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Full text of "Flight without formulae; simple discussions on the mechanics of the aeroplane"

THE LIBRARY 

OF 

THE UNIVERSITY 
OF CALIFORNIA 

LOS ANGELES 



FLIGHT WITHOUT FORMULA 



THE MECHANICS OF THE AERO- 
PLANE. A Study of the Principles of 
Flight. By COMMANDANT DUCHENE. 
Translated from the French by JOHN H. 
LEDEBOER, B.A., andT. O'B. HUBBARD. 
With 98 Illustrations and Diagrams. 
8vo. 8s. net. 



FLYING : Some Practical Experiences. 
By GUSTAV HAMEL and CHARLES C. 
TURNER. With 72 Illustrations. 8vo. 
I2s. 6d. net. 

LONGMANS, GREEN, AND CO., 

LONDON, NEW YORK, BOMBAY.CALCUTTA, MADRAS 



FLIGHT WITHOUT 
FORMULA 

SIMPLE DISCUSSIONS ON THE 
MECHANICS OF THE AEROPLANE 



BY 

COMMANDANT DUCHENE 



OF THE FRENCH GENIE 



TRANSLATED FROM THE FRENCH BY 

JOHN H. LEDEBOER, B.A. 

ASSOCIATE FELLOW, AERONAUTICAL SOCIETY; EDITOR 
"AERONAUTICS"; JOINT- AUTHOR OF "THE AEROPLANE" 
TRANSLATOR OF " THE MECHANICS OF THE AEROPLANE" 



SECOND EDITION 



LONGMANS, GREEN, AND CO. 

39 PATERNOSTER ROW, LONDON 

FOURTH AVENUE & 30 STREET, NEW YORK 

BOMBAY, CALCUTTA, AND MADRAS 

1916 

All rights reserved 



First Published . . July 1914 
Type Reset . . October 1916 



TL 

S" 

JD 



I (> 



TRANSLATOR'S PREFACE 



and equations are necessary evils ; they repre- 
sent, as it were, the shorthand of the mathematician and the 
engineer, forming as they do the simplest and most con- 
venient method of expressing certain relations between 
facts and phenomena which appear complicated when 
dressed in everyday garb. Nevertheless, it is to be feared 
that their very appearance is forbidding and strikes terror 
to the hearts of many readers not possessed of a mathematical 
turn of mind. However baseless this prejudice may be 
as indeed it is the fact remains that it exists, and has hi 
the past deterred many from the study of the principles of 
the aeroplane, which is playing a part of ever -increasing 
importance in the life of the community. 

The present work forms an attempt to cater for this class 
of reader. It has throughout been written in the simplest 
possible language, and contains in its whole extent not a 
single formula. It treats of every one of the principles of 
flight and of every one of the problems involved in the 
mechanics of the aeroplane, and this without demanding 
from the reader more than the most elementary knowledge 
of arithmetic. The chapters on stability should prove of 
particular interest to the pilot and the student, containing 
as they do several new theories of the highest importance 
here fully set out for the first time. 

In conclusion, I have to thank Lieutenant T. O'B. 
Hubbard, my collaborator for many years, for his kind and 
diligent perusal of the proofs and for many helpful 
suggestions. 

J. H. L. 



968209 



CONTENTS 

CHAPTER I 



PAGE 

FLIGHT IN STILL AIR SPEED 1 



CHAPTER II 

PLIGHT IN STILL AIR POWER 16 

CHAPTER III 
PLIGHT IN STILL AIR POWER (concluded) .... 35 

CHAPTER IV 

FLIGHT IN STILL AIR THE POWER-PLANT . . . . 53 

CHAPTER V 

FLIGHT IN STILL AIR THE POWER-PLANT (concluded) . . 70 

CHAPTER VI 

STABILITY IN STILL AIR LONGITUDINAL STABILITY . . . 90 

CHAPTER VII 

STABILITY IN STILL AIR LONGITUDINAL STABILITY (cotl- 

cluded) ..... . . . . 115 

CHAPTER VIII 

STABILITY IN STILL AIR LATERAL STABILITY . . .142 

CHAPTER IX 

STABILITY IN STILL AIR LATERAL STABILITY (concluded) 

DIRECTIONAL STABILITY, TURNING . . . . . 161 

CHAPTER X 

THE EFFECT OF WIND ON AEROPLANES . . . . 183 



Flight without Formulae 

Simple Discussions on the Mechanics 
of the Aeroplane 



CHAPTER I 
FLIGHT IN STILL AIR 

SPEED 

NOWADAYS everyone understands something of the main 
principles of aeroplane flight. It may be demonstrated 
in the simplest possible way by plunging the hand in 
water and trying to move it at some speed horizontally, 
after first slightly inclining the palm, so as to meet or 
" attack " the fluid at a small " angle of incidence." It 
will be noticed at once that, although the hand remains 
very nearly horizontal, and though it is moved horizon- 
tally, the water exerts upon it a certain amount of pressure 
directed nearly vertically upwards and tending to lift the 
hand. 

This, in effect, is the principle underlying the flight of 
an aeroplane, which consists in drawing through the air 
wings or planes in a position nearly horizontal, and thus 
employing, for sustaining the weight of the whole machine, 
the vertically upward pressure exerted by the air on these 
wings, a pressure which is caused by the very forward 
movement of the wings. 

Hence, the sustentation and the forward movement of 
an aeroplane are absolutely interdependent, and the former 

1 



2 FLIGHT WITHOUT FORMULAS 

can only be produced, in still air, by the latter, out of which 
it arises. 

But the entire problem of aeroplane flight is not solved 
merely by obtaining from the " relative " air current which 
meets the wings, owing to their forward speed, sufficient 
lift to sustain the weight of the machine ; an aeroplane, 
in addition, must always encounter the relative air current 
in the same attitude, and must neither upset nor be thrown 
out of its path by even a slight aerial disturbance. In 
other words, it is essential for an aeroplane to remain in 
equilibrium more, in stable equilibrium. 

This consideration clearly divides the study of aeroplane 
flight in calm air into two broad, natural parts : 

The study of lift and the study of stability. 

These two aspects will be dealt with successively, and 
will be followed by a consideration of flight in disturbed air. 

First we will proceed to examine the lift of an aeroplane 
in still air. 

Following the example of a bird, and in accordance with 
the results obtained by experiments with models, the wings 
of an aeroplane are given a span five or six times greater 
than then 1 fore-and-aft dimension, or " chord," while they 
are also curved, so that their lower surface is concave.* It 
is desirable to give the wings a large span as compared 
to the chord, in order to reduce as far as possible the 
escape or leakage of the air along the sides ; while it 
has the further advantage of playing an important part 
in stability. Again, the camber of the wings increases their 
lift and at the same time reduces their head-resistance or 
" drag." 

The angle of incidence of a wing or plane is the anerle. 

* In English this curvature of the wing is generally known as the 
" camber." On the whole, it would perhaps be more accurate to describe 
the upper surface as being convex, since highly efficient wings have been 
designed in which the camber is confined to the upper surface, the lower 
surface being perfectly flat. TRANSLATOR. 



FLIGHT IN STILL AIR 3 

expressed in degrees, made by the chord of the curve in 
profile with the direction of the aeroplane's flight. 

As stated above, the pressure of the air on a wing mov- 
ing horizontally is nearly vertical, but only nearly. For, 
though it lifts, a wing at the same time offers a certain 
amount of resistance known either as head-resistance or 
drag * which may well be described as the price paid for 
the lift. 

As the result of the research work of several scientists, 
and of M. Eiffel in particular, with scale models, unit figures, 
or " coefficients," have been determined which enable us to 
calculate the amount of lift possessed by a given surface 
and its drag, when moving through the air at certain angles 
and at certain speeds. 

Hereafter the coefficient which serves to calculate the 
lifting-power of a plane will be simply termed the lift, 
while that whereon the calculation of its drag is based will 
be known as the drag. 

M. Eiffel has plotted the results of his experiments in 
diagrams or curves, which give, for each type of wing, the 
values of the lift and drag corresponding to the various 
angles of incidence. 

The following curves are here reproduced from M. Eiffel's 
work, and relate to : 

A flat plane (fig. 1). 

A slightly cambered plane, a type used by Maurice 

Farman (fig. 2). 
A plane of medium camber, adopted by Breguet 

(fig. 3). 

A deeply cambered plane, used by Bleriot on his 
No. XI. monoplanes, cross-Channel type (fig. 4). 

* The word " drag " is here adopted, in accordance with Mr Archibald 
Low's suggestion, in preference to the more usual " drift," in order to 
prevent confusion, and so as to preserve for the latter term its more 
general, and certainly more appropriate meaning, illustrated in the ex- 
pression " the drift of an aeroplane from its course in a side-wind," or 
" drifting before a current." TRANSLATOR. 



4 FLIGHT WITHOUT FORMULA 

These diagrams are so simple as to render further 
explanation superfluous. 




.0.00. 



0.02 001 000 

Drag. Drag. 

FIG. 1. Flat plane. FIG. 2. Maurice Farman plane. 

The calculation of the lifting-power and the head-resist- 
ance produced by a given type of plane, moving through 



FLIGHT IN STILL AIR 



the air at a given angle of incidence and at a given speed, 
is exceedingly simple. To obtain the desired result all that 



J0.08 




0.00 



0.00 



0.02 : 01 

Drag. 

FIG. 3. Br6guet plane. 



O.QO 



02 0.01 0.00 

Drag. 
FIG. 4. Bleriot XI. plane. 



is needed is to multiply either the lift or the drag co- 
efficients, corresponding to the particular angle of incidence, 



6 FLIGHT WITHOUT FORMULA 

by the area of the plane (hi square metres, or, if English 
measurements are adopted, hi square feet) and by the square 
of the speed, hi metres per second (or miles per hour).* 

EXAMPLE. A Bleriot monoplane, type No. XI., has an 
area of 15 sq. m., and flies at 20 m, per second at an angle 
of incidence of 7. (1) What weight can its wings lift, and 
(2) what is the power required to propel the machine ? 

Referring to the curve in fig. 4, the lift of this particular 
type of wing at an angle of 7 is 0-05, and its drag 0-0055. 

Hence 

T -f. . Square of 

Lift. Area. the Speed. 

0-05 x 15 x 400 

gives the required value of the lifting-power, i.e. 300 kg. 
Again 

Drag. Are, * 

0-0055 x 15 x 400 
gives the value of the resistance of the wings, i.e. 33 kg. 

Let us for the present only consider the question of lift, 
leaving that of drag on one side. 

From the method of calculation shown above we may 
immediately proceed to draw some highly important de- 
ductions regarding the speed of an aeroplane. The fore- 
and-aft equilibrium of an aeroplane, hi fact, as will be 
shown subsequently, is so adjusted that the aeroplane can 
only fly at one fixed angle of incidence, so long as the 
elevator or stabiliser remains untouched. By means of the 
elevator, however, the angle of incidence can be varied 
within certain limits. 

In the previous example, let the Bleriot monoplane be 
taken to have been designed to fly at 7. It has already 
been shown that this machine, with its area of 15 sq. m. 
and its speed of 72 km. per hour, will give a lifting-power 
equal to 300 kg. Now, if this lifting-power be greater than 

* Throughout this work the metric system will henceforward be 
strictly adhered to. TRANSLATOR. 



FLIGHT IN STILL AIR 7 

the weight of the machine, the latter will tend to rise ; if the 
weight be less, it will tend to descend. Perfectly horizontal 
flight at a speed of 72 km. per hour is only possible if the 
aeroplane weighs just 300 kg. 

In other words, an aeroplane of a given weight and a 
given plane-area can only fly horizontally at a given angle 
of incidence at one single speed, which must be that at 
which the lifting-power it produces is precisely equal to 
the weight of the aeroplane. 

Now it has already been shown that the lifting-power 
for a given angle of incidence is obtained by multiplying 
the lift coefficient corresponding to this angle by the plane 
area and by the square of the speed. This, therefore, must 
also give us the weight of the aeroplane. It is clear that 
this is only possible for one definite speed, i.e. when the 
square of the speed is equal to the weight, divided by the 
area multiplied by the inverse of the lift. And since the 
weight of the aeroplane divided by its area gives the load- 
ing on the planes per sq. m., the following most important 
and practical rule may be laid down : 

The speed (in metres per second) of an aeroplane, flying 
at a given angle of incidence, is obtained by multiplying 
the square root of its loading (in kg. per sq. in.} by the square 
root of the inverse of the lift corresponding to the given 
angle. 

At first sight the rule may appear complicated. Actually 
it is exceedingly simple when applied. 

EXAMPLE. A Breguet aeroplane, with an area of 30 sq. m. 
and weighing 600 kg., flies with a lift of 0-04, equivalent 
(according to the curve in fig. 3) to an angle of incidence 
of about 4. What is its speed ? 

600 
The loading is =20 kg. per sq. m. 



Square root of the loading 4 '47. 

Inverse of the lift is =25. 
0-04 

Square root of inverse of the lift =5. 



FLIGHT WITHOUT FORMULA 



The speed required, therefore, in metres per second 4-47 
X 5=22- 3 m. per second, or about 80 km. per hour. But 
if a different angle of incidence, or a different figure for 
the lift which is equivalent, and, as will be seen here- 
after, more usual be taken, a different speed will be 
obtained. 

Hence each angle of incidence has its own definite speed. 

For instance, if we take the Breguet aeroplane already 
considered, and calculate its speed for a whole series of 
angles of incidence, we obtain the results shown in Table I. 
But before examining these results in greater detail, so far 
as the relation between the angles of incidence, or the lift, 
and the speed is concerned, a few preliminary observations 
may be useful. 

TABLE I. 









'Speed. 


In m.p.s. 


In km.p.h. 


Lift. 


Correspond- 
ing Angle of 
Incidence. 


! Square 
Inverse of i Root of 
Lift. ; Inverse of 
Lift. 






~* >j~o 50 


10 > 
|1^ . 










