a' z 2 dS . . . (162) 
 
 If the plane be rectangular and of constant curvature 
 throughout its span, the corrected expression for M p 
 becomes 
 
 M p = - KUTtf - i*p(a)alc} . . . . (163) 
 
 There is a small corresponding change in L p , which now 
 becomes 
 
 /c} .... (164) 
 
 but it will be seen that with n and /*' small this correction 
 is negligible, the correction in M p being alone important. 
 These modified forms are only based on the assumption 
 that a^(a)/c is constant throughout the span of the planes, 
 so that this factor can be removed outside the sign of in- 
 tegration. This would still be true if the planes were not 
 rectangular, provided that all sections parallel to the plane 
 of x, y formed circular arcs of the same angle. It must be 
 remembered that if the centre of pressure moves forward 
 as the angle of attack decreases, <'(a) is negative. 
 
 The correction does not affect the value of S3 when 
 jF=0, and under the same condition the effect on 6 
 
vir LATERAL STABILITY 149 
 
 is probably not very important. The most important 
 effects are in $) and the part of ( involving sin 6. The 
 portion of (g involving cos 6 will be unaffected. 
 
 Corrections for aspect ratio, for effects at 
 "extremities of main planes and for wash. 
 
 93. In order to simplify the algebra, we have in general 
 assumed that the resistance coefficient K is the same for all 
 the planes under consideration. As a matter of fact, how- 
 ever, K is a function of the aspect ratio, as well as of the 
 camber, and, therefore, in dealing with different surfaces 
 Si, S. 2 , we ought to have retained KiS i9 K 2 S 2 , where 
 we have written KSi and KS. 2 . It will be seen, however, 
 that the algebra is quite sufficiently complicated as it 
 stands, and the same results may be arrived at by waiting 
 till the final formulae have been obtained, and then 
 multiplying each area by a coefficient depending on 
 the aspect ratio. This is equivalent to reducing all the 
 areas to the same aspect ratio, or, more strictly, to the same 
 resistance coefficient K. The results will hold good pro- 
 vided that the " reduced areas " so obtained be substituted 
 for the actual areas of the planes in the formulae. 
 
 In connection with lateral stability a correction is 
 required owing to the fact that the conditions assumed for 
 narrow planes at small angles do not hold good at the 
 extremities of the main planes where the stream lines 
 flow over the tips instead of being, at any rate, approxi- 
 mately two-dimensional. This leads to a correction in 
 the value of / or I' dependent on the forms of the planes at 
 their extremities, and, therefore, to the substitution of a 
 corrected constant for the radius of gyration implied in 
 / and /'. According to Lanchester, two such constants 
 are required, which he calls the " aerodynamic and aero- 
 dromic radii." If this is the case, one will occur in Z^and 
 
I 5 
 
 STRAIGHT PLANES AND FINS CH. vn 
 
 M p , the other in L q and M q . If we denote by I\ and I\ 
 the corresponding values of /' corrected in accordance 
 with this hypothesis, and if we apply the results to the 
 approximate values given by (145) 79, we find that 
 I\ (depending on the aerodynamic radius) .replaces 1' 
 everywhere, except in the part of ( involving cos 0, 
 where / 2 occurs. In (g we must replace T (2/j, cos 6 si n 6) 
 by 7' 2 2ju cos 6 1\ sin 0, and assuming 9 to be small, the 
 effect of the difference between F 2 and I\ is equivalent 
 to an increase in the angle which the flight path makes 
 with the horizon from 6 to & ', where 
 
 ff = e + W. - A) (165) 
 
 * 1 
 
 If Lanchester's aerodromic and aerodynamic radii are 
 equal, this effect vanishes, and the only change required is 
 in the assumed value of T. 
 
 The correction for "wash" when two fins are placed 
 one behind the other at the same level should give little 
 difficulty, if it is needed ; but the effect could be avoided by 
 placing the fins at slightly different levels. 
 
CHAPTER VIII. 
 
 LATERAL STABILITY BENT UP PLANES. 
 
 Comparison of bent up planes with vertical fins. 
 
 94. We now consider such cases as that of a sup- 
 porting surface consisting of a pair of wings bent up 
 at a dihedral angle, or bent up at their tips. In all such 
 cases the general expressions (136) of 76 must be 
 adopted as the starting point, and the first point which 
 
 presents itself 'for inquiry is how far the effects of bending 
 up are equivalent, in their effect on lateral stability, to the 
 addition of vertical fins. 
 
 Consider a pair of elements dS of such planes situated 
 symmetrically at R, R f on opposite sides of the plane of 
 
 151 
 
152 BENT UP PLANES CHAP. 
 
 (x, y) 9 and let the normals to these elements meet the 
 plane of (x, y) in a point J (Fig. 35). Then in every 
 circumstance the resultant pressures on the two elements 
 will pass through J, and will be equivalent to a single 
 force acting through J. The only part of this force which 
 will affect the lateral stability will be its component perpen- 
 dicular to the plane of (x, y) 9 and thus it might appear 
 that the elements were equivalent to a single vertical fin 
 of suitable area placed at the point J. This, however, is 
 
 FIG. VIII. A MONOPLANE IN CONSTRUCTION BY BLERIOT IN 1907. 
 
 An aeroplane, of the " Antoinette" type, in the course of construction, showing 
 the bent up wings and vertical tail plane in the rear as also the arrangement 
 of the engines, the propeller in front being detached. The photograph was 
 taken by the author in July, 1907, on a visit to Bleriot's factory outside the 
 Porte Maillot, Paris, with the late Captain Ferber, before aeroplanes had 
 flown or Bleriot had become well known. 
 
 not strictly the case as the velocities of R and R f are in 
 general different from that of J. 
 
 If f, 77, be the co-ordinates of J, the equation of the 
 normal gives 
 
 ^-&- -^ (166) 
 
 whence Iz nx = n%, ny mz = n?) ; and therefore 
 
 2lz-nx = - 
 
viu LATERAL STABILITY 153 
 
 Substituting in the formula of 76 we get 
 
 = In*d8, ^j = triads, Z<1 
 
 KU J KU J KU 
 
 Lv_ 
 
 KU 
 
 A. .. /^.o _AL ___ fnWdS, 
 
 -TT-jj = ~ I 't-'gu/Aj, JTTT ~ ] '" S'/"^ _7T77 ~r I ' 5\"fe ~" SSJWo 
 
 (167) 
 
 The effect of the element cS is thus equivalent to that 
 of an element n 2 dS in the plane z = at the point (f , 77, 0) 
 in the terms of the first two columns, but in the third 
 column this is no longer the case, the factor 2 x occurring 
 in place of f. The equivalence will be exact in the case 
 when = #. This requires that Z = 0, i.e. that the 
 element should be placed parallel to the axis of x, and 
 therefore in a neutral direction to the line of flight, in 
 which case it will exert no lifting force in steady flight. 
 The simplest application of this property consists in 
 replacing a vertical fin placed above the main planes by 
 two small bent up planes situated at the extremities of 
 these planes, sloping up at an angle j3 to the horizon, and 
 parallel to the line of flight. In this case, we shall have 
 I = o, m = cos /3, n = sin j3, and if a is the area of either 
 plane, the two will be equivalent to a single fin of area 
 2<r sin 2 @ placed at a height z cot j3 above them. Such a 
 pair of terminal planes may be called " stabilizers," the 
 term being used whether they are neutral to the line of 
 flight or not. 
 
 A pair of tandem planes each fitted with neutral 
 stabilizers will thus be equivalent to a system with two 
 raised vertical fins, thus affording a convenient substitute 
 for the system of 84, which possesses such wide limits 
 of stability. The substitution of stabilizers for fins, of 
 course, presents obvious advantages not directly connected 
 with the present investigation. 
 
 95. In the general case, it is convenient to put 
 
 I = sin a m = cos a cos 3 n = cos a sin /3 . . (168) 
 
154 BENT UP PLANES CHAP. 
 
 In this case, a is the angle of attack or angle made by 
 the element with the direction of steady flight, ft the 
 angle through which the element is bent up about an axis 
 parallel to the axis of x. We shall also, in accordance 
 with our usual convention, write 
 
 dS' = dS cos 2 a, /i = tan a 
 
 and we get 
 
 ny - mz = cos a (y sin - z cos /3) 
 
 Iz - nx = cos a (pz - x sin j8) 
 2lz nx = cos a (2p.z - x sin /3), 
 
 and writing f =z cos y sin /3, the expressions for the 
 derivatives become 
 
 r z n r z 
 
 = / sin 2 j8 dtf, fcFTr = - / sin P &&> Wi 
 
 J KU J K( 
 
 - I sin jS dS', WT7 ' 
 
 J 
 
 - a; sin 3) (^ - x sin /3)d8T 
 (169) 
 
 These expressions and the subsequent analysis become 
 considerably simplified when x sin @ is negligible com- 
 pared with fjuz. 
 
 This occurs (l) in the case of a single lifting system 
 with bent up wings in which x = 0, (2) in the case of a 
 tandem system of bent up lifting planes, when the angle 
 of bending up @ is small compared with the angle of 
 attack a, and the span of the planes is not small compared 
 with the distance between them. 
 
 Single lifting plane with terminal stabilizers. 
 
 96. The following solution, due to Mr. Harper, applies to 
 the case of a single straight lifting surface S, furnished at 
 the ends with a pair of small stabilizers of total area T 
 
VIII 
 
 LATERAL STABILITY 
 
 '55 
 
 (each being of area J T), placed with their centres in the 
 plane x = 0, at the points R(Q,y, z) and R' (0, y t 2;), their 
 angles of attack being a and the stabilizers being bent up 
 at an angle /#. In these circumstances, the traces of 
 the stabilizers in the plane x = will make angles j3 
 with the axis of z, and if the direction of the plane at R 
 
 FIG. 36. 
 
 be produced to meet Gy in P, and GQ be drawn 
 perpendicular on RP (Fig. 36), we shall have 
 
 EQ = z cos j8 - i/ sin = 
 
 with the notation of the last article. The angle # must 
 obviously not be too small if the stabilizers are to be 
 efficient. 
 
 We shall, as usual, employ the notation T cos 2 a = T' , 
 // = tan a, which greatly simplifies the analysis, and 
 further to shorten matters, we shall assume, in the first 
 instance, that K is the same for and T, it being 
 distinctly understood that differences in this respect are to 
 be corrected for in any future formulae by multiplying T 
 or T in the final result by the ratio K'jK of the two 
 coefficients, and that this correction may, in certain 
 circumstances, be considerable. It only remains to write 
 fjfjT f for fjLdSf and x = Q in the results of the last article, 
 to remove the sign of integration, and to add the results 
 to those of 76 due to the lifting plane. 
 
156 BENT UP PLANES 
 
 Thus the equation in X becomes 
 
 CHAP 
 
 
 W\ T , . 2 Wcos0fKVcj\ 
 
 XT TTQ ' ^ [1 P) T^ TT-> 1 ^ I 
 
 z sin ft, 
 
 and the coefficients are given by 
 21 = W(AB - F 2 ) . 
 
 W W sin 6(KUg 
 Kg - -K^m\\ 
 
 + 2T'fi'z sin 
 
 = 0. 
 
 (170) 
 
 (171) 
 
 B(T 
 
 2) 
 
 2 
 
 W 
 ^ 
 
 W 
 
 ^jt 
 
 TI ' sin 
 
 B) 
 
 sin 0(4*'* - FO ..... (171c) 
 
 - F)sm0 - (Fp'z - E0 cos^] 
 
 cos ~ sn 
 
 The condition of stability ( positive in the case of 
 horizontal flight requires that f shall be positive and 
 greater than //z//*, that is, in the figure, 
 
 RS tan a < EQ tan a, ..... (172) 
 
 and stability necessarily ceases when the angle of descent 
 passes the limit given by 
 
 tan 6 = 2 tan a ...... (173) 
 
 The condition RQ positive is satisfied if the normals to 
 R and R> meet above the centre of gravity, not below. 
 
 The above condition makes the first term of *) negative. 
 In the case of = and F=0, the condition 5) positive 
 leads to 
 
 T> 
 
 + 
 
 that is, 
 
 a' > 
 
 / 
 
 .Re 
 
 tail a - 
 
vin LATERAL STABILITY 157 
 
 If 
 
 > I'KU* tan , 
 
 that is, 
 
 R > KlT 2 Icos a sin a . . . . (175) 
 
 this condition (174) is satisfied for all positive values of a 
 (RQ being positive) and for d = 0, in which case the 
 planes would be neutral. 
 
 Now, since by the condition of equilibrium 
 
 W = KU 2 S sin a cos a + KU*T sin a' cos a' cos |8, 
 
 it follows that when T is small compared with S this con- 
 dition may be written approximately, 
 
 and that this condition is sufficient even if T is not 
 neglected. This requires that the radius of gyration of 
 the mass of the aeroplane about the axis of j should 
 exceed that of the areas of its main planes about the axis 
 of x. 
 
 If this condition is not satisfied, stability cannot be 
 secured with the stabilizers in the neutral position, and 
 the conditions of stability may be written 
 
 ES tan a .. , , .. sq. of rad. of gyration of mass about Oy 
 EQ tana sq. of rad. of gyration of S about Ox 
 
 If the stabilizers are attached at the same level as the 
 centre of gravity (S and G coinciding), RQ = RS cos /5, 
 and RQ must therefore < RS and a < a. 
 
 It is further to be observed that by making the product 
 of inertia F negative, both (5 and 3) can be increased, a 
 condition favourable to increased stability. 
 
 97. The further condition necessary for stability <> 
 is somewhat complicated, and we shall only consider the 
 approximate form when both T and p are small, and 
 F= 0, = 0, // being also small from condition (172). In 
 this case 
 
 IS T sin 2 j8- 
 
 Wff~ ?B approximates to -- w~~ ' 
 
158 "ANTOINETTE" TYPE CHAP. 
 
 while the term 9l$) 2 becomes negligible, and the condition, 
 which may be written 8/93X8/D leads to 
 
 **w BC , 
 
 rl^p < I'M - p'z) - 
 
 which becomes 
 
 that is, 
 
 j) f?f) 
 
 EQ tan n ES tan a' < / = 
 
 jrj7/ 2 sin a cos a cot a\ . (178) 
 
 V^cos^a^sin 2 ^ 8 ) 
 
 (assuming the approximate condition of equilibrium). 
 
 It will be observed that the system here considered does 
 not include a tail plane. The addition of this would 
 further complicate the algebra, and the case may perhaps 
 stand, over for future investigation in view of the 
 discussion of the analogous problem of the next article. 
 The possibility of satisfying the conditions of stability with 
 a negative offers another interesting question, especially 
 in view of published accounts of the Dunne biplane. 
 
 Theory of the Antoinette type. 
 
 98. The following investigation is due to Mr. E. H. 
 Harper. By the " Antoinette " type is here meant a 
 machine supported by a single-lifting pair of bent up 
 wings, and provided with a vertical tail or rudder fin. 
 It is stated that in such arrangements it has been found 
 necessary to place the centre of gravity G lower than the 
 intersection P of the wings, and we shall assume that this 
 is done, the tail being, however, on a level with the centre 
 of gravity. We are not concerned with the effect of a 
 horizontal rudder, which only affects the longitudinal 
 stability. 
 
VIII 
 
 LATERAL STABILITY 
 
 Taking Fig. 37 as representing the section of the wings 
 in the plane of (y, z), R any point on the wing on the 
 positive side of the plane of (x, y), GQ a perpendicular on 
 the wing AP produced, let RP = p, PQ = e, so that 
 
 RQ = z cos j8 - y sin j8 ^- p + e, EP = z sec /3 = p. 
 
 Also we suppose the rudder of area T placed at (x, 0, 0), 
 x being negative if the rudder is behind. 
 
 W\ 
 
 KUg 
 
 With this notation the determinantal equation becomes 
 
 W cos 6(KUg\ 
 K*U*v \ A ) 
 
 - f(p 
 
 + e) sin 
 
 + 2* f pp sin /3 cos /3dS' - xT, 
 
 ' 
 
 - 2 l 
 
 ) cos 
 
 + 
 
 + e) cos fidS 1 , + 2 L 2 p 2 cos 2 j8dfiT, 
 
 = . . (179) 
 
 Assuming the hypotheses involved in the definition of 
 narrow planes, these expressions would hold good even if 
 the angles of attack and of bending up a and should 
 vary for different parts of the wing, as in the wings of 
 birds, provided p and e as well as a and j3 are treated as 
 varying at different distances from the plane of (x, y) and 
 are properly defined, p not being the distance of R from P, 
 
" ANTOINETTE " TYPE 
 
 CHAP. 
 
 but the length of the tangent at R measured to the plane 
 2 = 0. Except in the case of numerical calculations, 
 however, the investigation would probably be far too 
 complicated to lead to any simple results. We therefore 
 consider only the case where a, /3, e are constant. In this 
 case we may write 
 
 f 
 
 (p 
 
 = S'[(p 
 
 Photo.] 
 
 [The Sport and General Illustrations Co. 
 
 FIG. IX. LATHAM'S TRIAL FLIGHT. 
 