J'ilfcr 


lit" 










:(? S 3 "8 


|*a 


1 


2 


3 


4 


5 


6 


0-020 


(about) 


50 


7-07 


31-6 


113-6 


0-030 


2 ,, 


33-3 577 


25-8 


92-8 


0-040 


4 ,, 


25 


5-00 


22-3 


80-3 


0-050 


64 


20 


4-47 


20-0 


72-0 


0-060 


10 ,, 


16-6 4-08 


18-2 


65-6 


0-066 


15 ,, 


15-2 


3-90 


17-4 


62-6 



In the first place, it should be noted that when the 
Breguet whig has no angle of incidence, when, that is, the 
wind meets it parallel to the chord, it still has a certain 
lift. This constitutes one of the interesting properties of 



FLIGHT IN STILL AIR 9 

a cambered plane. While a flat plane meeting the air 
edge-on has no lift whatever, as is evident, a cambered 
plane striking the air in a direction parallel to its chord 
still retains a certain lifting-power which varies according 
to the plane section. 

Thus, in those conditions a Breguet wing still has a lift of 
0-019, and if figs. 4 and 2 are examined it will be seen that 
at zero incidence the Bleriot No. XI. would similarly have 
a lift of 0-012, but the Maurice Farman of only 0-006. It 
follows that a cambered plane exerts no lift whatever only 
when the wind strikes it slightly on the upper surface. In 
other words, by virtue of this property, a cambered plane 
may be regarded as possessing an imaginary chord if the 
expression be allowed inclined at a negative angle (that is, 
in the direction opposed to the ordinary angle of incidence) 
to the chord of the profile of the plane viewed in section. 

If the necessary experiments were made and the curves 
on the diagrams were continued to the horizontal axis, it 
would be found that the angle between this " imaginary 
chord " and the actual chord is, for the Maurice Farman 
plane section about 1, for that of the Bleriot XI. some 2, 
and for that of the Breguet 4. 

Let it be noted in passing that in the case of nearly 
every plane section a variation of 1 in the angle of incidence 
is roughly equivalent to a variation in lift of 0-005, at any 
rate for the smaller angles. One may therefore generalise 
and say that for any ordinary plane section a lift of 0-015 
corresponds to an angle of incidence of 3 relatively to the 
" imaginary chord," a lift of 0-020 to an angle of 4, a lift 
of 0-025 to 5, and so forth. 

Turning now to the upper portion of the curves in the 
diagrams, it will be seen that, beginning with a definite 
angle of incidence, usually in the neighbourhood of 15, the 
lift of a plane no longer increases. The curves relating to 
the Breguet and the Bleriot cease at 15, but the Maurice 
Farman curve clearly shows that for angles of incidence 
greater than 15 the lift gradually diminishes. Such coarse 



10 FLIGHT WITHOUT FORMULA 

angles, however, are never used in practice, for a reason 
shown in the diagrams, which is the excessive increase in 
the drag when the angle of incidence is greater than 10. 
In aviation the angles of incidence that are employed there- 
fore only vary within narrow limits, the variation certainly 
not surpassing 10. 

We may now return to the main object for which Table I. 
was compiled, namely, the variation in the speed of an 
aeroplane according to the angle of incidence of its planes. 

First, it is seen that speed and angle of incidence vary 
inversely, which is obvious enough when it is remembered 
that in order to support its own weight, which necessarily 
remains constant, an aeroplane must fly the faster the 
smaller the angle at which its planes meet the air. 

Secondly, it will be seen that the variation in speed is 
more pronounced for the smaller angles of incidence ; hence, 
by utilising a small lift coefficient great speeds can be 
attained. Thus, for a lift equal to 0-02, at which the 
Breguet wing would meet the air along its geometrical 
chord, the speed of the aeroplane, according to Table I., 
would exceed 113 km. an hour. 

If an aeroplane could fly with a lift coefficient of O'Ol, 
that is, if the planes met the air with their upper surface 
the imaginary chord would then have an angle of incidence 
of no more than 2 the same method of calculation would 
give a speed of over 160 km. per hour. 

The chief reason which in practice places a limit on the 
reduction of the lift is, as will be shown subsequently, the 
rapid increase in the motive -power required to obtain high 
speeds with small angles of incidence. And further, there 
is a considerable element of danger in unduly small angles. 
For instance, if an aeroplane were to fly with a lift of O'Ol 
so that the imaginary chord met the air at an angle of 
only 2 a slight longitudinal oscillation, only just exceeding 
this very small angle, would be enough to convert the fierce 
air current striking the aeroplane moving at an enormous 
speed from a lifting force into one provoking a fall. It is 



FLIGHT IN STILL AIR 11 

true that the machine would for an instant preserve its 
speed owing to inertia, but the least that could happen 
would be a violent dive, which could only end in disaster 
if the machine was flying near the ground. 

Nevertheless there are certain pilots, to whom the word 
intrepid may be justly applied, who deny the danger and 
argue that the disturbing oscillation is the less likely to 
occur the smaller the angle of incidence, for it is true, as 
will be seen hereafter, that a small angle of incidence is an 
important condition of stability. However this may be, 
there can be no question but that flying at a very small 
angle of incidence may set up excessive strains in the frame- 
work, which, in consequence, would have to be given 
enormous strength . Thus , if it were possible for an aeroplane 
to fly with a lift coefficient of 0-01, and if, owing to a wind 
gust or to a manoeuvre corresponding to the sudden " flatten- 
ing out " practised by birds of prey and by aviators at the 
conclusion of a dive, the plane suddenly met the air at an 
angle of incidence at which the lift reaches a maximum 
that is, from 0-06 to 0-07 according to the type of plane 
the machine would have to support, the speed remaining 
constant for the time being by reason of inertia, a pressure 
six or seven times greater than that encountered in normal 
flight, or than its own weight. 

In practice, therefore, various considerations place a limit 
on the decrease of the angle of incidence, and it would 
accordingly appear doubtful whether hitherto an aeroplane 
has flown with a lift coefficient smaller than 0-02.* 

It is easy enough to find out the value of the lift co- 
efficient at which exceptionally high speeds have been 
attained from a few known particulars relating to the 
machine in question. The particulars required are : 

The velocity of the aeroplane, which must have been 
carefully timed and corrected for the speed of the wind ; 

The total weight of the aeroplane fully loaded ; 

The supporting area. 

* See footnote on p. 12. 



12 FLIGHT WITHOUT FORMULA 

The lift may then be found by dividing the loading of 
the planes by the square of the speed in metres per second. 

EXAMPLE. An aeroplane with a plane area of 12 sq. m. 
and weighing, fully loaded, 360 kg. has flown at a speed of 
130 km. or 36-1 m. per second. What was its lift coefficient ? 

O f*(\ 

The loading= = 30 kg. per sq. m. 

12 

Square of the speed=1300. 

Ofi 

Required lift=-?"- =about 0-023.* 

Table I. further shows that when the angle of incidence 
reaches the neighbourhood of 15 (which cannot, as has 
been seen, be employed in practical flight) the lift reaches 
its maximum value, and the speed consequently its minimum. 

* At the time of writing (August 1913) the speed record, 171 '7 km. 
per hour or 47'6 m. per second, is held by the Deperdussin monocoque 
with a 140-h.p. motor, weighing 525 kg. with full load, and with a plane 
area of about 12 sq. m. (loading, 43'7 kg. per sq. m.). Another machine 
of the same type, but with a 100-h.p. engine, weighing 470 kg. in all, and 
with an area of 11 sq. m., has attained a speed of 168 km. per hour or 
46'8 m. per second. According to the above method of calculation, the 
flight in both cases was made with a lift coefficient of about 0'0195. 
AUTHOR. 

Since the above was written, all speed records were broken during 
the last Gordon-Bennett race in September 1913. The winner was 
Prevost, on a 160-h.p. Gnome Deperdussin monoplane, who attained a 
speed of a fraction under 204 km. per hour ; while Vedrines, on a 160- 
h.p. Gnome-Ponnier monoplane, achieved close upon 201 km. per hour. 
The Deperdussin monoplane, with an area of 10 sq. m., weighed, fully 
loaded, about 680 kg. ; the Ponnier, measuring 8 sq. m., weighed ap- 
proximately 500 kg. Adopting the same method of calculation, it is 
easily shown that the lift coefficients worked out at 0'021 and 0'020 
respectively. It is just possible that these figures were actually slightly 
smaller, since it is difficult to determine the weights with any consider- 
able degree of accuracy. However, the error, if there be any, is only 
slight, and the result only confirms the author's conclusions. Since that 
time Emile Vedrines is stated to have attained, during an official trial, 
a speed of 212 km. per hour, on a still smaller Ponnier monoplane 
measuring only 7 sq. m. in area and weighing only 450 kg. in flight. 
This would imply a lift coefficient of 0'0185, a figure which cannot be 
accepted without reserve. TRANSLATOR. 



FLIGHT IN STILL AIR 13 

If the angle surpassed 15 the lift would diminish and the 
speed again increase. 

A given aeroplane, therefore, cannot in fact fly below a 
certain limit speed, which in the case of the Breguet already 
considered, for instance, is about 63 km. per hour. 

It will be further noticed that in Table I. one of the 
columns, the second one, contains particulars relating only 
to the Breguet type of plane. If this column were omitted, 
the whole table would give the speed variation of any 
aeroplane with a loading of 20 kg. per sq. m. on its 
planes, for a variation in the lift coefficient of the planes. 
It was this that led to the above remark, made in passing, 
that it was more usual to take the lift coefficient than the 
angle of incidence ; for the former is independent of the 
shape of the plane. 

The speed variation of an aeroplane for a variation in its 
lift coefficient can easily be plotted hi a curve, which would 
have the shape shown in fig. 5, which is based on the figures 
in Table I. 

The previous considerations relate more especially to a 
study of the speeds at which a given type of aeroplane can 
fly. In order to compare the speeds at which different 
types of aeroplanes can fly at the same lift coefficient, we 
need only return to the basic rule already set forth (p. 7). . 
It then becomes evident that these speeds are to one 
another as the square roots of the loading. 

The fact that only the loading comes into consideration 
in calculating the speed of an aeroplane shows that the 
speed, for a given lift coefficient, of a machine does not 
depend on the absolute values of its weight and its plane 
area, but only on the ratio of these latter. The most heavily 
loaded aeroplanes yet built (those of the French military 
trials in 1911) were loaded to the extent of 40 kg. per 
sq. m. of plane area.* The square root of this number 
being 6-32, an aeroplane of this type, driven by a sufficiently 

* The 140-kp. Deperdussin monocoque had a loading of 43-8 kg. 

per sq. m. 



14 



FLIGHT WITHOUT FORMULA 



powerful engine to enable it to fly at a lift coefficient of 
0-02 (the square root of whose inverse is 7-07), could have 
attained a speed equal to 6-32 x 7-07, that is, it could have 
exceeded 44-5 m. per second or 160 km. per hour. 

It is therefore evident that there are only two means for 
increasing the speed of an aeroplane either to reduce the 



30 



20 



10 



00 



Lift 



0.0/0 0.020 0.030 0,0*0 0.050 0.060 0.076 
FIG. 5. 



lift coefficient or to increase the loading. Both methods 
require power ; we shall see further on which of the two is 
the more economical in this respect. 

The former has the disadvantages contested, it is true 
which have already been stated. The latter requires 
exceptionally strong planes. 

In any case, it would appear that, in the present stage of 
aeroplane construction, the speed of machines will scarcely 
exceed 150 to 160 km. per hour ; and even so, this result 



FLIGHT IN STILL AIR 15 

could only have been achieved with the aid of good engines 
developing from 120 to 130 h.p.* So that we are still far 
removed from the speeds of 200 and even 300 km. per hour 
which were prophesied on the morrow of the first advent 
of the aeroplane, f 

In concluding these observations on the speed of aero- 
planes, attention may be drawn to a rule already laid down 
in a previous work,! which gives a rapid method of calculat- 
ing with fair accuracy the speed of an average machine 
whose weight and plane area are known. 

The speed of an average aeroplane, in metres per second, 
is equal to five times the square root of its loading, in kg. 
per sq. m. 

This rule simply presupposes that the average aeroplane 
flies with a lift coefficient of 0-04, the inverse of whose 
square root is 5. The rule, of course, is not absolutely 
accurate, but has the merit of being easy to remember and 
to apply. 

EXAMPLE. What is the speed of an aeroplane weighing 
900 kg., and having an area of 36 sq. m. ? 

900 
Loading = r = 25 kg. per sq. m. 

OO 

Square root of the loading =5. 

Speed required=5x5=25 m. per second or 90 km. 
per hour. 

* In previous footnotes it has already been stated that the Deper- 
dussin monocoques, a 140-h.p. and a 100-h.p., have already flown at 
about 170 km. per hour. But these were exceptions, and, on the whole, 
the author's contention remains perfectly accurate even to-day. TRANS- 
LATOR. 

f The reference, of course, is only to aeroplanes designed for everyday 
use, and not to racing machines. TRANSLATOR. 

J The Mechanics of the Aeroplane (Longmans, Green & Co.). 



CHAPTER II 

FLIGHT IN STILL AIR 

POWER 

IN the first chapter the speed of the aeroplane was dealt 
with in its relation to the constructional features of the 
machine, or its characteristics (i.e. the weight and plane 
area), and to its angle of incidence. It may seem strange 
that, in considering the speed of a motor-driven vehicle, no 
account should have been taken of the one element which 
usually determines the speed of such vehicles, that is, of 
the motive-power. But the anomaly is only apparent, and 
wholly due to the unique nature of the aeroplane, which 
alone possesses the faculty denied to terrestrial vehicles 
which are compelled to crawl along the surface of the 
earth, or, in other words, to move hi but two dimensions of 
being free to move upwards and downwards, in all three 
dimensions, that is, of space. 

The subject of this chapter and the next will be to 
examine the part played by the motive-power in aeroplane 
flight, and its effect on the value of the speed. 

In all that has gone before it has been assumed that, in 
order to achieve horizontal flight, an aeroplane must be 
drawn forward at a speed sufficient to cause the weight of 
the whole machine to be balanced by the lifting power 
exerted by the planes. But hitherto we have left out of 
consideration the means whereby the aeroplane is endowed 
with the speed essential for the production of the necessary 
lifting-power, and we purposely omitted, at the time, to 



FLIGHT IN STILL AIR 17 

deal with the head-resistance or drag, which constitutes, as 
already stated, the price to be paid for the lift. 

This point will now be considered. 