 An example of the " Antoinette " type in which lateral stability is obtainable by 
 means of (i) bent up wings, (ii) centre of gravity below the dihedral angle 
 formed by the wings (iii) a vertical tail fin placed at not less than a certain 
 limiting distance behind the main planes ( 98-101). For longitudinal 
 stability the aeroplane may probably be regarded as a " single lifting 
 system with neutral tail," though it may be desirable to apply corrections 
 for displacement of centre of pressure and camber. 
 
 where p refers to the centroid of the positive wing, and K 
 is its radius of gyration about that point. It will easily be 
 seen that 
 
 and the other minors of the determinant all simplify in 
 
viii LATERAL STABILITY 161 
 
 the same way. With this simplification alone, the other 
 terms being left as integrals, the coefficients of the biquad- 
 ratic become 
 
 51 = W(AB - F*) . . . ; .' . . . . (18Qa) 
 -. W {2Ap* Jp 2 cos 2 (US' - 3 fy Jp(p + f ) cos ftdtf + sf(p + 
 
 + (AB - F*)S' sin 2 ft. + T{(AB - F*) + WA. (1806) 
 [(2Ap? cos 2 ft - SFfjL cos 3 + B) sin 2 ft + 2JT 2 " 22 
 
 sin 
 
 + T[ Wx z l (p + i*dS' + A J(2p V cos 2 ^ + 3/zpJc sin /3 cos 3 
 
 r TF" 
 
 - Fj (3/ip cos ^ + 2x sin /3) (p + f )d& + B \ (p + )*dS' - -%- Ax] 
 
 (180c) 
 2) W 
 
 W 
 
 - S^/zsin^cos/S + ai s sffi*/3)#V - ^- 
 
 *s 
 
 W 
 
 cos 
 
 pdS' + (Bco$0 - 
 
 r 
 / (p 
 
 J 
 
 cos B - A sin 0;] (180d) 
 
 cos ^ cos " sin ^^'^V sin ^ cos /3 
 
 / r 
 
 cos 6 sin /3 / (p + e)d' + 2^ cos 6 cos |3 / p(p + e)d' 
 
 (180e) 
 
 99. CASE I. Let T=0 (no rudder tail). 
 
 In 5t, 23, (, all the terms are positive except some 
 containing first powers of F. These vanish when F=0, 
 and, moreover, the terms underlined twice in F are small 
 compared with other terms in the coefficients, so that the 
 value of F has little effect on 33. On the other hand a 
 negative value of F will certainly increase both (5 and 
 <), and will consequently be favourable to stability. We 
 shall, however, here suppose that F=0. 
 
 To make (S positive when 6 = 0, we must have e sin ft 
 positive ; the condition 2) positive then requires that sin ft 
 should be positive, hence e must also be positive, that is, 
 
 M 
 
1 62 "ANTOINETTE" TYPE CHAP. 
 
 the wings must lend upwards, not downwards, and 
 their intersection must be above the centre of gravity, 
 not below. 
 
 The range of inclination of flight path consistent with 
 stability will be limited by the condition (from d > 0) 
 
 tan B < 2 tan a cos /3. 
 
 The condition <) positive becomes with F= 
 
 JBcos 6 + Au. cos 8 sin rB cos & -i 
 
 ~~KU* ~ P + \ in7 2 ~ " sin a cos cos > (is*) 
 If 
 
 ^^ > S* 2 sin a cos a cos ^ . . . (182) 
 
 the coefficient of e will be positive and the condition 
 <) >0 will be satisfied for all values of e. 
 This condition can be written 
 
 B cos & > KU 2 S sin a cos a cos /3/c 2 
 
 that is from the conditions of equilibrium 
 
 B > WK Z . . . . . . (182a) 
 
 or radius of gyration of mass about OT/ 
 
 > radius of gyration of each plane about its centroid, 
 in every case, or for rectangular planes of span about 2b 
 
 * B > Wtf/12 . . . . . (1826) 
 
 If, however, .this condition is not satisfied, there is a 
 superior limit to the value of e beyond which the con- 
 dition 5) > fails. 
 
 The condition < > 0, for which the approximation 
 (<) (E S 3 > is probably sufficient in most cases, imposes 
 a further limitation on the value of e. It is obvious that 
 Es (33 is positive when (1 = 0, which happens if = 
 and is negative if 2) = 0. The conditions may be satisfiable 
 for all values of by making B WK? sufficiently large, 
 otherwise the value of e cannot exceed a certain limit e 2 
 less than the limit e l determined by the condition $) > 0. 
 
 The general conclusion is that while the condition of 
 
viii LATERAL STABILITY 163 
 
 stability ( > requires the centre of gravity to be below 
 the dihedral angle, too low a position may lead to the failure 
 of the condition > 0. The effect of this failure would 
 be that the modulus of decay of the oscillations would 
 become negative, and the oscillations would therefore 
 increase in amplitude. This conclusion is entirely in 
 accordance with the results of experience as stated in 
 aeronautical books and journals, although, owing to the 
 somewhat ill-defined notions that commonly prevail 
 regarding the meaning of stability, the production of 
 dangerous oscillations has up till now not been always 
 attributed to its probable cause, namely, inherent lateral 
 instability. 
 
 100. CASE 1 1. Let = 0, stability being due to the rudder 
 T. Let x be negative (rudder behind centre of gravity, & 
 positive. 
 
 In this case all the terms in 2(, S3, 6, ), with the 
 exception of a few containing F, are positive. To make 
 the part containing T in @ positive in the case of = 0, 
 it is necessary to make 
 
 2 tan a f p(p + t)dS' 
 
 (.-.-) > - -4 .... (183) 
 
 tan/3/(p + f)dS' 
 
 so that, assuming as we do that e = 0, and supposing the 
 wings rectangular and of span 26, we must have 
 
 46 tan a 
 
 that is, the rudder, if placed behind the centre of gravity, 
 must be J at a distance behind greater than J b tan a cot ft 
 in order to ensure stability. 
 
 With this arrangement there is usually not much diffi- 
 culty in securing the additional condition of stability 
 $ > 0. For in (180) when T and ft are small ^jKUg 
 reduces to the singly-underlined part W Bf (p + e) 2 c', and 
 the positive part of (, also underlined, when multiplied 
 
 M 2 
 
1 64 "ANTOINETTE" TYPE CH.VIII 
 
 by this, cancels out with the product of the two singly- 
 underlined terms in (5). The rest of 62) ($8 is essentially 
 positive, at least with this approximate value of $8. 
 
 101. It will thus be observed that raising the wings 
 above the centre of gravity is favourable to the condition ( 
 positive, but the possibilities in this direction are limited 
 by the conditions 2) and positive. On the other hand 
 the addition of a tail plane is favourable to the latter 
 conditions, but there is an inferior limit to the length 
 imposed by the former. A combination of the two may 
 evidently be advantageous. Another arrangement of con- 
 siderable interest is that in which two rudder planes are 
 placed one in front of and one behind the main planes so 
 as to make 2jT# = 0, and in which, moreover, e = 0. In 
 this case the conditions 2) > and ( > will certainly be 
 satisfied. 
 
 Range of stability. So far as e is concerned, ( 
 diminishes when the aeroplane is descending, while so 
 far as T is concerned, ( diminishes when ascending. By 
 a suitable combination of raised planes and rudder the 
 two effects may be made to counteract each other. 
 
 In (, the part containing sin 6 is 
 
 - Tx 
 
 ((p + e)W] sin 6 . (184) 
 
 and this will vanish if 
 
 T( - x) = (S sin a cos a sin )8 cos /3 /- + f \ 2 + K 2 ( 185 ) 
 
 giving for rectangular planes and e small, 
 
 T( - a?) = J c8 sin a cos a sin /3 cos /3 (185a) 
 
 In this case ( will be independent of 6, except for the 
 factor cos 6. If the coefficient of sin 0, instead of being 
 zero, is made small, the range of inclination consistent 
 with stability will still be correspondingly extended. 
 
CHAPTER IX. 
 
 GENERAL CONCLUSIONS. 
 
 102. It may, I suppose, be now taken as proved that 
 an aeroplane can be constructed which possesses inherent 
 stability, both longitudinal and lateral, when propelled 
 horizontally, and also when the inclination of the flight- 
 path to the horizon does not exceed certain determinate 
 limits. Further, these limits can be extended in various 
 different ways, and the limitations regarding the angle of 
 descent can be altogether removed. It will be evident, 
 however, that an aeroplane constructed on these lines 
 differs in certain points of detail from those adopted by 
 many of the most successful of present-day aviators ; 
 indeed, it does not appear to be denied that some of these 
 are unstable. Thus arises a divergence of opinion regard- 
 ing the advantage or otherwise of inherent stability, and 
 all that can be done now is to state the case for the side 
 which is supported by the present investigation. 
 
 In the case of longitudinal stability, there is probably 
 very little need now to advocate the claims of auxiliary 
 surfaces, for the use of these, has by this time become 
 almost, if not quite, universally recognised. Practically 
 every modern aeroplane is provided with tail planes, and 
 many are provided with both front and rear " controls," 
 an arrangement which, as is here shown, gives increased 
 longitudinal stability, and in this connection is especially 
 useful in ascending. The original Wright biplane was 
 
 165 
 
1 66 GENERAL CONCLUSIONS CHAP. 
 
 provided only with auxiliary planes in front, and this 
 arrangement could only be made longitudinally stable by 
 throwing the weight forward and inclining these rudder 
 planes at a steeper angle to the line of flight than the 
 main supporting planes, an arrangement which would 
 certainly give increased " drift " or resistance in the 
 direction of motion. Since then a stabilizing tail plane 
 has been added in the machines put on the market. 
 