Reverting to the concrete case first examined, that of the 
horizontal flight at an angle of incidence of 7 of a Bleriot 
monoplane weighing 300 kg. and possessing a wing area of 
15 sq. m., it has been seen that the speed of this machine 
flying at this angle would be 20 m. per second or 72 km. 
per hour, and that the drag of the wings at the speed 
mentioned would amount to 33 kg. 

Unfortunately, though alone producing lift in an aero- 
plane, the planes are not the only portions productive of 
drag, for they have to draw along the fuselage, or inter- 
plane connections, the landing chassis, the motor, the 
occupants, etc. 

For reasons of simplicity, it may be assumed that all these 
together exert the same amount of resistance or drag as 
that offered by an imaginary plate placed at right angles 
to the wind, so as to be struck full in the face, whose area 
is termed the detrimental surface of the aeroplane. 

M. Eiffel has calculated from experiments with scale 
models that the detrimental surface of the average 
single-seater monoplane amounted to between f and 1 
sq. m., and that of an average large biplane to about 
1| sq. m.* But it is clear that these calculations can 
only have an approximate value, and that the detrimental 
surface of an aeroplane must always be an uncertain 
quantity. 

But in any case it is evident that this parasitical effect 
should be reduced to the lowest possible limits by stream- 
lining every part offering head-resistance, by diminishing 
exterior stay wires to the utmost extent compatible with 
safety, etc. And it will be shown hereafter that these 
measures become the more important the greater the 
speed of flight. 

The drag or passive resistance can be easily calculated 
* These figures have since been undoubtedly reduced. 

2 



18 PLIGHT WITHOUT FORMULA 

for a given detrimental surface by multiplying its area 
in square metres by the coefficient 0-08 (found to be the 
average from experiments with plates placed normally), and 
by the square of the speed in metres per second. 

Thus, taking once again the Bleriot monoplane, let us 
suppose it to possess a detrimental surface of 0-8 sq. m. ; 
its drag at a speed of 72 km. per hour or 20 m. per second 
will be: 

n ? * Detrimental Square of 
Coefficient. gurface> ^ gpeed> 

0-08 x 0-8 x 400 =26 kg. (about). 

As the drag of the planes alone at the above speed 
amounts to 33 kg., it is necessary to add this figure of 
26 kg., in order to find the total resistance, which is 
therefore equal to 59 kg. The principles of mechanics 
teach that to overcome a resistance of 59 kg. at a speed 
of 20 m. per second, power must be exerted whose amount, 
expressed in horse-power, is found by dividing the product 
of the resistance (59 kg.) and the speed (20 m. per second) 
by 75.* We thus obtain 16 h.p. But a motor of 16 h.p. 
would be insufficient to meet the requirements. 

I^or the propelling plant, consisting of motor and pro- 
peller, designed to overcome the drag or air resistance of 
the aeroplane, is like every other piece of machinery subject 
to losses of energy. Its efficiency, therefore, is only a 
portion of the power actually developed by the motor. 
The efficiency of the power-plant is the ratio of useful, 
power that is, the power capable of being turned to effect 
after transmission to the motive power. 

Thus, in order to produce the 16 h.p. required for 
horizontal flight in the above case of the Bleriot mono- 

* This is easily understood. The unit of power, or horse-power, is 
the power required to raise a weight of 75 kg. to a height of 1 in. in 
1 second, so that, to raise in this time a weight of 59 kg. to a height of 

59 x 20 
20 m., we require =^ h.p. Exactly the same holds good if, instead 

of overcoming the vertical force of gravity, we have to overcome the 
horizontal resistance of the air. 



FLIGHT IN STILL AIR 19 

plane, it would be necessary to possess an engine develop- 
ing 32 h.p. if the efficiency is only 50 per cent., 26-6 h.p. 
for an efficiency of 60 per cent., etc. 

But if the aeroplane were to fly at an angle of incidence 
other than 7 which, as already stated, would depend on 
the position of the elevator the speed would necessarily 
be altered. If this primary condition were modified, the 
immediate result would be a variation in the drag of the 
planes, in the head-resistance of the aeroplane, in the 
propeller-thrust, which is equal to the total drag, and lastly, 
in the useful power required for flight. 

Each value of the angle of incidence and consequently 
of the speed therefore has only one corresponding value 
of the useful power necessary for horizontal flight. 

Returning to the Breguet aeroplane weighing 600 kg. 
with a plane area of 30 sq. m., on which Table I. was based, 
we may calculate the values of the useful powers required 
to enable it to fly along a horizontal path for different 
angles of incidence and for different lifts. The detrimental 
surface may be assumed, for the sake of simplicity, to be 
1-2 sq. m. 

The values of the drag corresponding to those of the 
lift will be obtained from the polar diagram shown in 
fig. 3. 

Table II., p. 20, summarises the results of the calcula- 
tion required to find the values of the useful powers for 
horizontal flight at different lift coefficients. 

Various and interesting conclusions may be drawn from 
the figures in columns 8 and 9 of this Table. 

In the first place, it will be noticed from the figures in 
column 8 that the propeller-thrust (equivalent to the drag 
of the planes added to the head-resistance of the machine, 
i.e. column 6 and column 7) has a minimum value of 91 kg., 
corresponding to a lift coefficient of 0-05, and to the angle 
6. This angle, which, in the case under consideration, 
is that corresponding to the smallest propeller-thrust, is 
usually known as the optimum angle of the aeroplane. 



20 



FLIGHT WITHOUT FORMULA 



TABLE II. 





| 




Speed Values. 


fl 


B fe 

g^g 


1^' -S 
g g ^|| g 


it-is 

*-:- 




&C ^" 




3 ^ o 5T 1 




^ 


o ~~ '**. ^a 


Lift. 


II 

ic 
s *s 




11 

n 

o | 


Drag of P 
(drag x area, 3 
x square of 


il^ 

llfi 

|sl i 1 

w sl x 


Propeller- T 


Useful Power f 
zontal Flight ( 
product of figs 
3 and 8 divide 


m.p.s. 


kni.p.h. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


0-020 





31-6 


113-6 


0-0022 


66kg. 


96kg. 


162kg. 


68 


0-030 


2 


25-8 


92-8 


0-0024! 48 


64 


112 


38 


0-040 


4 


22-3 


80-3 


0-0032 


48 


48 


96 


29 


0-050 


64* 


20-0 


72-0 


0-0044 


53 


38 


91 


24 


0-060 


10 


18-2 


65-6 


0-0063 


63 


32 


95 


23 


0-066 


15 


17-4 62-6 


0-0118 


107 


29 


136 


31 



When the lift coefficient is small, the requisite thrust, 
it will be seen, increases very rapidly, and the same holds 
good for high lift coefficients. 

Secondly, the figures in column 9 show that, together 
with the thrust, the useful power required for flight reaches 
a minimum of 23 h.p., corresponding to a lift value of 
0-06. The angle of incidence at which this minimum of 
useful power can be achieved, about 10 in the present case, 
can be termed the economical angle. 

This angle is greater than the optimum angle, which can 
be explained by the fact that, though the thrust begins 
to increase again, albeit very slowly, when the angle of 
incidence is raised above the optimum angle, the speed still 
continues to decrease to an appreciable extent, and for the 
time being this decrease in speed affects the useful power 
more strongly than the increasing thrust ; and the minimum 
value of the useful power is, consequently, not attained 
until, as the angle of incidence continues to grow, the 



FLIGHT IN STILL AIR 21 

increase in the thrust exactly balances the decrease in the 



The figures in column 9 again show the great expenditure 
of power required for flight at a low lift coefficient. Thus, 
the Breguet aeroplane already referred to, driven by a 
propelling plant of 50 per cent, efficiency, flying at a lift of 
0-05 that is, at a speed of 72 km. per hour only requires 
an engine developing 46 h.p. ; but it would need a 136-h.p. 
engine to fly with a lift of O02, or at about 113 km. per 
hour. It is mainly on this account that, as we have already 
stated, the use of low lift coefficients is strictly limited. 

The variations in power corresponding to variations in 
speed (and in lift) can be plotted in a simple curve. 

Fig. 6 is of exceptional importance, for it may be said to 
determine the character of the machine, and will hereafter 
be referred to as the essential aeroplane curve. 

After these preliminary considerations on the power 
required for horizontal flight, we may now proceed to 
examine the precise nature of the effect of the motive- 
power on the speed, which will lead at the same time to 
certain conclusions relating to gliding flight * 

For this, recourse must be had to one of the most elemen- 
tary principles of mechanics, known as the composition 
and decomposition of forces. The principle is one which 
is almost self-evident, and has, in fact, already been used in 
these pages, when at the beginning of Chapter I. it was 
shown that in the air pressure, which is almost vertical, on 
a plane moving horizontally, a clear distinction must be 
made between the principal part of this pressure, which is 
strictly vertical (the lift), and a secondary part, which is 
strictly horizontal (the drag). 

And, conversely, it is evident that for the action of two 
forces working together at the same time may be substituted 

* There is really no excuse for the importation into English of the 
French term " vol plane," and still less for the horrid anglicism 
" volplane," since " gliding flight " is a perfect English equivalent of the 
French. TRANSLATOR. 



22 



FLIGHT WITHOUT FORMULA 



that of a single force, termed the resultant of these two 
forces. This proceeding is known as the composition of 
forces. So, in compounding the vertical reaction con- 
stituting the lift, and the horizontal reaction which forms 



71) 



40 



3* 



id 



- 



10 



Flying Speeds (m/s). 



FIG. 6. 

The figures on the curve indicate the lift. 

the drift, one obtains the total air pressure, which is simply 
their resultant. 

Both the composition and decomposition of forces is 
accomplished by way of projection. Thus (fig. 7), the force 
Q,* which is inclined, can be decomposed into two forces, 

* A force is represented by a straight line, drawn in the direction in 
which the force operates, and of a length just proportional to the 
magnitude of the force. 



FLIGHT IN STILL AIR 



F and r, vertical and horizontal respectively, by projecting 
in the horizontal and vertical directions the end point A on 
two axes starting from the point 0, where the forces are 
applied. Conversely, these two forces F and r may be 




FIG. 7. 




FligU-Path. ._ 



FIG. 8. 

compounded into one resultant Q, by drawing the diagonal 
of the parallelogram or rectangle of which they form two 
of the sides. We may now return to the problem under 
consideration. 

If we take the aeroplane as a whole, instead of dealing 
with the planes alone, it will be readily seen that the 
horizontal component of the air pressure on the whole 



24 FLIGHT WITHOUT FORMULA 

machine is equal to the drag of the planes added to the 
passive or head-resistance, the while the vertical component 
remains practically equal to the bare lift of the planes, 
since the remaining parts of the structure of an aeroplane 
exert but slight lift, if at all.* The entire pressure of the 
air on a complete aeroplane in flight is therefore farther 
inclined to the perpendicular than that exerted on the planes 
alone. 

If (see fig. 8) the aeroplane is assumed to be represented 
by a single point O, in horizontal flight, the air pressure Q 
exerted upon it may be decomposed into two forces, of 
which the lift F is equal and directly opposite to the 
weight P, and the drag r, or total resistance to forward 
movement, which must be exactly balanced by the thrust 
t of the propeller. 

But, supposing the engine be stopped and the propeller 
consequently to produce no thrust (fig. 9), the aeroplane 
will assume a descending flight-path such that the planes 
still retain the single angle of 7, for instance, which we 
have assumed, so long as the elevator is not moved, and 
such that the air pressure Q on the planes becomes 
absolutely vertical, in order to balance the weight of the 
machine, instead of remaining inclined as heretofore. This 
is gliding flight. 

Relatively to the direction of flight, the air pressure Q 
still retains its two components, of which r is simply the 
resistance of the air opposed to the forward movement of 
the glider. The second component F is identical to the 
lifting power in horizontal flight, and its value is obtained 
by multiplying the lift coefficient corresponding to the 
angle 7 by the plane area, and by the square of the speed 
of the aeroplane on its downward flight-path. 

Fig. 9 shows that, by the very fact of being inclined, the 
force F is slightly less than the weight of the machine, but, 
since the gliding angle of an aeroplane is usually a slight 

* For the sake of simplicity, we may consider that the tail plane, 
which will be hereinafter dealt with, exerts no lift. 



FLIGHT IN STILL AIR 



25 



one, the lifting power F may still be deemed to be equal to 
the weight of the aeroplane. 

Clearly, therefore, every consideration in the first chapter 
which related to the speed in horizontal flight is equally 
applicable to gliding flight, so that it may be said that 





FIG. 9. 

when an aeroplane begins to glide, without changing its 
angle, the speed remains the same as before. 

In fact, horizontal flight is simply a glide in which the 
angle of the flight-path has been raised by mechanical 
means. 

On comparing figs. 8 and 9 it will be seen that this angle 
is that which, in fig. 8, is marked by the letter p. If this 



26 



FLIGHT WITHOUT FORMULAE 



angle is represented, as in the case of any gradient, in 
terms of a decimal fraction, it will be found to depend on 
the ratio which the forces r and F bear to one another. 
Hence, the following rule may be stated : 

RULE. The gliding angle assumed at a given angle of 
incidence by any aeroplane is equal to the thrust required 
for its horizontal flight at the same angle, divided by the 
weight of the machine. 

Thus the Bleriot monoplane dealt with in the first 
instance, which requires for horizontal flight at an angle 
of 7 a thrust of 59 kg., and weighs 300 kg., would assume 
on its glide, at the same angle of incidence, a descending 

59 

flight-path equal to , or 0-197, which is equivalent to 
oOO 

nearly 20 cm. in every metre (1 in 5). The Breguet 
aeroplane on which Tables I. and II. were based, weighing 
600 kg., would assume at different angles (or lift coefficients) 
the gliding angles shown in Table III. 

TABLE III. 



Angle 
corre- 
Lift. spending 
to the 
Lift. 


Speed. 


Propeller- 
Thrust in 
Horizontal 

Flight. 


Gliding Angle 
Weight 
(600kg.) 
divided by 
figures in 
col. 5. 


m/s. 


km/li. 