 It is in connection with lateral stability, however, that 
 different counsels prevail, some authorities considering 
 it preferable to avoid the use of bent-up planes, to reduce 
 vertical auxiliary surfaces to the minimum necessary for 
 steering, and, in short, to abandon inherent lateral 
 stability altogether. Among the reasons given for this 
 course it is stated that (l) devices such as fins or bent-up 
 planes are very uncertain in their action ; (2) they are 
 liable to set up oscillations ; (3) they may cause serious 
 rolling when the aeroplane is suddenly struck by a side 
 wind. 
 
 Now it will be seen that the problem of lateral stability 
 is one of considerable complexity, especially when bent-up 
 surfaces are concerned. It is easy to see how an attempt 
 to deal with the matter by trial might lead to uncertain 
 results, the significance of which it would be difficult to 
 understand, without some kind of theory to act as a guide. 
 In attempting to avoid one condition of instability, we 
 run the risk of introducing another. If a single vertical 
 rudder only is provided, it introduces one kind of 
 instability when in front and another when behind. 
 
 The production of oscillations indicates that the 
 aeroplane is dynamically unstable ; in other words, that 
 the arrangement instead of producing stability has had 
 the reverse effect. We may illustrate this point in 
 connection with the Antoinette type. 
 
 It is stated that the bent up wings necessitate a low 
 centre of gravity, but if this is put too low oscillations 
 
ix GENERAL CONCLUSIONS 167 
 
 are set up. This is entirely in accordance with Mr. 
 Harper's investigations given here, which show that one 
 condition of stability can be satisfied by placing the centre 
 of gravity below the dihedral angle formed by the wings, 
 but that another condition sometimes requires that its 
 depth should be less than a certain limit. By placing the 
 centre of gravity too far below the dihedral angle the 
 machine may again become unstable, and the kind of 
 dynamical instability that is indicated by the conditions is 
 that in which oscillations of increasing amplitude are 
 set up. 
 
 103. The behaviour of an aeroplane when struck by 
 gusts of wind depends to a very great extent on its 
 inherent dynamical stability. 
 
 A sudden gust of wind which quickly subsides will 
 disturb the motion of the aeroplane, and the important 
 requisite is that it shall quickly settle down into its 
 original steady motion. For this to happen it is essential 
 that the state of steady motion in question shall be 
 dynamically stable ; if this is not the case the aeroplane 
 will either deviate further from the equilibrium position, 
 or oscillations will be set up which will continually 
 increase. 
 
 A permanent change of wind velocity, say V, will 
 cause an alteration in the conditions of equilibrium, and 
 the aeroplane will have to acquire the steady motion con- 
 sistent with the new conditions. If the relative velocity 
 of the aeroplane when moving steadily is 17, the aeroplane 
 will in future have to move with a relative velocity U 
 and therefore a resultant velocity compounded of Z7and V. 
 This again involves the condition that the new steady 
 motion shall be stable. The subsequent problem is just 
 the same as if the aeroplane had remained in still air, but 
 had from some cause or other been jerked in such a way 
 as to impart an initial velocity to it equal and opposite 
 to V. 
 
1 68 GENERAL CONCLUSIONS CHAP. 
 
 Periodic gusts of wind would, of course, give rise to 
 forced oscillations, thus presenting a further problem for 
 the mathematician, as we have only here considered free 
 oscillations. If the period of the gusts happened to 
 coincide with a period of free oscillation, a swaying 
 motion would be set up which might become dangerous. 
 Now the theory of forced oscillations shows that when 
 the free oscillation is a damped oscillation, the forced 
 oscillations do not in these cases increase beyond a certain 
 limit. The damping of the oscillations thus becomes 
 important, and this is secured by giving the aeroplane 
 adequate dynamical stability. 
 
 104. One difficulty still remains, and shows the need of 
 making the motion of aeroplanes the subject of further 
 mathematical investigations than have been possible in the 
 limits of this book. The investigation of stability may, from 
 what has been said above, be fairly well sufficient when we 
 have to deal with the effects of light gusts of wind where 
 the change of wind velocity V is small compared with the 
 velocity U of the aeroplane itself. 
 
 It is, however, important that investigations should be 
 made of the dynamical effects of a sudden squall of wind 
 of the most violent character that an aeroplane is liable 
 to encounter in the course of its flight. A further 
 feature of such an investigation should be the calculation 
 of the stresses set up in the framework of the aeroplane 
 by the wind pressures. The problem is one of the many 
 unsolved problems awaiting solution at the hands of 
 mathematicians, of which a list is given at the end. 
 
 The important condition to be satisfied in such a case 
 is that the motions initially imparted to the aeroplane 
 shall not be of such a character as to render the machine 
 unmanageable. In this respect, I suppose, we may take 
 it for granted that rotations are the most objectionable 
 motions, and that it is essential that these shall be small. 
 If this be assumed, it is probably an easy matter to 
 
ix GENERAL CONCLUSIONS 169 
 
 reconcile this condition with the conditions of inherent 
 stability. 
 
 Thus if an aeroplane is fixed with a vertical rudder 
 plane in the rear, a side gust of wind will cause it to 
 swerve round and face the wind. If it has the vertical 
 rudder in front, it will turn away from the wind. If, 
 however, it is provided with fins both fore and aft, and 
 their combined centre of pressure is at the centre of 
 gravity, a side wind will cause no rotation about a 
 vertical axis. 
 
 If the fins be raised above the centre of gravity, as in 
 the arrangement of 84, a side wind will cause the 
 aeroplane to heel over about the line of flight, but the 
 rotation thus produced will be damped by the resistance 
 to rotation depending on the term containing / cos 2 a in 
 our equations, which is large. In any case, the rotation 
 can be minimised by making the height of the fins not 
 too great, and it will be seen that this is exactly what the 
 conditions of stability require. 
 
 If, as in 86, the centre of pressure of the fins is 
 placed slightly in front of the centre of gravity, the 
 conditions of stability require that its distance in front 
 shall not be too large, and this again is in accordance 
 with the requirement that the rotation set up by a side 
 wind shall be small. 
 
 The construction of aeroplanes with fore and aft 
 auxiliary surfaces for longitudinal balance and stability, 
 moreover, provides the framework on which it only 
 remains to add the necessary vertical fins required for 
 lateral stability. The Curtiss biplane already has a 
 triangular vertical fin in front and a vertical rudder in 
 the rear. 
 
 The use of two raised fins placed fore and aft is no new 
 thing, for such an arrangement is shown by Lanchester in 
 connection with models constructed by him in 1894. 
 
 The " Langley Aerodrome " consisted of two pairs of bent- 
 
i yo GENERAL CONCLUSIONS CH. ix 
 
 up wings placed tandem, with a rudder behind. Having 
 regard to the partial analogy between the effects of bent-up 
 wings and vertical fins, this type possessed some re- 
 semblance in its properties to the arrangement with two 
 raised fins, and was probably laterally stable. This view 
 is confirmed by the records of its flights. At the same 
 time, the stability of such a type deserves more detailed 
 study, and it is with regret that it is now crowded out 
 of our text and placed in our list of unsolved problems. 
 
CHAPTER X. 
 
 COMPARISON WITH OTHER THEORIES. 
 
 105. We are here only concerned with investigations 
 directly related to the present work, so that the following 
 remarks only refer in many cases to the particular sections 
 of the writings in question which deal with stability. 
 
 BRYAN-WILLIAMS (1903). 1 
 
 The resistance derivatives were here divided by the 
 mass of the aeroplane. The X u of this book was therefore 
 then denoted by mX u . There was a term denoted in our 
 present notation by N r (X u + Y v ) omitted in the expression 
 for 6, which was incorrect to this extent. 
 
 CAPTAIN FERBER. 2 
 
 106. Captain Ferber assumes in the first place that 
 the surfaces of an aeroplane can be replaced by three 
 equivalent " fictive surfaces " S, s, a in three planes 
 mutually at right angles. This is true in certain cases, 
 but not in the sense implied in his footnote, where he 
 assumes that a surface can be replaced by its projections, 
 and that the error will be small. If instead of an inclined 
 surface, we consider its projections on the three co-ordinate 
 planes, then when the wind is blowing parallel to one of 
 
 1 Proc. Royal Society, 73, June, 1903. 
 
 2 Revue d'Artillerie 67, i, ii, October and November, 1905. 
 
 171 
 
172 OTHER THEORIES CHAP. 
 
 the axes there will be no pressure in a perpendicular 
 direction such as there was on the original surface. 
 