1 2 


3 4 


5 


6 


0-02 


31-6 


113-6 


162 


0-270 


I 










0-03 | 2 


25-8 


92-8 


112 


0-187 


0-04 ! 4 


22-3 


80-3 


96 


0-160 






| 




6i 


20-0 


72-0 91 


0-151 


0-06 10 


18-2 


65-6 


9f> 


0-158 


0-066 15 


17-4 


62-6 


136 


0-22C 



It will now be seen that the best gliding angle is obtained 
when the angle of incidence is the same as the optimum 



FLIGHT IN STILL AIR 27 

angle of the aeroplane. The latter, therefore, is the best 
from the gliding point of view, so far as the length of the 
glide is concerned. 

In fig. 10, starting from a point 0, are drawn dotted lines 
corresponding to the gliding angle given in column 6 of 
Table III., and on these lines are marked off distances 
proportional to the speed values set out in columns 3 or 4 ; 
the diagram will then give, if the points are connected 
into a curve, the positions assumed, in unit time, by a 
glider, launched at the various angles from the point 0. 

It will be observed in the first place that any given 
gliding path, such as OA, for instance, cuts the curve at 
two points, A and B, thus showing that this gliding path 
could have been traversed by the aeroplane at two different 
speeds, OA and OB, corresponding to the two different 
angles of incidence, 1 and 15 in the present case. 

Only for the single gliding path OM, corresponding to 
the smallest gliding slope and the optimum angle of inci- 
dence, do these two points coincide. 

But it is not by following this gliding path that an aero- 
plane will descend best in the vertical sense during a given 
period of time ; for this it will only do by following the 
path OE corresponding to the highest point on the curve, 
and the angle of incidence to be adopted to achieve this 
result is none other than the economical angle. But the 
difference in the rate of fall is only slight for the example 
in question. 

It will be noted that as the angle of incidence diminishes, 
the gliding angle rapidly becomes steeper. If the curve 
were extended so as to take in very small angles of 
incidence, it would be found that at a lift coefficient of 
0-015 the gliding path would already have become very 
steep, that this steepness would increase very rapidly for 
the coefficient 0-010, and that at 0-005 it approached a 
headlong fall. The fall, in fact, must become vertical when 
the lift disappears, that is, when the plane meets the air 
along its imaginary chord. 



28 FLIGHT WITHOUT FORMULA 




FLIGHT IN STILL AIR 29 

In these conditions, a slight variation in the lift there- 
fore brings about a very large alteration in the gliding 
angle, and this effect is the more intense the smaller the 
lift coefficient. The glide becomes a dive. Hence it is 
clear that this is another danger of adopting a low lift 
coefficient. 

This brief discussion on gliding flight, interesting enough 
in itself, was necessary to a proper understanding of the 
part played by power in the horizontal flight of an aero- 
plane, for we can now regard the latter in the light of 
a glide in which the gliding path has been artificially 
raised. 

And this raising of the gliding path is due to the power 
derived from the propelling plant. 

This will be better understood if we assume that, during 
the course of a glide, the pilot started up his engine again 
without altering the position of the elevator, so that the 
planes remained at the same angle as before ; the gliding 
path would gradually be raised until it attained and even 
surpassed the horizontal, while the aeroplane (as has been 
seen) would approximately maintain the same speed 
throughout. 

Hence it may be said that when the angle of incidence 
remains constant, the speed of an aeroplane is not produced 
by its motive power, as in the case of all other existing 
vehicles, since, when the motor is stopped, this speed is 
maintained. 

The function of the power-plant is simply to overcome 
gravity, to prevent the aeroplane from yielding, as it in- 
evitably must do in calm air, to the attraction of the earth ; 
in other words, to govern its vertical flight. 

In the case now under consideration, the speed therefore 
is wholly independent of the power, since, as has been seen, 
it is entirely determined by the angle of incidence, and if 
this remains constant, as assumed, any excess of power will 
simply cause the aeroplane to climb, while a lack of power 



30 FLIGHT WITHOUT FORMULA 

will cause it to coine down, but without any variation in the 
speed. 

But this must not be taken to imply that the available 
motive-power cannot be transformed into speed, for such, 
happily, is not the case. Hitherto the elevator has been 
assumed to be immovable so that the incidence remained 
constant. 

As a matter of fact, the incidence need only be diminished 
through the action of the elevator in order to enable the 
aeroplane to adopt the speed corresponding to the new 
angle of planes, and in this way to absorb the excess of 
power without climbing. 

Nevertheless and the point should be insisted upon as 
it is one of the essential principles of aeroplane flight the 
angle of incidence alone determines the speed, which cannot 
be affected by the power save through the intermediary of 
the incidence. 

Hitherto we have constantly alluded to the different 
speeds at which an aeroplane can fly, as if, in practice, 
pilots were able to drive their machines at almost any 
speed they desired. In actual fact, a given aeroplane 
usually only flies at a single speed, so that we are in 
the habit of referring to the X biplane as a 70 km. 
per hour machine, or of stating that the Y monoplane 
does 100 km. per hour. This is simply because up to 
now, and with very few exceptions, pilots run their engines 
at their normal number of revolutions. In these conditions 
it is evident that the useful power furnished by the 
propelling plant determines the incidence, and hence the 



Thus, referring once again to Table II., it will be seen 
for example that, if the Breguet biplane receives 29 h.p. 
in useful power from its propelling plant, the pilot, in 
order to maintain horizontal flight, will have to manipulate 
his elevator until the incidence of the planes is approxi- 
mately 4, which corresponds to the lift 0-040. 

The speed, then, would only be about 80 km. per hour. 



FLIGHT IN STILL AIR 31 

Experience teaches the pilot to find the correct position 
of the elevator to maintain horizontal flight. Should the 
engine run irregularly, and if the aeroplane is to maintain 
its horizontal flight, the elevator must be slightly actuated 
in order to correct this disturbing influence. 

Horizontal flight, therefore, implies a constant mainten- 
ance of equilibrium, whence the designation equilibrator, 
which is often applied to the elevator, derives full justifi- 
cation. 

But if the engine is running normally, the incidence, and 
consequently the speed, of an aeroplane remain practically 
constant, and these constitute its normal incidence and 
speed. 

Generally the engine is running at full power during 
flight, and so in the ordinary course of events the normal 
speed of an aeroplane is the highest it can attain. 

But there is a growing tendency among pilots to 
reserve a portion of the power which the engine is 
capable of developing, and to throttle down in normal 
flight. In this case the reserve of power available may 
be saved for an emergency, and be used the case will be 
dealt with hereafter for climbing rapidly, or to assume 
a higher speed for the time being. In this case the 
normal speed is, of course, no longer the highest possible 
speed. 

In the example already considered, the Breguet biplane 
would fly at about 80 km. per hour, if it possessed useful 
power amounting to 29 h.p. 

But by throttling down the engine so that it normally 
only produced a reduced useful power equivalent to 24 h.p., 
the normal speed of the machine, according to Table II., 
would only be 72 km. per hour (the normal incidence being 
6| and the lift 0-050). 

The pilot would therefore have at his disposal a surplus 
of power amounting to 5 h.p., which he could use, by, 
opening the throttle, either for climbing or for temporarily 
increasing his speed to 80 km. per hour. 



32 FLIGHT WITHOUT FORMULA 

Although, therefore, an aeroplane usually only flies at 
one speed, which we call its normal speed, it can perfectly 
well fly at other speeds, as was shown in Chapter I. But, 
in order to obtain this result, it is essential that on each 
occasion the engine should be made to develop the precise 
amount of power required by the speed at which it is 
desired to fly. 

Speed variation can therefore only be achieved by 
simultaneously varying the incidence and the power, or, in 
practice, by operating the elevator and the throttle together. 
This may be accomplished with greater or less ease 
according to the type of motor in use, but certain pilots 
practise it most cleverly and succeed in achieving a very 
notable speed variation, which i? of great importance, 
especially in the case of high-speed aeroplanes, at the 
moment of alighting. 

As has already been explained, the horizontal flight of an 
aeroplane may be considered in the light of gliding flight 
with the gliding angle artificially raised. From this point 
of view it is possible to calculate in another way the power 
required for horizontal flight. 

For instance, if we know that an aeroplane of a given 
weight, such as 600 kg., has, for a given incidence, a glid- 
ing angle of 16 cm. per metre (approximately 1 in 6) at 
which its speed is 22-3 m. per second, we conclude that 
in 1 second it descended 0-16 x 22-3=3-58 m. Hence, 
in order to overcome its descent and to preserve its hori- 
zontal flight, it would be necessary to expend the useful 
power required to raise a weight of 600 kg. to a height 
of 3-58 m. in 1 second. Since 1 h.p. is the unit required 
to raise a weight of 75 kg. to a height of 1 m. in 1 second, 

the desired useful power = about 29 h.p. This, 

as a matter of fact, is the amount given by Table II. for 
the Breguet biplane which complies with the conditions 
given. 

In order to find the useful power required for the 



FLIGHT IN STILL AIR 33 

horizontal flight of an aeroplane flying at a given incidence, 
and hence at a given speed, multiply the weight of the machine 
by this speed and by the gliding angle corresponding to the 
incidence, and divide by 75. 

By a similar method one may easily calculate the useful 
power required to convert horizontal flight into a climb at 
any angle. 

Thus, if the aeroplane already referred to had to climb, 
always at the same speed of 22-3 m. per second, at an angle 
of 5 cm. per metre (1 in 20), it would be necessary to expend 
the additional power 

0-05x600x22-3 , 

=about 9 h.p. 

75 

Of course, this expenditure of surplus power would be 
greater the smaller the efficiency of the propeller, and 
would be 12 h.p. for 75 per cent, efficiency, and 18 h.p. 
for 50 per cent, efficiency. 

Clearly, this method of making an aeroplane climb by 
increasing the motive power can only be resorted to if 
there is a surplus of power available, that is, if the engine 
is not normally running at full power, which until now is 
the exception. 

For this reason, when, as is generally the case, the engine 
is running at full power, climbing is effected in a much 
simpler manner, which consists in increasing the angle of 
incidence of the planes by means of the elevator. 

Let us once more take our Breguet biplane which, with 
motor working at full power, flies at a normal speed of 
22-3 m. per second (80-3 km. per hour) at 4 incidence (or 
a lift coefficient of 0-040). The useful power needed to 
achieve this speed (see Table II.) is 29 h.p. 

Assume that, by means of his elevator, the pilot increases 
the angle of incidence to 10 (lift coefficient 0-060). Since 
horizontal flight at this incidence, which must inevitably 
reduce the speed to 18-2 m. per second or 65-6 km. per 
hour, would only require 23 h.p., there will be an ex- 

3 



34 FLIGHT WITHOUT FORMULAE 

cess of power amounting to 6 h.p.,* and the aeroplane 
will rise. 

The climbing angle can be calculated with great ease. 
The method is just the converse of the one we have just 
employed, and thus consists in dividing 6x75 (representing 
the surplus power) by 600 x20 (weight multiplied by speed), 
which gives an angle of 3*75 cm. per metre (1 in 27 
about). 

This climbing rate may not appear very great ; still, for 
a speed of 18-2 m. per second, it corresponds to a climb 
of 68 cm. per second=41 m. per minute=410 m. in 10 
minutes, which is, at all events, appreciable. 

The aeroplane, therefore, may be made to climb or to descend 
by the operation of the elevator by the pilot. 

More especially is the elevator used for starting. In 
this case the elevator is placed in a position corresponding 
to a very slight incidence of the main planes, so that these 
offer very little resistance to forward motion when the 
motor is started and the machine begins to run along the 
ground. As soon as the rolling speed is deemed sufficient, 
the elevator is moved to a considerable angle, which causes 
the planes to assume a fairly high incidence, and the aero- 
plane rises from the ground. 

* This is not strictly correct, since, as will be seen hereafter, the 
propeller efficiency varies to some extent with the speed of the aeroplane ; 
still, we shall not make a grievous error in assuming that the efficiency 
remains the same. 



CHAPTER III 

FLIGHT IN STILL AIR 

POWER (concluded] 

THE second chapter was mainly devoted to explaining how 
one may calculate the useful power required for horizontal 
flight, at the various angles of incidence and at the different 
lift coefficients in other words, at the various speeds of a 
given aeroplane. 

In addition, gliding flight has been briefly touched on, 
and has served to show the precise manner in which the 
power employed affects the speed of the aeroplane. 

In the present chapter this discussion will be completed ; 
it will be devoted to finding the best way of employing the 
available power to obtain speed. Incidentally, we shall 
have occasion to deal briefly with the limits of speed which 
the aeroplane as we know it to-day seems capable of 
attaining. 

It has been shown that the flight of a given aeroplane 
requires a minimum useful power, and that this is only 
possible when the angle of incidence is that which we have 
termed the economical angle. 

The power would therefore be turned to the best account, 
having regard merely to the sustentation of the aeroplane, 
by making it fly normally at its economical angle. 

But, on the other hand, this method is most defective 
from the point of view of speed, for as fig. 6 (Chapter II.) 
clearly shows, when the machine flies at its economical 
angle, a very slight increase in power will increase the 

35 



36 FLIGHT WITHOUT FORMULA 

speed to a considerable extent. Besides, the method in 
question would be worthless from a practical point of view, 
since it is evident that an aeroplane flying under these 
conditions would be endangered by the slightest failure of 
its engine. 

Such, in fact, was the case with the first aeroplanes which 
actually rose from the ground ; they flew " without a 
margin," to use an expressive term. And even to-day the 
same is true of machines whose motor is running badly : 
in such a case the only thing to be done is to land as soon 
as possible, since the aeroplane will scarcely respond to the 
controls. 

The other characteristic value of the angle of incidence 
referred to in Chapter II., there called the optimum angle, 
corresponds to the least value of the ratio between the 
propeller-thrust and the weight of the aeroplane, or to its 
equivalent the best gliding angle. 

For the best utilisation of the power in order to obtain 
speed, which alone concerns us for the moment, there is a 
distinct advantage attached to the use of the optimum 
angle for the normal incidence of the machine ; Colonel 
Renard, indeed, long ago pointed out that by using the 
optimum angle for normal flight in preference to the 
economical angle, one obtained 32 per cent, increase in speed 
for an increase in power amounting to 13 per cent. only. 

In any case, when the incidence is optimum the ratio 
between the speed and the useful power required to obtain 
it is largest. This is easily explained by reference to 
Chapter II., which showed that the useful power required 
for horizontal flight at a given incidence is proportional to 
the speed multiplied by the gliding angle of the aeroplane 
at the same incidence. 