 Ferber's theorem is true in the following sense. We 
 assume that an ideal aeroplane is a system built up of 
 non-interfering surface elements, the air resistances on 
 which follow rigorously the sine law for all angles of 
 incidence. Then by applying a similar transformation to 
 that used in elasticity or in dealing with moments of inertia, 
 we can show that when the system is moving ivithout 
 rotation there are three mutually perpendicular planes, 
 such that the component forces due to air resistance are 
 the same in magnitude and direction as would be produced 
 if the system were replaced by three areas, S lt S. 2 , S 3 , 
 situated in these planes. Calling these the principal 
 planes and their intersections principal axes, we can 
 show that if certain conditions are satisfied the 
 equivalence can be extended to the moments of the 
 forces by suitably choosing the positions of the areas 
 Si t S 2 , $ 3 in their planes. The conditions require that 
 a wind blowing parallel to a principal axis shall produce 
 no couple about that axis, the exceptions thus being 
 surfaces analogous to screw propellers. For an aeroplane 
 with one plane of symmetry these conditions are satisfied. 
 
 When the effects of rotation are taken into account 
 this equivalence no longer holds good. This is shown by 
 our proof that two fins give lateral stability that cannot 
 be obtained with one. Captain Ferber thus neglects the 
 most important effects of rotation in his treatment of 
 lateral stability, namely the terms depending on /, the 
 moment of inertia of the main plane areas. As a con- 
 sequence, probably of this, he gets a cubic instead of a 
 biquadratic for lateral stability, or as he calls it, an 
 equation of the fifth degree with two zero roots. The 
 extra degree is accounted for by his including the azimuth 
 among the variables, and, as has been pointed out, stability 
 in azimuth does not come into the question here any more 
 
x OTHER THEORIES 173 
 
 than in other forms of locomotion. The presence of the 
 other zero root shows that there is something wrong with 
 the stability. The neglected term / really is of first 
 importance in the problem when we are dealing with 
 aeroplanes, the surfaces of which have considerable span. 
 
 The use of Euler's angular co-ordinates is also somewhat 
 unfortunate, for, as we have pointed out, this system is 
 unsuitable for investigating oscillations about a position of 
 equilibrium, Captain Ferber's equations breaking down if 
 we put = (see 14 of this book). 
 
 LANCHESTER. 1 
 
 107. In most points of fundamental importance there 
 is substantial accord between the present conclusions and 
 Lanchester's, and the main differences are in the method of 
 treatment. 
 
 Lanchester's condition of longitudinal stability is based 
 essentially on what we here call the hypothesis of " narrow 
 planes flying at small angles," and it agrees with the 
 result here obtained in the " simplest case M of horizontal 
 flight, although it is probably correct to say that the 
 method of obtaining it has an appearance of being want- 
 ing in rigour. Lanchester starts with a study of the 
 " phugoid " curves described by an ideal plane which 
 always places itself tangentially to the direction of 
 motion, drift and rotatory inertia, thus, being neglected. 
 He then applies corrections for resistance and moment 
 of inertia, and in the latter connection takes account of 
 the effects of the auxiliary tail on the rotatory motion. 
 In doing so he assumes the motion to be harmonic, and 
 takes the mean effect over each oscillation instead of 
 writing down a differential equation. 
 
 The reason why Lanchester's method leads to correct 
 
 1 Aerodonetics, London, 1908. 
 
174 OTHER THEORIES CHAP. 
 
 results will be evident on examining the discussion of the 
 long oscillations and their trajectories given in 52, 53 of 
 this book. In applying approximate methods to separate 
 the long oscillations, two successive approximations are 
 shown to be necessary. The first approximation determines 
 the period of oscillation, and the trajectory of the aeroplane 
 is shown to this order to be of the character described by 
 Lanchester, and to be unaffected by the rotatory inertia 
 of the mass. To determine the modulus of decay a 
 second approximation taking account of the rotatory 
 inertia is necessary, and the fact that the modulus of 
 decay is of a higher order than the frequency justifies the 
 averaging method adopted by Lanchester. But as a 
 necessary consequence, of course, Lanchester's method 
 does not give the short oscillations or even indicate their 
 existence. The sufficiency of his condition depends on 
 the fact, shown in 51, that the conditions of stability 
 for the long oscillations necessarily involve the conditions 
 of stability for the short ones. 
 
 In the case of " Lateral and Directional Stability," 
 Lanchester dties not use the three equations of motion, 
 but he is careful to emphasize the interdependence of 
 the three motions which would be determined by these 
 equations. He also emphasizes the necessity of fins or 
 bent-up planes, or both if an aeroplane is to be stable. 
 Whether the stability is to be described as " rotative " or 
 " asymmetric " is a question of terminology. Under the 
 heading of " Fin Resolution " Lanchester gives a discussion 
 practically amounting to the same thing as our applica- 
 tion of the principle of Parallel Axes to the second 
 moments of the fins. This method shows the essential 
 difference between a single fin and a pair of fins, one 
 in front and one behind, in the matter of producing 
 stability. The coefficients which we denote by / are 
 expressed in terms of two quantities called the aero- 
 dynamic and aerodromic radii. (See 93 above for a 
 
x OTHER THEORIES 175 
 
 discussion of the effect of assuming them unequal.) The 
 stability in non-horizontal flight is not discussed, but 
 an experiment is described in which a model turned 
 suddenly round sideways at particular points of its flight- 
 path, and the conclusion is suggested that owing to the 
 longitudinal oscillations the straight path became laterally 
 unstable at those points where the inclination of the 
 flight path exceeded a certain limit for stability. 
 
 BRILLOUIN. 1 
 
 108. Prof. Marcel Brillouin approaches the question of 
 balance and stability from a different point of view. His 
 investigations refer mainly to static instead of dynamic 
 instability, and he seeks by the study of metacentric 
 curves to. discuss the conditions that an aeroplane shall 
 have a unique position of statically stable equilibrium, 
 and thus be self-rightiug in all positions. He proceeds to 
 study the effect of manoeuvring on the balance of an 
 aeroplane, and his investigation thus becomes independent 
 of the problem of inherent dynamical stability. 
 
 REISSNER, 2 
 
 109. In the first of two. recent articles of considerable 
 interest, Dr. H. Reissner, of Aachen, places the problem of 
 lateral steering of aeroplanes on an exact and formal 
 basis. In a subsequent article, he points out the in- 
 sufficiency of the conditions for lateral stability previously 
 proposed, and the necessity of a further condition, thus 
 confirming the conclusions of the present investigation. 
 
 Cuocco. 3 
 
 110. Lieutenant Crocco's principal paper refers to 
 dirigibles, not aeroplanes/ In investigating the longi^ 
 
 1 Stability des Aeroplanes, surface metacentrique. Revue de Mecanique, 1909. 
 
 2 Zeitschrift fur Flugtechnik und Motorluftsctii/ahrt, 1910, 9, 10, and 
 Flugsp'irt, 1910-11. 
 
 3 Bollettino della Societa aeronautica italiana, April, 1907, and June, 1907. 
 Government Blue Book, 1909, 10, pp. 41, 161. 
 
176 OTHER THEORIES CHAP. 
 
 tuclinal stability, he obtains a cubic instead of a biquadratic, 
 based on the equations of rotation and of vertical motion. 
 The equation of horizontal motion is thus ignored. This 
 is certainly incorrect in the case of an aeroplane. Both 
 Lan Chester's and the present investigation show that in 
 the long longitudinal oscillations the variations in 
 horizontal velocity play a predominating part, and, 
 moreover, we have proved that the conditions of stability 
 depend more on the long than on the short oscillations, 
 stability for the former in general sufficing for the latter. 
 For directional stability, Crocco again obtains a cubic, 
 which is certainly inapplicable to aeroplanes. Even in 
 the case of a dirigible, Crocco's fundamental assumption, 
 " that the velocity is constant within the time considered," 
 is open to serious question. In another note, attention is 
 called to the stability condition which requires that for a 
 single lifting aeroplane the neutral plane must be behind. 
 It is pointed out that for a double lifting aeroplane the 
 condition a^ a 2 positive has an equivalent effect. 
 
 SORE Au. 1 
 
 111. For longitudinal stability, Soreau certainly obtains 
 a biquadratic, and deduces conditions analogous to those 
 obtained in the present investigation. In connection 
 with asymmetric stability, he falls into the common 
 error of not recognizing the interdependence of stability 
 for rolling and directional stability, thus applying the 
 term lateral stability in its most restricted sense. He is 
 thus led to conclusions with which the present writer 
 altogether disagrees, and which are further in conflict 
 with the trend of Lanchester's chapter on the subject. 
 Soreau would do away with vertical auxiliary surfaces and 
 bent-up planes, and would, in particular, avoid vertical 
 
 1 Etat actuel et VAvenir de I'Aviation. (Metnoires de la Societe des 
 Ingenieurs cimles de France, 1908, II. Blue Book, 1909-10, pp. 54, 154.) 
 
x OTHER THEORIES 177 
 
 auxiliary surfaces raised above the centre of gravity (see 
 Blue Book, p. 66). In other words, he would leave his 
 machine inherently unstable, and would use a gyroscope 
 or other similar contrivance to counteract automatically 
 this defect. 
 