When the gliding angle is least (i.e. flattest), that is, 
when the incidence is that of the optimum angle, the ratio 
of power to speed is also smallest, and hence the ratio of 
speed to maximum power. 

It would therefore appear that by using the optimum 



FLIGHT IN STILL AIR 37 

angle as the normal incidence we would obtain the best 
results from the point of view with which we are at present 
concerned, which is that of the most profitable utilisation 
of the power to produce speed. This, in fact, is generally 
accepted as the truth, and in his scale model experiments 
M. Eiffel always recorded this important value of the angle 
of incidence, together with the corresponding flattest gliding 
angle. 

Nevertheless we are not prepared to accept as inevitably 
true that the optimum angle is necessarily the most ad- 
vantageous for flight, so far as the transmutation of power 
into speed is concerned. This will now be shown by 
approaching the question in a different manner, and by 
finding the best conditions under which a given speed can 
be attained. 

The power required for flight is proportional, as has been 
shown, to the propeller-thrust multiplied by the speed. 
Hence, on comparing different aeroplanes flying at the same 
speed, it will be found that the values of the power ex- 
pended to maintain flight will have the same relation to one 
another as the corresponding values of the propeller-thrust. 

If we assume that the detrimental surface of each one 
of these aeroplanes is identical, the head-resistance will be 
the same in each case, since it is proportional to the detri- 
mental surface multiplied by the square of the speed (which 
is identical in every case). 

It follows that the speed in question will be attained 
most economically by the aeroplane whose planes exert 
the least drag. Now, it was shown in Chapter II. that 
the drag of the wings of an aeroplane is a fraction of the 
weight of the machine equal to the ratio between the 
drag coefficient and the lift coefficient corresponding to 
the incidence at which flight is made. 

If we assume, therefore, that the weight of each aero- 
plane is identical, it follows that the best results are given 
by that machine whose planes in normal flight have the 
smallest drag-to-lift ratio. 



38 FLIGHT WITHOUT FORMULA 

Reference to the polar diagrams (Chapter I., figs. 1, 2, 3, 
and 4) shows that the minimum drag-to-lift ratio occurs 
at the angle of incidence corresponding to the point on 
the curve where a straight line rotated about the centre 
0-00 comes into contact with the curve. This angle of 
incidence is beyond all question, for any aeroplane provided 
with planes of the types under consideration, the most 
profitable from our point of view ; this angle, in other 
words, is that at which an aeroplane of given weight can 
fly at a given speed for the least expenditure of power, 
and this for any weight and speed. Hence this is the 
angle at which an aeroplane possessing one of these wing 
sections should always fly in theory. Accordingly, it may be 
termed the be t angle of incidence, and the corresponding 
lift coefficient the best lift coefficient. 

The value of the best incidence only depends on the 
wing section, but it is always smaller than the optimum 
angle, which in its turn depends not only on the wing 
section but also on the ratio of the detrimental surface to 
the plane area. 

A straight line rotated from the centre 0-00 in figs. 2, 3, 
and 4 indicates that the best lift coefficients for M. Farman, 
Breguet, and Bleriot XI. plane sections are respectively 
0-017, 0-035, and 0-047, corresponding to the best angles 
of incidence 1|, 2, and 6. These values can only be de- 
termined with some difficulty, however, since the curves 
are so nearly straight at these points that the rotating line 
would come into contact with the curves for some distance 
and not at one precise point alone. 

On the other hand, it is evident that the drag-to-lift ratio 
only varies very slightly for a series of angles of incidence, 
the range depending on the particular plane section, so 
that one is justified in saying that each type of wing pos- 
sesses not only one best incidence and one best lift, but 
several good incidences and good lifts. 

Thus, for the Maurice Farman section, the good lifts lie 
between 0-010 and 0-025 approximately, and the corre- 



FLIGHT IN STILL AIR 39 

spending good incidences extend from 1 to 4, while the 
drag-to-lift ratio between these limits remains practically 
constant at 0-065. 

For the Breguet wing, the good lifts are between 0-030 
and 0-045, the good incidences between 3 and 6, and the 
drag-to-lift ratio remains about 0-08. 

Lastly, for the Bleriot XI. the same values read as 
follows : 0-030 and 0-055, 3 and 6, and about 0-105. 

Even at this point it becomes evident that the use of 
.slightly cambered wings is the more suitable for flight 
with a low lift coefficient, and that for a large lift a heavily 
cambered wing is preferable. 

If the optimum angle of an aeroplane, which depends, 
as already shown, on the ratio between the detrimental 
surface and the plane area, is included within the limits 
of the good incidences, its use as the normal angle of in- 
cidence remains as advantageous as that of any other 
" good " incidence. But if it is not included,* flight at the 
optimum angle would require, in theory at all events, a 
greater expenditure of power than would be required under 
similar conditions if flight took place at any of the good 
incidences. 

This shows that the optimum angle is not necessarily 
that at which an aeroplane should fly normally in order to 
use the power most advantageously. 

To sum up : the normal speed should always correspond 
to a " good " angle of incidence. 

Should this not be the case in fact, it would be possible 
to design an aeroplane which, for the same weight and 
detrimental surface as the one under consideration, could 
achieve an equal speed for a smaller expenditure of power. 

A concrete example will render these considerations 
clearer. 

In Table II. (Chapter II.) there was set out the variation 

* This would be possible more particularly in the case of aeroplanes 
with very slightly cambered planes and small wing area and considerable 
detrimental surface. 



40 FLIGHT WITHOUT FORMULA 

of the useful power required for the horizontal flight of a 
Breguet aeroplane weighing 600 kg., with a plane area of 
30 sq. m. and a detrimental surface of 1-20 sq. m., according 
to its speed. 

Let us assume that the useful power 24 h. p. developed 
by the propeller makes the aeroplane fly normally at 
0-050 lift, or at its optimum incidence. The speed will 
then be 72 km. per hour or 20 m. per second. This lift 
coefficient 0-050, be it noted, is slightly greater than the 
highest of the good incidences peculiar to the Breguet 
section. 

Now let us take another aeroplane of the same type, also 
weighing 600 kg. and with the same detrimental surface 
of 1-20 sq. m., but with 40 sq. m. plane area, which should 
still fly at the same speed of 20 m. per second. 

The lift coefficient may be obtained (cf. Chapter I.) by 
dividing the loading of the planes (15 kg.) by the square of 
the speed in metres per second (400), which gives 0-0375. 
Now this is one of the good lift coefficients of the Breguet 
plane. In these conditions, therefore, the drag-to-lift ratio 
will assume the constant value of about 0-08 common to all 
good incidences. 

It follows that the drag of the planes will be equal to 
the weight, 600 kg. x 0-08=48 kg. 

The head-resistance, on the other hand, will remain the 
same as in the original aeroplane whose speed was 72 km. 
per hour, since head-resistance is dependent simply on the 
amount of detrimental surface and on the speed (neither of 
which undergoes any change). The head-resistance, there- 
fore (cf. Chapter II.), equals 38 kg. 

The propeller-thrust, equal to the sum of head-resistance 
and drag of the planes, will be 86 kg., and the useful power 
required for flight = 

Thrust (86) XqeedjjO^ ^ 

75 

The figure thus obtained is less than the 24 h.p. of useful 



FLIGHT IN STILL AIR 41 

power required to make the aeroplane first considered fly 
at 72 km. per hour. 

Therefore, in theory at all events, the optimum angle is 
not necessarily the most advantageous from the point of 
view of the least expenditure of power to obtain speed. 
But in practice the small saving in power would probably 
be neutralised owing to the difficulty of constructing two 
aeroplanes of the same type with a plane area of 30 and 
40 sq. m. respectively without increasing the weight and 
the detrimental surface of the latter. Hence the advantage 
dealt with would appear to be purely a theoretical one in 
the present case. 

But this would not be so with an aeroplane whose normal 
angle of incidence was smaller than the good incidences 
belonging to its particular plane section. For instance, let 
us assume that the propeller of the Breguet aeroplane 
(vide Table II.) furnishes normally 68 useful h.p., which 
would give the machine a speed of 113-6 km. per hour or 
31-6 m. per second, at the lift 0-020, which is less than the 
good lifts for this plane section. 

Now take another Breguet aeroplane of the same weight 
and detrimental surface, but with a plane area of only 
20 sq. m. Calculating as before, it will be found that in 
order to achieve a speed of 113-6 km. per hour, this machine 
Avould have to fly with a lift of 0-030, which is one of the 
good lifts, and that useful power amounting to only 60 h.p. 
would be sufficient to effect the purpose. This time the 
advantage of using a good incidence as the normal angle is 
clearly apparent. 

As a matter of fact, in practice the advantage would 
probably be even more considerable, since a machine with 
20 sq. m. plane area would probably be lighter and have 
less detrimental surface than a 30 sq. m. machine. 

Care should therefore be taken that the normal angle of 
an aeroplane is included among the good incidences belonging 
to its plane section, and, above all, that it is not smaller than 
the good incidences. 



42 PLIGHT WITHOUT FORMULA 

This manner of considering good incidences and lifts 
provides a solution of the following problem which was 
referred to in Chapter I. : 

Since there are only two means of increasing the speed 
of an aeroplane either by increasing the plane loading 
or by reducing the lift coefficient which of these is the more 
economical ? 

To begin with, the question will be examined from a 
theoretical point of view, by assuming that the adoption of 
either means will have the same effect in each case on the 
weight and the detrimental surface, since the values of 
these must be supposed to remain the same in the various 
machines to enable our usual method of calculation to be 
applied. 

This being so, it will be readily seen that as long as the 
normal lift remains one of the good lifts, both means of 
increasing the speed are equivalent as far as the expenditure 
of useful power is concerned. 

On the one hand, since the drag-to-lift ratio retains 
approximately the same value for all good lifts, the drag 
of the planes will remain for every angle of incidence a 
constant fraction of the weight, which is assumed to be in- 
variable. On the other hand, at the speed it is desired to 
attain, the head-resistance, proportional to the detrimental 
surface, which is also assumed to be invariable, will remain 
the same in both cases. Consequently, the propeller-thrust, 
equal to the sum of the two resistances (drag of the planes 
4-head-resistance), and hence the useful power, will retain 
the same value by whichever of the two methods the increase 
in speed has been obtained. 

But if the lift had already been reduced to the smallest 
of the good lift values, and it was still desired to increase 
the speed, the most profitable manner of doing this would 
be to increase the loading by reducing the plane area. So 
much for the theoretical aspect of the problem. 

Purely practical considerations strengthen these theor- 
etical conclusions, in so far as they clearly prove the ad- 



FLIGHT IN STILL AIR 43 

vantage of increasing the speed by the reduction of plane 
area, even where the lift remains one of the good lift 
values. 

Indeed, in practice the two methods are no longer equiva- 
lent in the latter case, since, as already mentioned, the 
reduction of the wing area is usually accompanied by a 
decrease in the weight and detrimental surface. 

Generally speaking, it is therefore preferable to take the 
highest rather than the lowest of the good lifts as the 
normal angle of incidence, and this conclusion tallies, 
moreover, with that arising from the danger of flying 
at a very low lift. Finally, the normal angle would thus 
remain in the neighbourhood of the optimum angle, 
which is an excellent point so far as a flat gliding angle 
is concerned.* 

Obviously, the advantage of the method of increasing 
the speed by reducing the plane area over that consisting 
in reducing the lift becomes greater still in the case where 
the latter method, if applied, would lead to the lift being 
less than any of the good lift values. 

The disadvantage of greatly reducing the plane area 
to obtain fast machines is the heavy loading which it 
entails and the lessening of the gliding qualities. The best 
practical solution of the whole problem would therefore 
appear to consist in a judicious compromise between these 
two methods. 

As usual, a concrete example will aid the explanation 
given above. 

Let the Breguet aeroplane already referred to be supposed 
to fly at a speed of 92-8 km. per hour with a lift of 0-030, 
which is the lowest of its good lift values. Table II. shows 
that this would require 38 h.p. 

Another machine of the same type, and having the same 
weight and detrimental surface, but with an area of only 
20 sq. m. (instead of 30), in order to attain the same speed 

* Chapter X. will show that this conclusion is strengthened still 
further by the effect of wind on the aeroplane. 



44 FLIGHT WITHOUT FORMULA 

would have to fly at 0-040 lift, which is also one of the 
good lift values. 

The necessary calculations would show that the latter 
machine, like the former, would also require 38 h.p. This is 
readily explicable on the score that the drag of the planes 
is 0-08 of the weight, or 48 kg., while the head-resistance 
also remains constant and equal to 64 kg. (Table II.). 

In theory, therefore, there is nothing to choose between 
either solution. But in practice the latter is preferable, 
since the 20 sq. m. machine would in all likelihood be lighter 
and possess less detrimental surface. 

But if a speed of 113-6 km. per hour were to be attained, 
the 20 sq. m. aeroplane has a distinct advantage both in 
theory, and even more in practice, for the machine with 
30 sq. m. area would have to fly at 0-020 lift, which is 
lower than the good lift values belonging to the Breguet 
plane section, which would, as already shown, require 
useful power amounting to 68 h.p., whereas 60 h.p. would 
suffice to maintain the smaller machine in flight at the 
same speed. 

We have already set forth the good lift values belonging 
to the Maurice Farman, Breguet, and Bleriot XI. plane 
sections, and the corresponding values of the drag-to-lift 
ratio or, its equivalent, the ratio of the drag of the planes 
to the weight of the machine. . 

Reference to these values has already shown that slightly 
cambered planes are undoubtedly more economical for low 
lift values, which are necessary for the attainment of high 
speeds, especially in the case of lightly loaded planes, as in 
some biplanes. 

But the good lift values of very flat planes are usually 
very low from 0-010 to 0-025 in the case of the Maurice 
Farman which greatly restricts the use of these values, 
since, as already stated, it is doubtful whether hitherto an 
aeroplane has flown at a lower lift value than 0-020. 

The advantages and disadvantages of these three wing 
sections, from the point of view at issue, will be more readily 



FLIGHT IN STILL AIR 



45 



BL 


?RIOT 
0^ 


n XI 










s 

\ 






Bf 


EGUE 


T O^ 


\ 










5 




/'~ 


~N 


1 


M.I 


'ARM* 


I/V 


\ 








1 


















\ 










1 










I 










! 










1 o 1 
1 
9 










, i 

4 ! 










A 













O.Q8 



0.07 



0.08 



seen by plotting their polar curves in one diagram, as 
shown in fig. 11. 

The Breguet and Maurice Farman curves intersect at a 
point corresponding to the 
lift value 0-030, whence we 
may conclude that for all lift 
values lower than this, the 
Maurice Farman section is 
the better,* but for all lift 
values higher than 0-032 
(which at present are more 
usual), the Breguet wing 
has a distinct advantage. 
In the same way, the 
Maurice Farman is better 
than the Bleriot XI. for 
lift values below 0-042, 
whereas the latter is better 
for all higher lift values. 

Finally, the Bleriot XI. 
only becomes superior to 
the Breguet for lift values 
in excess of 0-065, which 
are very high indeed, and 
little used owing to the 
fact that they correspond 
to angles in the neighbour- 
hood of the economical 
angle. 

To apply these various 
considerations, we will now 
proceed to fix the best con- 
ditions in which to obtain 



A 0.04 



0.03 



0.02 



0.01 



0.02 
Drag. 



0.01 



o'uo 



o.oq 



FIG. 11 



a speed of 160 km. per 
hour or about 44-5 m. per second, which appears to be 
the highest speed which it seems at present possible to 
* Since it has a smaller drag for the same lift. 



46 FLIGHT WITHOUT FORMULA 

reach,* that is, by assuming it to be possible to have a 
loading of 40 kg. per sq. m. of surface and to fly at a lift- 
value of 0-020. 

In laying down this limit to the speed of flight we also 
stated our belief that, in order to enable it to be attained, 
engines developing from 120 to 130 effective h.p. would 
have to be employed. 

This opinion was founded on the results of M. Eiffel's 
experiments, from which it was concluded that an aeroplane 
to attain this speed would have to possess a detrimental 
surface of no more than 0-75 sq. m. 

Now, the last two Aeronautical Salons, those of 1911 
and 1912, have shown a very clearly marked tendency 
among constructors to reduce all passive resistance to the 
lowest possible point, especially in high-speed machines, 
and it would appear that in this direction considerable 
progress has been and is being made. 

One machine in particular, the Paulhan-Tatin " Torpille," 
specially designed with this point in view, is worthy of 
notice. 

Its designer, the late M. Tatin, estimated the detrimental 
surface of this aeroplane at no more than 0-26 sq. m., and 
its resistance must in fact have been very low, since it had 
the fair-shaped lines of a bird, every part of the structure 
capable of setting up resistance being enclosed in a shell-like 
hull from which only the landing wheels, reduced to the 
utmost verge of simplicity, projected. 

Taking into account the slightly less favourable figures 
obtained by M. Eiffel from experiments with a scale model, 
the detrimental surface of the " Torpille " may be estimated 
at 0-30 sq. m. 

According to information given by M. Tatin himself, the 
weight of this monoplane was 450 kg., and its plane area 
12-5 sq. m. 

* It should, however, be remembered that this limit has actual^ 
been exceeded, with a loading of 44 kg. per sq. m. and a lift value of 
slightly less than 0*020. See also Translator's note on p. 12. 



FLIGHT IN STILL AIR 47 

Let us assume that the planes, which were only very 
slightly cambered, were about equivalent to those of the 
Maurice Farman, and that they flew at a good lift coefficient. 
In that case the drag of the planes would be equal to 0-065 
of the weight of the machine, or to 29-5 kg. 

On the other hand, at the speed of 44-5 m. a second, the 
head-resistance = 

n ffi Detrimental Square of 

Coefficient. ^^ ^ s ^ 

0-08 x 0-3 X 1980 - 47-5 kg. 

The propeller-thrust, consequently, the sum of both 
resistances, would =17 kg. 

The useful power required would thus= 

77x44-5 , 

=about 45 h.p. 

75 

Propeller efficiency in this case must have been exception- 
ally high (as will be seen hereafter), and was probably in 
the region of 80 per cent. 

The engine -power required to give the " Aero-Torpille " a 

A* 

speed of 160 km. per hour must therefore have been = 

0*8 

57 h.p., or approximately 60 h.p. 

M. Tatin considered that he could obtain the same result 
with even less motive-power, and that some 45 h.p. would 
suffice. If this proves to be the case, the detrimental 
surface of the aeroplane would have to be less than 0-30 
sq. m. and the propeller efficiency even higher than 80 per 
cent., or else and this was M. Tatin's own opinion the 
coefficients derived from experiments with small scale 
models must be increased for full-size machines, their 
value possibly depending in some degree on the speed.* 

* No proof, as a matter of fact, was possible owing to the short life of 
the machine. But the results obtained from other machines in which 
stream-lining had been carried out to an unusual degree, such as the 
Deperdussin " monocoque " which, with an engine of 85-90 effective 
h.p., only achieved 163 km. per hour would appear to show that the 



48 FLIGHT WITHOUT FORMULA 

It should also be noted that, in order to attain 160 km. 
per hour, the Tatin " Torpille " would have to fly at a lift 
coefficient equal to 

36 (loading) =0-018 

1980 (square of the speed) 

Perhaps it will seem strange that simply by estimating 
the value of the detrimental surface at 0-30 instead of the 
previous estimate of 0-75, the motive power required for 
flight at 160 km. per hour should have been reduced by 
one-half. Yet there is no need for surprise ; for if the 
method for calculating the useful power necessary for 
horizontal flight (set forth in Chapter II., and since applied 
more than once) is carefully examined, it becomes evident 
that, whereas that portion of the power required only for 
lifting remains proportional to the speed, the remaining 
portion, used to overcome all passive resistance, is propor- 
tional to the cube of the speed. 

For this reason it is of such great importance to cut down 
the detrimental surface hi designing a high-speed machine. 

Thus, in the present case, of the 46 h.p. available, only 
18 h.p. are required to lift the machine. The remain- 
ing 28 h.p., therefore, are necessary to overcome passive 
resistance. 

Had the detrimental surface been 0-75 sq. m. instead of 
0-30, the useful power absorbed in overcoming passive 
resistance would have been 

Q.75x28 =7() h mgtead 2g 
0-30 

To complete our examination of the high-speed aeroplane, 
Table IV. has been drawn up, and includes the values of the 
useful power required on the one hand for the flight of a 
Maurice Farman plane at a good incidence, and weighing 

estimate of 0'30 sq. m. for the detrimental surface was too low, a con- 
clusion supported by M. Eiffel's experiments. 

It is doubtful whether an aeroplane has yet been built with a detri- 
mental surface of much less than half a square metre. 



FLIGHT IN STILL AIR 



49 



1 ton (metric), and on the other for driving through 
air a detrimental surface of 1 sq. m. at speeds from 
to 200 km. per hour. 

TABLE IV. 



the 
150 



Speed. 




lllL 


II. J^s 

|3*g*-i 


|||.si 


Km. 


Metres 


Drag of 
Planes (kg.) ; 
per aeroplane 
ton. 


-sll^l 


1151II 

fsflll 


Ijjn 


per hr. 


per see. 


"* ^ >> " 


3 f *" ^oo g 


^ f, 2 -S ^ 






g * ^*> 


(2S ^6 


}S se^ 8 


1 


2 


3 


4 


5 


6 


150 


41-6 


65 


36 


138 


76 


160 


44-4^ 


slsf 


38 


157 


93 


170 


47-2 


l?|ol 


41 


178 


112 


180 
190 


50-0 
52-8 




43 
46 


200 
232 


132 
157 


200 


55-6, 


-** K 5 


48 


248 


184 






^ o 







According to this Table, an aeroplane weighing 500 kg., 
and possessing, as we supposed in the case of the Tatin 
" Torpille," a detrimental surface of 0-30 sq. m., would re- 
quire a useful power of about 80 h.p. to attain a speed of 
200 km. per hour. This high speed could therefore be 
achieved with a power-plant consisting of a 100-h.p. motor 
and a propeller of 80 per cent, efficiency. It could only be 
obtained just as the " Torpille " could only achieve 160 
km. per hour at a lift coefficient of 0-018 with a plane 
loading of about 56 kg. per sq. m. Consequently, the 

area of the planes would be only -=^- = 9 sq. m. 

OD 

If the theoretical qualities of design of machines of the 
" Torpille " type are borne out by practice * our present 

* But, according to what has already been said, this does not seem to 
be the case. Hence, a speed of 200 km. per hour is not likely to be 



50 FLIGHT WITHOUT FORMULA 

motors would appear to be sufficient to give them a speed 
of 200 km. per hour. But this would necessitate a very 
heavy loading and a lift coefficient much lower than any 
hitherto employed a proceeding which, as we have seen, is 
not without danger. Moreover, one cannot but be uneasy 
at the thought of a machine weighing perhaps 500 or 600 
kg. alighting at this speed. 

This, beyond all manner of doubt, is the main obstacle 
which the high-speed aeroplane will have to overcome, and 
this it can only do by possessing speed variation to an 
exceptional degree. We will return to this aspect of the 
matter subsequently. 

To-day an aeroplane, weighing with full load a certain 
weight and equipped with an engine giving a certain power, 
in practice flies horizontally at a given speed. 

These three factors, weight, speed, and power, are always 
met with whatever the vehicle of locomotion under con- 
sideration, and their combination enables us to determine 
as the most efficient from a mechanical point of view that 
vehicle or machine which requires the least power to attain, 
for a constant weight, the same speed. 

Hence, what we may term the, mechanical efficiency 
of an aeroplane may be measured through its weight 
multiplied by its normal speed and divided by the motive- 
power. 

If the speed is given in metres per second and the power 
in h.p., this quotient must be divided by 75. 

RULE. The mechanical efficiency of an aeroplane is 
obtained by dividing its weight multiplied by its normal 
speed (in metres per second) by 75 times the power, or, what 
is the same thing, by dividing by 270 times the power the 
product of the weight multiplied by the speed in kilometres 
per hour. 

EXAMPLE. An aeroplane weighing 950 kg., and driven 

attained with a 100-h.p. motor. Whether an engine developing 140 h.p. 
or more will succeed in this can only be shown by the future, and perhaps 
at no distant date. See footnote, p. 12. 



FLIGHT IN STILL AIR 51 

by a, lOQ-h.p. engine, flies at a normal speed of 117 km. per 
hour. What is its mechanical efficiency ? 

950x117 



Reference to what has already been said will show that 
mechanical efficiency is also expressed by the propeller 
efficiency divided by the gliding angle corresponding to 
normal incidence. This is due to the fact that, firstly, the 
useful power required for horizontal flight is the 75th part 
of the weight multiplied by the speed and the normal 
gliding angle, and, secondly, because the motive power is 
obtained by dividing the useful power by the propeller 
efficiency. Accordingly, a machine with a propeller efficiency 
of 70 per cent., and with a normal gliding angle of 0-17, 

would have a mechanical efficiency =4-12. 

This conception of mechanical efficiency enables us to 
judge an aeroplane as a whole from its practical flying 
performances without having recourse to the propeller 
efficiency and the normal gliding angle, which are difficult 
to measure with any accuracy. 

Even yesterday a machine possessing mechanical efficiency 
superior to 4 was still, aerodynamically considered, an 
excellent aeroplane. But the progress manifest in the 
last Salon entitles us, and with confidence, to be more 
exacting in the future. 

Hence, the average mechanical efficiency of the ordinary 
run of aeroplanes enables us in some measure to fix definite 
periods in the history of aviation. In 1910, for instance, 
the mean mechanical efficiency was roughly 3-33, on which 
we based the statement contained in a previous work that, 
in practice, 1 h.p. transports 250 kg. in the case of an average 
aeroplane at 1 m. per second. 

This rule, which obviously only yielded approximate 
results, could be applied both quickly and easily, and enabled 
one, for instance, to form a very fair idea of the results 



52 FLIGHT WITHOUT FORMULA 

that would be attained in the Military Trials of 1911. In 
fact, according to the rules of this competition, the aero- 
planes would have to weigh on an average 900 kg. To 

give them a speed of 70 km. per hour or - m. per second, 

3'6 

c i A 90 v 70 

for instance, the rule quoted gives x ^-^- 

250 o'b 

But 250 X 3-6 remains the denominator whatever the speed 
it is desired to attain, and is exactly equal to 900, the weight 
of the aeroplane. From this, one deduced that in this case 
the power required in h.p. was equivalent to the speed in 
kilometres per hour : 

70 km. per hour 70 h.p. 

80 ..... 80 h.p. 
100 ,, 100 h.p. 

If, on the other hand, certain machines during these trials, 
driven by engines developing less than 100 effective h.p., 
flew at over 100 km. per hour, this was due simply to their 
mechanical efficiency being better than the 3-33 which 
obtained in 1910, and was already too low for 1911. 

At the present day, therefore, accepting 4 as the average 
mechanical efficiency, the practical rule given above should 
be modified as follows : 

RULE. 1 h.p. transports 300 kg. of an average aeroplane 
at 1 m. per second. 



CHAPTER IV 

FLIGHT IN STILL AIR 
THE POWER-PLANT 

BOTH this chapter and the next will be devoted to the 
power-plant of the aeroplane as it is in use at the present 
time. This will entail an even closer consideration of the 
part played by the motive-power in horizontal and oblique 
flight, and will finally lead to several important conclusions 
concerning the variable-speed aeroplane and the solution of 
the problem of speed variation. 

The power-plant of an aeroplane consists in every case 
of an internal combustion motor and one or more propellers. 
Since the present work is mainly theoretical, no description 
of aviation motors will be attempted, and only those of 
their properties will be dealt with which affect the working 
of the propeller. 

Besides, the motor works on principles which are beyond 
the realm of aerodynamics, so that from our point of view 
its study has only a minor interest. It forms, it is true, an 
essential auxiliary of the aeroplane, but only an auxiliary. 
If it is not yet perfectly reliable, there is no doubt that it 
will be in a few years, and this quite independently of any 
progress in the science of aerodynamics. 

Deeply interesting, on the other hand, are the problems 
relating to the aeroplane itself, or to that mysterious 
contrivance which, as it were, screws itself into the air 
and transmutes into thrust the power developed by the 
engine. 