 LECORNU. 1 
 
 112. The Government abstracts 2 contain a reference to 
 Lecornu's paper on " Graphic Statics," which, like our 
 discussion in 36-47, deals with the application of 
 graphic methods to the forces acting on an aeroplane, the 
 sine law of resistance being assumed. 
 
 1 Sur la statique graphique de I'aeroplane. (Comptes rendus, February 22, 
 1909.) 
 
 2 Blue Book, 1909-10. 
 
CHAPTER XL 
 
 PROBLEMS. 
 
 113. The following problems refer partly to questions 
 not coming within the scope of this book, and partly to 
 points which it has been considered scarcely desirable to 
 discuss in further detail in the preceding pages. Some 
 are pretty simple, and could be treated as exercises or 
 research subjects for a science student of fair ability ; 
 others would involve mathematical investigations or 
 experiments, or both, of a prolonged character. 
 
 1. The " characteristics " of a system of ideal planes 
 following the sine law of resistance, i.e. the determination 
 of tbe simplest possible scheme of equivalent planes (see 
 note on Captain Ferber, 106) a purely mathematical 
 investigation. 
 
 2. The steering of an aeroplane in a horizontal circle. 
 A study of Dr. Reissner's paper on this subject ( 109) 
 will certainly show possibilities of further investigation. 
 
 3. The stability of a dirigible, and in this connection 
 the discussion of formulae that shall best represent the 
 resistances due to inertia combined with head resistance, 
 in view of the fact that the hydrodynamical formula) for 
 inertia effects refer, of course, to motion in an incom- 
 pressible, unresisting medium. The six equations of 
 motion to be used, and not to be assumed independent 
 unless shown to be so. 
 
 4. A determination of the initial motions set up when 
 an aeroplane is suddenly struck by a gust of wind in any 
 
 178 
 
CH. xi PROBLEMS 
 
 179 
 
 direction, longitudinal, lateral, or vertical, or compounded 
 of all three. 
 
 5. The calculation of the stresses set up in the frame- 
 work of the aeroplane in such a case. 
 
 6. An experimental investigation on the effects of 
 rotation on broad planes, and, if possible, the determina- 
 tion of formulae, empirical or otherwise, which shall 
 represent these effects at least approximately. 
 
 7. Experiments on the wash caused by front planes on 
 back planes, also on the effects' of wash caused by 
 propellers. 
 
 8. The effect of a low centre of gravity on longitudinal 
 stability, i.e. a further development of the methods sug- 
 gested in 69, for an aeroplane the planes and propeller 
 of which are both raised to the same height. 
 
 9. An experimental study of the effects of inclination of 
 flight path on stability. 
 
 10. A more detailed examination of the effects of the 
 " product of inertia " F on lateral stability. It will be 
 seen that in our investigation F has been retained in most 
 of the fundamental formulae, but that in the final deduc- 
 tions we have usually put F = 0, thus assuming the line 
 of flight to be a principal axis of inertia. As a deviation 
 from this condition may either increase or decrease the 
 stability, the investigation may become important. 
 
 11. The lateral stability of types like the Langley 
 aerodrome, consisting of two pairs of bent-up wings 
 arranged tandem. The mathematics is in this case bound 
 to be fairly heavy. 
 
 12. A more extended descriptive examination of the 
 nature of the asymmetric oscillations and the motions 
 associated with them. 
 
 13. The investigation of the forced oscillations set up 
 in an aeroplane by periodic gusts of wind, with especial 
 reference to the case of synchrony with the free oscillations 
 and the influence of damping in this case. 
 
 N 2 
 
i8o PROBLEMS CHAP. 
 
 14. The possibility, or otherwise, of helicoidal steady 
 motion in the case of a symmetrical aeroplane, in par- 
 ticular when steady motion in a straight line is laterally 
 unstable. 
 
 15. A further discussion of the effects of camber on 
 both longitudinal and lateral stability. 
 
 16. A complete mathematical investigation (in the form 
 of a thesis) on the stability and oscillations of a kite, 
 with special reference to the question as to how far the 
 resistance derivatives and stability of an aeroplane could 
 be investigated experimentally by flying the model as a 
 kite. 
 
 17. Experimental determinations of the resistance de- 
 rivatives of aeroplane models by attaching them to 
 whirling tables or pendulums, with especial reference to 
 those derivatives which depend on rotation, comparison 
 with calculated values, experimental flights with the same 
 models, and comparison of their behaviour as regards 
 stability with the results of calculation. 
 
 18. Further photographic determinations of the paths 
 and oscillations of model aeroplanes and gliders, with 
 special reference to their lateral oscillations. 
 
 19. The graphic statics of longitudinal equilibrium for 
 systems of three or more planes having different abscissae ; 
 effect of altering the inclination of one of them relative to 
 the aeroplane. 
 
 20. Finally, the discussion of the equations of motion 
 of an aeroplane in their most general form, and the 
 search for cases in which these equations admit of exact 
 integration, is probably a mathematical problem which, 
 like the well-known Problem of Three Bodies, may well, 
 occupy the attention of pure mathematicians for the 
 next half century, and still leave plenty of work to be 
 done. 
 
xi NOTES 181 
 
 Examples. 
 
 114. Particulars of the construction and dimensions of 
 the leading modern aeroplanes will be found in most 
 aeronautical journals and manuals and engineering papers. 
 An abundant supply of " examples " can be obtained by 
 calculating their stability by the methods of this book ; 
 determining the coefficients of the biquadratics for longi- 
 tudinal and lateral oscillations ; approximate solutions of 
 the biquadratics ; periods and logarithmic decrements (or, 
 in the case of instability, logarithmic increments) of the 
 principal oscillations. 
 
 NOTES. 
 Gyrostatic action of propellers. 
 
 115. The gyrostatic effects due to the angular momentum 
 of a single propeller not only affect the steering of the 
 aeroplane, as is well known, but introduce interdependence 
 between the equations of rotation about the axes of y and 
 2, thus mixing up the longitudinal and lateral oscillations. 
 In a rigorous investigation, all six equations of motion 
 would have to be considered simultaneously. 
 
 The best plan is undoubtedly to destroy the gyrostatic 
 couples either by the use of twin screws rotating in 
 opposite directions, or by following the suggestion of 
 a recent French inventor who makes the rotary engine 
 and the propeller rotate in opposite directions, so that 
 their gyrostatic effects cancel. A pair of cog wheels is all 
 that is required. 
 
 If the gyrostatic couple is small, a solution of the 
 problem could be obtained, if it were worth while to 
 do so, by approximate methods. For a first approximation, 
 we should neglect the effect altogether, so that the oscilla- 
 tions would fall into the two "symmetric" and "asym- 
 
182 NOTES CHAP. 
 
 metric " groups. We should then apply small corrections 
 for the asymmetric motions set up by the symmetric 
 oscillations and vice versa. If an asymmetric and a 
 symmetric oscillation happened to have nearly the same 
 period, " resonance," effects would be set up with which 
 physicists are familiar, and the aeroplane would probably 
 not be a very comfortable one to travel on. 
 
 A proposition on automatic control. 
 
 116. The name " automatic stability " is now applied to 
 devices for regulating the balance of aeroplanes by means 
 of gyrostats or other mechanical contrivances. In this 
 connection, the following proposition may be useful : 
 
 An aeroplane formed of ideal straight planes with 
 no vertical fins or bent-up surfaces cannot be rendered 
 laterally stable by the use of gyrostats or pen- 
 dulums alone. 
 
 This applies even to the case where the aeroplane is 
 provided with warping devices or ailerons, if these are only 
 brought into action by the operations of the gyrostat. 
 
 Proof. In order that the plane of symmetry should 
 tend to assume a vertical direction when the machine is 
 disturbed, the gyrostat must be sensitive to variations in 
 the direction of gravity relative to the machine, since the 
 vertical direction is entirely defined by gravity. 
 
 Now, suppose such a machine (initially moving hori- 
 zontally) to be heeled over sideways through an angle $. 
 By assumption, there are no tangential resistances, there- 
 fore the machine will slip sideways with acceleration 
 g sin $. The only effect which the gyrostat is able to detect 
 is equivalent to a change in the intensity of gravity from 
 g to g cos <, the direction of the latter component relative 
 to the machine being unaltered. There is nothing, there- 
 fore, to make the gyrostat bring the machine back to 
 the vertical. 
 
xi NOTES 183 
 
 The only remedy is to retard the acceleration g sin <, 
 or make it evident in some way or other. This can only 
 be done by the use of some kind of fins or stabilizers 
 or other auxiliary surfaces attached to the machine or 
 to the gyrostat. In any case, it is impossible to do this 
 without in an equal degree making the gyrostat sensitive 
 to changes in wind velocity. When this is admitted, we 
 might just as well do away with the gyrostats altogether 
 and rely on the auxiliary surfaces instead. 
 