54 FLIGHT WITHOUT FORMULA 

The power developed by an internal combustion engine 
varies with the number of revolutions at which the resist- 
ance it encounters enables it to turn. There is a generally 
recognised ratio between the power developed and the speed 
of revolution. 

Thus, if a motor, normally developing 50 h.p. at 1200 
revolutions per minute, only turns at 960 revolutions per 

50 X 960 
minute, it will develop no more than =40 h.p. 

The rule, however, is not wholly accurate, and the 
variation of the power developed by a motor with the 
number of revolutions per minute is more accurately shown 
in the curve in fig. 12. It should be clearly understood 
that the curve only relates to a motor with the throttle 
fully open, and where the variation in its speed of rotation 
is only due to the resistance it has to overcome. 

For the speed of rotation may be reduced in another 
manner by shutting off a portion of the petrol mixture by 
means of the throttle. The engine then runs "throttled 
down," which is the usual case with a motor car. 

In such a case, if the petrol supply is constant, the curve 
in fig. 12 grows flatter, with its crest corresponding to a 
lower speed of rotation the more the throttle is closed and 
the explosive mixture reduced. 

Fig. 13 shows a series of curves which were prepared at 
my request by the managing director of the Gnome Engine 
Company ; these represent the variation in power with the 
speed of rotation of a 50-h.p. engine, normally running at 
1200 revolutions per minute, with the throttle closed to a 
varying extent. 

In practice, it is easier to throttle down certain engines 
than others ; with some it is constantly done, with others 
it is more difficult. 

Even to-day the working of a propeller remains one of 
the most difficult problems awaiting solution in the whole 
range of aerodynamics, and the motion, possibly whirling, 
of the air molecules as they are drawn into the revolving 



FLIGHT IN STILL AIR 

Horse-Power. 



55 



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56 



FLIGHT WITHOUT FORMULA 



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FLIGHT IN STILL AIR 57 

propeller has never yet been explained in a manner satis- 
factory to the dictates of science. 

All said and done, the rough method of likening a propeller 
to a screw seems the most likely to explain the results 
obtained from experiments with propellers. 

The pitch of a screw is the distance it advances in one 
revolution in a solid body. The term may be applied in a 
similar capacity to a propeller. The pitch of a propeller, 
therefore, is the distance it would travel forwards during 
one revolution if it could be made to penetrate a solid body. 
But a propeller obtains its thrust from the reaction of an 
elusive tenuous fluid. Clearly, therefore, it will not travel 
forward as great a distance for each revolution as it would 
if screwing itself into a solid. 

The distance of its forward travel is consequently always 
smaller than the pitch, and the difference is known as the 
slip. But, contrary to an opinion which is often held, this 
slip should not be as small as possible, or even be altogether 
eliminated, for the propeller to work under the best con- 
ditions. 

Without attempting to lay down precisely the phenomena 
produced in the working of this mysterious contrivance, we 
may readily assume that at every point the blade meets the 
air, or " bites " into it, at a certain angle depending, among 
other things, on the speed of rotation and of forward travel 
of the blade and of the distance of each point from the 
axis. 

Just as the plane of an aeroplane meeting the air along 
its chord would produce no lift, so a propeller travelling 
forward at its pitch speed that is, without any slip 
would meet the air at each point of the blades at no angle 
of incidence, and consequently would produce no thrust. 

The slip and angle of incidence are clearly connected 
together, and it will be easily understood that a given 
propeller running at a given number of revolutions will 
have a best slip, and hence a lest forward travel, just as 
a given plane has a best angle of incidence. 



58 FLIGHT WITHOUT FORMULA 

When the propeller rotates without moving forward 
through the air, as when an aeroplane is held stationary on 
the ground, it simply acts as a ventilator, throwing the air 
backwards, and exerts a thrust on the machine to which 
it is attached. But it produces no useful power, for in 
mechanics power always connotes motion. 

But if the machine were not fixed, as in the case of an 
aeroplane, and could yield to the thrust of the propeller, 
it would be driven forward at a certain speed, and the 
product of this speed multiplied by the thrust and divided 
by 75 represents the useful power produced by the pro- 
peller. 

On the other hand, in order to make the propeller rotate 
it must be acted upon by a certain amount of motive power. 
The relation between the useful power actually developed 
and the motive power expended is the efficiency of the 
propeller. 

But the conditions under which this is accomplished 
vary, firstly, with the number of revolutions per minute 
at which the propeller turns, and secondly, with the speed 
of its forward travel, so that it will be readily understood 
that the efficiency of a propeller may vary according to the 
conditions under which it is used. 

Experiments lately conducted notably by Major 
Dorand at the military laboratory of Chalais-Meudoii and 
by M. Eiffel have shown that the efficiency remains 
approximately constant so long as the ratio of the forward 
speed to its speed of revolution, i.e. the forward travel per 
revolution, remains constant. 

For instance, if a propeller is travelling forward at 15 
m. per second and revolving at 10 revolutions per second, 
its efficiency is the same as if it travelled forward at 30 
m. per second and revolved at 20 revolutions per second, 
since in both cases its forward travel per revolution is 
1-50 m. 

But the propeller efficiency varies with the amount of 
its forward travel per revolution. 



FLIGHT IN STILL AIR 59 

Hence, when the propeller revolves attached to a 
stationary point, as during a bench test, so that its forward 
travel is zero, its efficiency is also zero, for the only 
effect of the motive power expended to rotate the propeller 
is to produce a thrust, which in this instance is exerted 
upon an immovable body, and therefore is wasted so far 
as the production of useful power is concerned. 

Similarly, when the forward travel of the propeller per 
revolution is equal to the pitch, and hence when there is 
no slip, it screws itself into the air like a screw into a 
solid ; the blades have no angle of incidence, and therefore 
produce no thrust.* 

Between the two values of the forward travel per 
revolution at which the thrust disappears, there is a value 
corresponding on the other hand to maximum thrust. 
This has already been pointed out, and has been termed 
the best forward travel per revolution. 

This shows that the thrust of one and the same propeller 
may vary from zero to a maximum value obtained with a 
certain definite value of the forward travel. The variation 
of the thrust with the forward travel per revolution may 
be plotted in a curve. A single curve may be drawn to 
show this variation for a whole family of propellers, geometri- 
cally similar and only differing one from the other by their 
diameter. 

Experiments, in fact, have shown that such propellers 
had approximately the same thrust when their forward 
travel per revolution remained proportional to their 
diameter. 

Thus two propellers of similar type, with diameters 
measuring respectively 2 and 3m., would give the same 
thrust if the former travelled 1-2 m. per revolution and 

* This could never take place if the vehicle to which the propeller 
was attached derived its speed solely from the propeller ; it could only 
occur in practice if motive power from some outside source imparted 
to the vehicle a greater speed than that obtained from the propeller- 
thrust alone. 



60 



FLIGHT WITHOUT FORMULA 



the latter 1-8 m., since the ratio of forward travel to 
diameter =0-60. 

This has led M. Eiffel to adopt as his variable quantity 
not the forward travel per revolution, but the ratio of this 
advance to the diameter, which ratio may be termed 
reduced forward travel or advance. 

Fig. 14, based on his researches, shows the variation in 
thrust of a family of propellers when the reduced advance 
assumes a series of gradually increasing values.* 

The maximum thrust efficiency (about 65 per cent, in 
this case) corresponds to a reduced advance value of 0-6. 



Reduced Advance. 
FIG. 14. 

Hence a propeller of the type under consideration, with a 
diameter of 2-5 m., in order to give its highest thrust, 
would have to have a forward travel of 2-5 xO'6=l-2 m. 
Consequently, if in normal flight it turned at 1200 
revolutions per minute, or 20 revolutions per second, the 
machine it propelled ought to fly at 1-20x20=24 m. per 
second. 

For all propellers belonging to the same family there 
exists, therefore, a definite reduced advance which is more 

* Actually, M. Eiffel found that for the same value of the reduced 
advance the thrust was not absolutely constant, but rather that it 
tended to grow as the number of revolutions of the propeller increased. 
Accordingly, he drew up a series of curves, but these approximate very 
closely one to another. 



FLIGHT IN STILL AIR 



61 



favourable than any other, and may thence be termed the 
best reduced advance, which enables any of these propellers 
to produce their maximum thrust. 

It has been shown that all geometrically similar pro- 
pellers in other words, belonging to the same family 
give approximately the same maximum thrust efficiency. 

But when the shape of the propeller is changed, this 
maximum thrust value also varies. 

It depends more especially on the ratio between the 
pitch of the propeller and its diameter, which is known as 
the pitch ratio. 

But, as the value of the highest thrust varies with the 
pitch ratio, so does that of the best reduced advance corre- 
sponding to this highest thrust. 

In the following Table V., based on Commandant Dorand's 
researches at the military laboratory of Chalais-Meudon 
with a particular type of propeller, are shown the values 
of the maximum thrust and the best reduced advance 
corresponding to propellers of varying pitch ratio. 

TABLE V. 



Pitch ratio . 


0-5 


0-6 


0-7 


0-8 


0-9 


1-0 


I'M 


Maximum thrust efficiency 


0-45 


0'53 


0'61 


0-70 


0-76 


0-80 


0-84 


Best reduced advance 


0"29 


0-38 


0-47 


0-55 


0-63 


072 


0-84 



EXAMPLE. A propeller of the Chalais-Meudon type with 
2-5 m. diameter and 2 m. pitch turns at 1200 revolutions 
per minute. 

1. What is the value of its highest thrust efficiency ? 

2. What should be the speed of the aeroplane it drives in 
order to obtain this highest thrust ? 

The pitch ratio is ~=0-8. 
Table V. immediately solves the first question : the 



62 FLIGHT WITHOUT FORMULA 

highest thrust efficiency is 0-7. Further, this table shows 
that to obtain this thrust the reduced advance should 
=0-55. In other words, the speed of the machine divided 
by 50 (the number of revolutions per second, 50 x diameter, 
2-5) should =0-55. 

Hence the speed =0-55 x 50=27-5 m. per second, or 
99 km. per hour. 

Again, Table V. proves, according to Commandant 
Dorand's experiments, that even at the present time it is 
possible to produce propellers giving the excellent efficiency 
of 84 per cent, under the most favourable running conditions, 
but only if the pitch ratio is greater than unity that is, 
when the pitch is equal to or greater than the diameter. 

It is further clear that, since the best reduced advance 
increases with the pitch ratio, the speed at which the 
machine should fly for the propeller (turning at a constant 
number of revolutions per minute) to give maximum 
efficiency is the higher the greater the pitch ratio. This is 
why propellers with a high pitch ratio, or the equivalent, 
a high maximum thrust, are more especially adaptable 
for high-speed aeroplanes. At the same time, they are 
equally efficient when fitted to slower machines, provided 
that the revolutions per minute are reduced by means of 
gearing. 

These truths are only slowly gaining acceptance to-day 
although the writer advocated them ardently long since, 
and this notwithstanding the fact that the astonishing 
dynamic efficiency of the first motor-driven aeroplane 
which in 1903 enabled the Wrights, to their enduring 
glory, to make the first flight in history, was largely due to 
the use of propellers with a very high pitch ratio, that is, 
of high efficiency, excellently well adapted to the relatively 
low speed of the machine by the employment of a good 
gearing system. 

The only thing that seemed to have been taught by this 
fine example was the use of large diameter propellers. 

This soon became the fashion. But, instead of gearing 



FLIGHT IN STILL AIR 63 

down these large propellers, as the Wrights cleverly did, 
they were usually driven direct by the motor, and so that 
the latter could revolve at its normal number of revolutions 
the pitch had perforce to be reduced. 

As the pitch decreased, so the maximum efficiency and 
the best reduced advance that is, the most suitable flying 
speed fell off, while at the same time the development of 
the monoplane actually led to a considerable increase in 
flying speed. 

The result was that fast machines had to be equipped 
with propellers of very low efficiency which, even so, they 
were unable to attain, as the flying speed of the aeroplane 
was too high for them. At most these propellers might 
have done for a dirigible, but they would have been poor 
even at that. 

Fortunately, a few constructors were aware of these 
facts, and to this alone we may ascribe the extraordinary 
superiority shown towards the end of 1910 by a few types 
of aeroplanes, among which we may name, without fear 
of being accused of bias, those of M. Breguet and the late 
M. Nieuport. 

But, since then, progress has been on the right lines, 
and those who visited the last three Aero shows must have 
been struck with the general decrease in propeller diameter, 
which has been accompanied by an increase in efficiency 
and adaptability to the aeroplanes of to-day. 

To take but one final example : the fast Paulhan-Tatin 
" Torpille," already referred to, had a pitch ratio greater 
than unity. For this reason its efficiency was estimated in 
the neighbourhood of 80 per cent. 

The foregoing considerations may be summed up as 
follows : 

1. The same propeller gives an efficiency varying accord- 
ing to the conditions in which it is run, depending on its 
forward travel per revolution. 

2. Each propeller has a speed of forward travel or advance 
enabling it to produce its highest efficiency. 



64 FLIGHT WITHOUT FORMULA 

3. For propellers of identical type but different diameters 
the various speeds of forward travel corresponding to the 
same thrust are proportional to the diameters, whence 
arises the factor of reduced advance, which, in other words, 
is the ratio between the forward travel per revolution and 
the diameter. 

4. The maximum efficiency of a propeller and its best 
reduced advance depend on its shape, and more especially 
on its pitch ratio. 

Hitherto the propeller has been considered as a separate 
entity, but in practice it works in conjunction with a 
petrol motor, whether by direct drive or gearing. 

But the engine and propeller together constitute the 
power-plant, and this new entity possesses, by reason of the 
peculiar nature of the petrol motor, certain properties 
which, differing materially from those of the propeller by 
itself, must therefore be considered separately. 

First, we will deal with the case of a propeller driven 
direct off the engine. 

Let us assume that on a truck forming part of a railway 
tram there has been installed a propelling plant (wholly 
insufficient to move the tram) consisting of a 50-h.p. motor 
running at 1200 revolutions per minute, and of a propeller, 
while a dynamometer enables the thrust to be constantly 
measured and a revolution indicator shows the revolutions 
per minute. 

The tram being stationary, the motor is started. 

The revolutions will then attain a certain number, 950 
revolutions per minute for instance, at which the power 
developed by the motor is exactly absorbed by the propeller. 
The latter will exert a certain thrust upon the train (which, 
of course, remains stationary), indicated by the dynamo- 
meter and amounting to, say, 150 kg. 