 One further point should, however, be considered. An 
 aeroplane might be provided with auxiliary surfaces 
 calculated to make it more inherently unstable, and 
 gyrostats might be used to convert this instability into 
 stability, as is, indeed, actually done in the Brennan mono- 
 rail. What is here implied is that if an aeroplane, when 
 tilted over sideways, tends either to return to the vertical 
 position or to deviate further from it, gyrostats may be 
 used with effect, but if it possesses neither tendency, 
 they become ineffectual, and some kind of auxiliary 
 surfaces must necessarily be introduced. 
 
 Note on the critical inclinations. 
 
 117. In discussing the effects of inclination of flight 
 path on stability we came upon the two limiting relations 
 tan 6 + 2 tan a = for longitudinal, and tan 2 tan a = 
 for lateral stability. It is useful therefore to get some 
 rough idea as to the why and wherefore of such condi- 
 tion, and their dependence on the law of resistance 
 
 Consider a single plane area the pressure on which 
 obeys this law. Draw the curve r 2 sin 6 = constant. 
 Then as long as the relative wind velocity is represented 
 by a radius vector of this curve, the pressure remains 
 constant, and if the plane receives a small additional 
 velocity dv in a direction tangential to the curve the same 
 
184 
 
 NOTES 
 
 CHAP. 
 
 thing occurs. If the direction of dv falls within the 
 curve the pressure decreases, if without, it increases. 
 The angle < (Fig. 38) between 27 and the direction of dv, 
 when the latter is tangential, is tan"^ rdOjdr) = 
 tan- x (2 tan 0). 
 
 If now the area is a kite placed in a horizontal wind, a 
 being the inclination of its face to the horizon, and if the 
 kite is kept in place by a string whose inclination to the 
 vertical is less than the angle whose tangent = 2 tan a, 
 it will be seen that a small additional upward velocity 
 perpendicular to the string will increase the pressure and 
 cause the kite to rise further, while a small velocity 
 
 FIG. 38. 
 
 in the opposite sense will reduce the pressure and cause 
 the kite to fall. This result indicates instability. For 
 stability the inclination of the string to the vertical must 
 be greater than tan" 1 (2 tan a). A complete discussion 
 of the conditions of stability would, however, necessitate 
 taking account of equations of rotation as well as trans- 
 lation. 
 
 We may similarly show, with the assumptions of 76, 
 that if the main plane of an aeroplane receives a small 
 rotation in a plane whose inclination to the line of flight 
 is given by tan" 1 (2 tan a) no couples are set up. The 
 limiting condition tan = 2 tan a for lateral stability thus 
 represents the fact that no couples are set up if the 
 aeroplane receives a small rotation in a horizontal plane. 
 
xi NOMENCLATURE 185 
 
 The same thing can be seen by means of the general 
 expressions (23) of 20. The condition @j = may be 
 written 
 
 0, cos 0, - sin 6 
 
 L w L p L q = 0, 
 
 M w M p M q 
 
 and is the condition that the equations 
 
 p cos 6 q sin 6 = 
 wL w + pL p + qL q = 
 wM w + pM p + qM q = 
 
 should be compatible ; in other words, that the motion 
 characterised by w t p, q which gives rise to no couples 
 L, M, should be a rotation about a vertical axis, as 
 indicated by the condition p cos q sin = 0. 
 
 The limiting condition, tan 6 = 2 tan a, for longitudinal 
 stability arising out of the expression for $), is more 
 difficult to explain. If we revert to the general equa- 
 tions (18) assuming A = and JV" M = the condition 
 becomes W sin 6 + UX U = or W sin 6 + 2 X = 0, and 
 leads to the statement that for stability the resistance due 
 to the component of gravity in the line of flight must be 
 less than twice the resistance due to the air. The effect 
 of head resistance is to increase the latter, and this is why 
 it increases the angle at which the aeroplane can rise. 
 
 NOMENCLATURE. 
 
 118. Abscissa. In accordance with our system of co- 
 ordinates, the abscissa of any plane is its distance in front 
 of the centre of gravity, being negative if the plane is 
 behind. 
 
 Aeroplane and plane. In accordance with the Aero- 
 nautical Society Committee's recommendation, aeroplane 
 = a flying machine, while plane = one of its surfaces. 
 In most cases in this book a pair of superposed planes 
 counts as a single plane for reasons explained. 
 
1 86 NOMENCLATURE CHAP. 
 
 Angle of attack This is the angle at which the air 
 blows on any given surface ; in an aeroplane moving 
 steadily it is the angle which a plane makes with the 
 direction of flight. It is sometimes called the angle of 
 incidence, but the optical analogy suggests that this name 
 would better be applied to the angle which the air current 
 makes with the normal to the surface. 
 
 Axes of co-ordinates The centre of gravity G is taken 
 as origin. The line of flight in steady motion is the axis 
 of x ; a perpendicular line in the plane of symmetry 
 measured downwards is the axis of y, and a perpendicular 
 to the plane of symmetry is the axis of 2. the three 
 forming a right-handed system. 
 
 Fin. A vertical auxiliary plane. 
 
 Inherent and automatic stability. The term automatic 
 stability is now frequently applied to mechanical devices 
 with movable parts, such as gyrostats, which are so 
 arranged as to restore equilibrium when an aeroplane or 
 other system is displaced. The Brennan monorail affords 
 one of the simplest illustrations of automatic stability. 
 An aeroplane is inherently stable when stability is secured 
 without the aid of movable parts, and the only kind of 
 stability considered in this book is inherent dynamical 
 stability. 
 
 Narrow planes. Planes of which the chord or breadth 
 is so small that when the plane rotates about an axis 
 parallel to the axis of z the difference of velocity of the 
 front and back edges is negligible, i.e. the rotary 
 derivatives are negligible. 
 
 Plane of symmetry Aeroplanes, practically without 
 exception, are symmetrical. When flying in a straight 
 line the plane of symmetry is a vertical plane through the 
 centre of gravity containing the line of flight. 
 
xi NOMENCLATURE 187 
 
 Resistance derivatives. If small changes be made in 
 the velocity components of an aeroplane, the ratios to 
 these of the corresponding changes in the values of the 
 components of air resistance I call the resistance 
 derivatives. 
 
 Rotary derivatives. The resistance derivatives arising 
 from a small rotation of a plane about an axis through 
 its centre of pressure parallel to the axis of z. 
 
 Single and double lifting systems. Neutral surfaces 
 
 Where an aeroplane has two surfaces or sets of superposed 
 surfaces, and the weight of the aeroplane is borne partly 
 by the front and partly by the rear surfaces, I call it a 
 double lifting system. If the whole weight is borne by 
 a single surface or set of superposed surfaces, I call it a 
 single lifting system. In such cases any other auxiliary 
 surfaces, such as tail planes, must be parallel to the 
 direction in which the wind blows on them, and I call this 
 position neutral. 
 
 Small angles. When the angle of attack is called 
 small, it is implied that the resistance may be taken to 
 follow the " sine law." 
 
 Stability, static and dynamic. A system in equilibrium 
 is said to be statically stable if, when displaced, it initially 
 tends to approach its original state of equilibrium. It 
 may, however, go past its equilibrium position and 
 oscillate, and if the oscillations continually grow larger 
 and larger it will be dynamically unstable. For dyna- 
 mical stability it is necessary that the system should 
 either return to its equilibrium position without oscillating, 
 or that the oscillations should gradually die out. In the 
 present book " stability" means dynamical stability. 
 
 Stabilizer In this book the term is applied to small 
 bent up planes attached at the extremities of the main 
 planes. 
 
1 88 NOTATION CHAP. 
 
 Straight and bent up planes. When the wings of an 
 aeroplane form a dihedral angle, I call them "bent up." 
 ''Straight" planes, on the other hand, project perpen- 
 dicularly on both sides of the plane of symmetry. 
 
 Symmetric and asymmetric stability. Symmetric 
 stability is practically synonymous with longitudinal 
 stability, and represents stability for motions in the plane 
 of symmetry. Asymmetric stability is stability for rota- 
 tions about the axis of x and y and displacements along 
 the axis of 2, and includes what are sometimes separated 
 as " lateral " and " directional '' stability. It is Lanchester's 
 " rotative stability." 
 
 119. NOTATION. 
 
 A, B, C, moments of inertia of aeroplane. F, product 
 of inertia. 
 
 9(, 93, (5, 5), d, co -efficients of biquadratic, with suffix 
 for symmetrical, l for asymmetric oscillations where a 
 distinction is necessary. 
 
 a, angle of attack. 
 
 a lt a 2 , angles of attack in a double lifting system, and in 
 this case a = a v a 2 . 
 
 /3, angle at which a plane is bent up. 
 
 a, semi-chord of plane. 
 
 ?>, semi-span. 
 
 c, radius of curvature of cambered plane. 
 
 d, distance of centroid of areas of a pair of transverse 
 planes from centre of gravity. 
 
 f, corresponding distance of centre of pressure of the 
 pair. 
 
 e, a constant determining the displacements of the 
 centre of pressure of a transverse plane. 
 
 6, a certain constant referring to the "Antoinette" 
 type (under lateral stability). 
 
xi NOTATION 189 
 
 F, product of inertia of aeroplane as above. 
 f(a), function of angle of attack determining law of 
 resistance ; $(), corresponding function determining 
 position of centre of pressure. 
 
 f'(a), 4>'( a )> differential coefficients with respect to a ; 
 f r (a), <f> r (a), rotary derivatives corresponding to a small 
 angular velocity r. 
 
 H, propeller thrust ; h, its perpendicular distance from 
 origin ; vj, its inclination to axis of x when different from 
 zero. 
 
 /, moment of inertia of a transverse plane with respect 
 to plane of symmetry F = I cos 2 a. 
 
 K, coefficient of resistance in R = KSU' 2 f(a), or R 
 = KSU 2 sin a. 
 
 k, radius of gyration about axis of z, C = Wk 2 . 
 
 /, distance between front and rear planes, in longi- 
 tudinal stability. 
 
 /, m, n, direction cosines of normal to a bent up surface- 
 element. 
 
 L, M, N, couples about axes due to resistance, in direc- 
 tions opposed to rotation. 
 
 L , M , N , their values in steady motion. 
 
 L, M, N, with suffixes, e.g. L p the corresponding resist- 
 ance derivatives ; L p = dL/dp. 
 
 M l9 M 2 , P moments and products of inertia of a system 
 of two or more vertical fins with respect to axes through 
 their centre of pressure parallel to the co-ordinate axes, the 
 co-ordinates of this centre of pressure being x, y, z. 
 
 x, coefficient in a small oscillation or disturbance pro- 
 portional to e M , hence a root of the biquadratic. 
 
 P, tangent of angle of attack ^ = tan a. 
 
 p l9 ft 2 , tangents of angles of attack in double lifting 
 systems. 
 
 v, the difference of these tangents = ^ /* 2 . 
 
 p, q, r, components of angular velocity of aeroplane. 
 
 R, resultant thrust of air on a plane. 
 
190 NOTATION 
 
 CH. XI 
 
 R, radius of gyration of a pair of planes (where this 
 notation causes no confusion with the previous one). 
 
 p distance measured from dihedral angle on a bent up 
 plane. 
 
 S, area of any plane surface, S' = S cos 2 a. 
 
 T, area of a vertical fin or pair of stabilizers, also total 
 area of several fins. 
 
 U, velocity in steady flight. 
 
 u, v, w, added velocity components in a small oscil- 
 lation. 
 
 W, weight of aeroplane. 
 
 x, y, 2, co-ordinates. 
 
 X, Y, Z, components of force due to air resistance, 
 taken positive when acting in the opposite direction to 
 the axes. X , Y , Z oi values in steady motion. X u , etc., 
 corresponding resistance derivatives X u = dX/du. 
 
 , a certain constant in connection with stabilizers. 
 
 0, inclination of flight path to horizon, positive when 
 descending ; in small symmetric oscillations 6 = 6 Q + e 
 where 6 is the value corresponding to steady motion. 
 This notation causes no confusion with the other meaning 
 of e mentioned above. 
 
 u, ratio of head resistance to drift. 
 
 <, angle through which plane of symmetry has rotated 
 about axis of x y measured from vertical position. 
 
INDEX 
 
 AEROSTAT, 54 
 
 " Antoinette type," 158-164 
 
 Approximate solutions of biquadratics, 
 
 79, 132 
 
 Aspect ratio, 149 
 Asymmetric derivatives, 123-128 
 Asymmetric oscillations and stability, 
 
 5, 12, 27, 31-33, 123-164 
 "Automatic" stability, vi, 182, 186 
 
 BENT-UP planes, 151-164 
 B16riot aeroplane, 152 
 Brillouin, 4, 70, 74, 175 
 Broad planes, 106-113, 121-122 
 Bryan and Williams, 4, 35, 171 
 
 CAMBER (curvature), 50, 114, 148 
 Centre of pressure, variations of, 11, 
 
 68, 69, 106-113 
 Chanute glider, 58 
 Conic, application of geometry of, 72- 
 
 74 
 
 Critical inclinations, 183 
 Crocco, 4, 175 
 Curtiss, 117, 169 ; aeroplane, 130 
 
 DAMPED oscillations, 30 
 Directional stability, 5, 27, 174, 176 
 Dirigibles, 4, 36 
 Discontinuous motion, 1 
 Double-lifting systems, 61-74, 95-116, 
 143-144 
 
 EDWARD MINES aeroplane, 110 
 Effect of tilting planes, 64-66 
 Equations of motion, 19-23 
 Equivalent systems, 10, 95-101, 119 
 p]uler's angular co-ordinates unsatisfac- 
 tory, 20, 173 
 
 Examples, material for, 181 
 Extremities of planes, correction for, 
 149 
 
 FARMAN, 117, aeroplane, 118 
 
 Ferber, 4, 171 
 
 Ferguson, vii 
 
 Fins, 13, 128-132, 135-143 
 
 191 
 
 Formulae of transformation, 48, 49 
 Friction, 113, 147 
 
 GLIDER, 10, 89, 90, 111 
 Glossary (see Nomenclature) 
 Government advisory committee, Re- 
 ports, 2, 4, 8, 46, 175, 176, 177 
 Graphic Statics, 9, 57-74, 177 
 Gravity, effects of, 135 
 Greenhill, 2 
 Gregorv, R.A., viii 
 Gusts of wind, 167-170 
 Gyrostatic action, 181, 182 
 
 HARPER, vii, 10, 14, 88, 94, 104, 129, 
 
 140, 142, 154, 158 
 Head resistance, effects, 90-93, 144- 
 
 147 
 
 Helmholtz, 1 
 Hydrodynamics, 17 
 
 INCLINATION of Flight Path, effects of, 
 9, 88-90, 111, 120, 121, 138, 139-141, 
 142, 156, 162, 164, 183-185 
 
 Inclination of Propeller axis, 87, 88 
 
 Independence of Height, Principle of, 
 9, 53-56 
 
 Invariants, 99-101 
 
 JONES, ROBERT, vii 
 KIRCHHOFF, 2 
 
 LAG, 117 
 
 Lanchester, 4, 9, 77, 115, 136, 169, 173 
 Lateral stability, 12, 31-33, 123-164 
 Lecornu, 177 
 Lefebvre aeroplane, 96 
 Longitudinal Derivatives, 38-56 
 Longitudinal Equilibrium, 57-74 
 Longitudinal stability, 27-31, 75-122 
 Long oscillations, 80, 84-87 
 
 METACENTRIC curves, 70 
 
 Modified conditions of stability, 35, 
 
 77-79 
 Modulus of Decay, 30, 80, 134 
 
192 
 
 INDEX 
 
 NARROW plane?, 8, 51-53, 123-126 
 Neutral tail plane, 57, 75, 109 
 Newton, 1 
 
 Nomenclature, 185-188 
 Notation, 188-190 
 
 PARACHUTE, plane, 54 
 
 Pendulum experiments, 46, forces on, 54 
 
 Primitive conditions of stability, 35, 77 
 
 Problems, 178-180 
 
 Propeller thrust, 22 
 
 Propeller thrust, central, 58, 61-66, 
 
 75-100 
 Propeller thrust, non-central, 60, 71-74, 
 
 101-106 
 Propeller thrust, variable, 26, 93, 145 
 
 RAISED fins, 135 
 
 Raised planes, 113 
 
 Reissner, 4, 175 
 
 Resistance derivatives, 6, 33-36, 38- 
 
 48, 52, 53 
 
 Rolling, resistance to, 133, 135 
 Rotary derivatives, 8, 41, 44-48, 112, 
 
 121-122 
 
 Rotative stability, 5, 27, 174 
 Rougier, aeroplane, 143 
 Routh's discriminant, 6, 30 
 
 SHORT oscillations, 80, 81-84 
 Simplest case of longitudinal stability, 
 75-79 
 
 Sine-law 52, 124; deviations from, 114 
 
 Single fin, effects of, 128 
 
 Single lifting systems, 9, 56, 57-61, 
 
 75-94, 109, 154-158 
 Single plane without rudder, 110-113 
 Small angles, 8, 51, 75-77, 103, 124, 131 
 Soreau, 4, 176 
 Speed limits consistent with stability, 
 
 77, 112 
 
 Stabilizers, 153-158 
 Steady motion, 23 
 Straight planes, 126 
 Symmetric or Longitudinal Derivatives, 
 
 38-56 
 Symmetric oscillations and stability, 
 
 5, 27-31, 75-122 
 
 THREE plane system, 12, 117-121 
 Trajectories, forms of, 83, 87 
 Transformation formulae, 48-49 
 Two-fin systems, 135, 138, 141 
 Turnbull, 50 
 Twin screws, 145 
 
 VARIABLE propeller thrust, effects of, 
 26, 93, 145 
 
 WASH, 95, 115, 150 
 Whirling table method, 45 
 Wright Brothers, 3, 83, aeroplane, 
 139 
 
 1/7 
 
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