The power developed by the motor at 950 revolutions per 
minute is shown by the power curve of the motor, which 
we will assume to be that shown in fig. 12. This would 
give about 43 h.p. at 950 revolutions per minute. 



FLIGHT IN STILL AIR 65 

The useful power, on the other hand, is zero, since no 
movement has taken place. 

Now let the train be started and run at, say, 10 km. per 
hour or 5 m. per second, the motor still continuing to run. 

The revolutions per minute of the propeller would 
immediately increase, and finally amount to, say, 1010 
revolutions per minute. 

The power developed by the motor would therefore have 
increased and would now amount, according to fig. 12, to 
45-5 h.p. 

But at the same time the dynamometer would show a 
smaller thrust about 130 kg. 

But this thrust would, though in only a slight degree, 
have assisted to propel the train forward and the useful 

power produced by the propeller would be =8*7 h.p. 

The acceleration in rotary velocity and the decrease in 
thrust which are thus experienced are to be explained on 
the score that the blades, travelling forward at the same 
time that they revolve, meet the air at a smaller angle 
than when revolving while the propeller is stationary. 
In these conditions, therefore, the propeller turns at 
a greater number of revolutions, though the thrust falls 
off. 

If the speed of the train were successively increased to 
10, 15 and 20 m. per second, the following values would be 
established each time : 

The normal number of revolutions of the power-plant ; 

The corresponding power developed by the motor ; 

The useful power produced by the propeller. 

We could then plot curves similar to that shown in fig. 15, 
giving for every speed of the train the corresponding 
motive power (shown in the upper curve) and the useful 
power (lower curve). The dotted lines and numbers give 
the number of revolutions. 

The lower curve representing the variation in the useful 
power produced by the propeller according to the forward 



66 



FLIGHT WITHOUT FORMULA 



speed of travel is of capital importance, and will hereafter 
be referred to as the power-plant curve. 

Usually the highest points of the two curves, L and M, 
do not correspond. This simply means that generally, and 
unless precautions have been taken to avoid this, the pro- 
peller gives its maximum thrust, and accordingly has its 
best reduced advance, at a forward speed which does not 



950 
40 




20 





k-d of fliffkt (in 



Q i 



FIG. 15. 



enable the motor to turn at its normal number of revolu- 
tions, 1200 in the present case, and consequently to develop 
its full power of 50 h.p. 

It is even now apparent, therefore, that one cannot mount 
any propeller on any motor, if direct-driven, and that there 
exists, apart altogether from the machine which they drive, 
a mutual relation between the two parts constituting the 
power-plant, which we will term the proper adaptation of 
the propeller to the motor. 



FLIGHT IN STILL AIR 67 

Its characteristic feature is that the highest points in the 
two curves representing the values of the motive power 
and the useful power at different speeds of flights lie in a 
perpendicular line (see fig. 16). 

The highest thrust efficiency is then obtained from the 
propeller at such a speed that the motor can also develop 
its maximum power. 



30 



20 



^ 



Spekd of fh$k 



10 (5 

FIG. 16. 



25 



The expression maximum power-plant efficiency will be 
used to denote the ratio of maximum useful power Mm 
(see fig. 15) developed at the maximum power LJ of which 
the motor is capable (50 h.p. in the case under consideration). 

The maximum power-plant efficiency, it is clear, corre- 
sponds to a certain definite speed of flight Om. This may 
be termed the best speed suited to the power-plant. 

If the adaptation of the propeller to the motor is good 
(as in the case of fig. 16), the maximum power-plant 
efficiency is the highest that can be obtained by mounting 



68 FLIGHT WITHOUT FORMULA 

direct-driven propellers belonging to one and the same 
family and of different diameters on the motor. 

Hence there is only one propeller in any family or series 
of propellers which is well adapted to a given motor. 

We already know that in a family of propellers the 
characteristic feature is a common value of the pitch ratio 
supposing, naturally, that the propellers are identical in 
other respects. The conclusion set down above can there- 
fore also be expressed as follows : 

There can be only one propeller of given pitch ratio that 
is well adapted to a given motor. The diameter of the pro- 
peller depends on the pitch ratio, and vice versa. 

Propellers well adapted to a given motor consequently 
form a single series such that each value of the diameter 
corresponds to a single value of the pitch, and vice versa. 

According to the results of Commandant Dorand's experi- 
ments with the type of propellers which he employed, the 
series of propellers properly adapted to a 50 h.p. motor 
turning at 1200 revolutions per minute can be set out as 
in Table VI., which also gives the best speed suited to the 
power-plant in each case, and the maximum useful powers 
developed obtained by multiplying the power of the motor, 
50 h.p., by the maximum efficiency as given in Table V. 

To summarise : 

1. The useful power developed by a given power-plant 
varies with the speed of the aeroplane on which it is mounted. 
The variation can be shown by a curve termed the char- 
acteristic power-plant curve. 

2. To obtain from the motor its full power and from the 
propeller its maximum efficiency the propeller must be 
well adapted to the motor, and this altogether independently 
of the aeroplane on which they are mounted. 

3. There is only a single series of propellers well adapted 
to a given motor. 

4. For a power-plant to develop maximum efficiency the 
aeroplane must fly at a certain speed, known as the best 
speed suited to the power-plant under consideration. 



FLIGHT IN STILL AIR 



TABLE VI. 



Pitch 
Ratio. 


Propeller 
Diameters 
(in metres). 


Propeller 
Pitch (in 
metres ; 
product 
of cols. 1 
and 2). 


Best Suitable Speed. 


Maximum 
Power- 
plant 
Efficiency 
(from 
Table V.) 


Maximum 
Useful 
Power 
developed 
(product of 
50 h.p. 
and col. 6). 


m.p.s. 


km.p.h. 


1 


2 


3 4 


5 


6 


7 


0-5 


2-46 


1-23 


14-2 


51 


0-45 


22-5 


0-6 


2-33 


1-40 


17-7 


64 


0-53 


26-5 


0-7 


2-24 


1-57 


21-1 


76 


0-61 


30-5 


0-8 


2-16 


1-73 


23-8 


86 


0-70 


35 


0-9 


2'09 


1-88 


26-4 


95 


0-76 


38 


1-0 


2-04 


2-04 


29-5 


106 


0-80 


40 


1-15 


1-98 


2-28 33-3 


126 


0-84 


42 



In conclusion, it will be advisable to remember that 
the conclusions reached above should not be deemed to 
apply with rigorous accuracy. Fortunately, practice is 
more elastic than theory. Thus we have already seen in 
the case of the angle of incidence of a plane that there is, 
round about the value of the best incidence, a certain 
margin within whose limits the incidence remains good. 
Just so we have to admit that a given power-plant may 
yield good results not only when the aeroplane is flying at 
a single best speed, but also when its speed does not vary 
too widely from this value. 

In other words, a certain elasticity is acquired in applying 
in practice purely theoretical deductions, though it should 
not be forgotten that the latter indicate highly valuable 
principles which can only be ignored or thrust aside with 
the most serious results, as experience has proved only 
too well. 



CHAPTER V 

FLIGHT IN STILL AIR 
THE POWER-PLANT (concluded) 

IN the last chapter we confined ourselves mainly to the 
working of the power-plant itself, and more particularly to 
the mutual relations between its parts, the motor and the 
propeller, without reference to the machine they are 
employed to propel. The present chapter, on the other 
hand, will be devoted to the adaptation of the power-plant 
to the aeroplane, and incidentally will lead to some con- 
sideration of the variable- speed aeroplane and of the greatest 
possible speed variation. 

In Chapter II. particular stress was laid on the graph 
termed the essential curve of the aeroplane, which enables 
us to find the different values of the useful power required 
to sustain in flight a given aeroplane at different speeds, 
that is, at different angles of incidence and lift coefficients. 

In fig. 17 the thin curve (reproduced from fig. 6, 
Chapter II.) is the essential aeroplane curve of a Breguet 
biplane weighing 600 kg., with an area of 30 sq. m. and 
a detrimental surface of 1-2 sq. m. 

But in the last chapter particular attention was also 
drawn to the graph termed the power-pla-nt curve, which 
gives the values of the useful power developed by a given 
power-plant when the aeroplane it drives flies at different 



In fig. 17 the thick curve is the power-plant curve, in the 
case of a motor of 50 h.p. turning at 1200 revolutions per 



FLIGHT IN STILL AIR 



71 



minute and a propeller of the Chalais-Meudon type, direct- 
driven, well adapted to the motor, and with a pitch ratio 
of 0-7. 

Table VI. (p. 69) gives the diameter and pitch of the 
propeller as 2-24 m. and 1-57 m. respectively. The 
maximum power-plant efficiency corresponds to a speed of 

60 




FIG. 17. 

22-1 m. per second. The maximum useful power is 30-5 h.p. 
These are the factors which enable us to fix M, the highest 
point of the curve. 

It will be clear that, by superposing in one diagram (as in 
fig. 17, which relates to the specific case stated above) the 
two curves representing in both cases a correlation between 
useful powers and speeds, and referring, in one case to the 
aeroplane, in the other to its power-plant, we should obtain 



72 FLIGHT WITHOUT FORMULA 

some highly interesting information concerning the adapta- 
tion of the power-plant to the aeroplane. 

The curves intersect in two points, Rj and R 2 , which 
means that there are two flight speeds, Oj. and O 2 , at 
which the useful power developed by the power-plant is 
exactly that required for the horizontal flight of the 
aeroplane. These two speeds both, therefore, fulfil the 
definition (see Chapter II.) of the normal flying speeds. 

From this we deduce that a power-plant capable of 
sustaining an aeroplane in level flight can do so at two 
different normal flying speeds. But in practice the machine 
flies at the higher of these two speeds, for reasons which 
will be explained later. 

These two normal flying speeds will, however, crop up 
again whenever the relation between the motive power and 
the speed of the aeroplane comes to be considered. Thus, 
when the motive power is zero, that is, when the aeroplane 
glides with its engine stopped, the machine can, as already 
explained, follow the same gliding path at two different 
speeds. The same, of course, applies to horizontal flight, 
since, as has been seen, this is really nothing else than an 
ordinary glide in which the angle of the flight-path has 
been raised by mechanical means, through utilising the 
power of the engine. 

Let us assume that the ordinary horizontal flight of the 
aeroplane is indicated by the point R l5 which constitutes its 
normal flight. 

The speed ORj will be roughly 23 m. per second, and the 
useful power required, actually developed by the propeller, 
about 30 h.p. 

According to Table II. (Chapter II.), the normal angle of 
incidence will be about 4, corresponding to a lift coefficient 
of 0-038. 

Let it be agreed that in flight, which is strictly normal, 
the pilot suddenly actuates his elevator so as to increase 
the angle of incidence to 6| (lift coefficient 0-05), and hence 
necessarily alters the speed to 20 m. per second. 



FLIGHT IN STILL AIR 73 

From the thin curve in fig. 17 (and from Table II., on 
which it is based) it is clear that the useful power required 
to sustain the aeroplane at this speed will be 24 h.p. 

On the other hand, according to the thick curve in the 
same figure, the power-plant at this same speed of 20 m. 
per second will develop a useful power of 30-3 h.p., giving 
a surplus of 6-3 h.p. over and above that necessary to 
sustain the machine. The latter will therefore climb, and 
climb at a vertical speed such that the raising of its weight 
absorbs exactly the surplus, NN' or 6-3 h.p., useful power 

developed by the power-plant, that is, at a speed of 

bOO 

=about 0-79 m. per second. 

Since this vertical speed must necessarily correspond to a 
horizontal speed of 20 m. per second, the angle of the climb, 
as a decimal fraction, will be the ratio of the two speeds, i.e. 

0-79 

=:0 < 0395=about 4 centimetres per metre=l in 25. 

As a matter of fact, we have already seen that by using 
the elevator the pilot could make his machine climb or 
descend ; but by considering the curves of the aeroplane 
and of the power-plant at one and the same time, we gain 
a still clearer idea of the process. 

Should the pilot increase the incidence to more than 6| 
the speed would diminish still more, and fig. 17 shows that, 
in so doing, the surplus power, measured by the distance 
dividing the two curves along the perpendicular correspond- 
ing to the speed in question, would increase. And with it 
we note an increase both in the climbing speed and in the 
upward flight-path. 

Yet is this increase limited, and the curves show that 
there is one definite speed, 01, at which the surplus of 
useful power exerted by the power-plant over and above 
that required for horizontal flight has a maximum 
value. 

If, by still further increasing the angle of incidence, 
the speed were brought below the limit 01, the climb- 



74 FLIGHT WITHOUT FORMULA 

ing speed of the aeroplane would diminish instead of 
increasing. 

Nevertheless, the upward climbing angle would still 
increase, but ever more feebly, until the speed attained 
another limit, Op, such that the ratio between the climbing 
speed to the flying speed, which measures the angle of the 
flight-path, attained a maximum. 

Thus, there is a certain angle of incidence at which an 
aeroplane climbs as steeply as it is possible for it to climb. 

If, when the machine was following this flight-path, 
the angle of incidence were still further increased by 
the use of the elevator, in order to climb still more, the 
angle of the flight-path would diminish. Relatively to 
its flight-path the aeroplane would actually come down, 
notwithstanding the fact that the elevator were set for 
climbing. 

The same inversion of the effect usually produced from 
the use of the elevator would arise if the aeroplane were 
flying under the normal conditions represented by the point 
R 2 in fig. 17. For a decrease in the angle of incidence 
through the use of the elevator would have the immediate 
and inevitable result of increasing the speed of flight, which 
would pass from O 2 to Og, for instance. But this would 
produce an increase QQ' in the useful power developed by 
the power-plant over and above that required for horizontal 
flight, so that even though the elevator were set for descend- 
ing, the aeroplane would actually climb. 

This inversion of the normal effect produced by the 
elevator has sometimes caused this second condition of 
flight to be termed unstable. 

For if a pilot flying hi these conditions, and not aware 
of this peculiar effect, felt his machine ascending through 
some cause or other, he would work his elevator so as to 
come down. But the aeroplane would continue to ascend, 
gathering speed the while. The pilot, finding that his 
machine was still climbing, would set his elevator still 
further for descending until the speed exceeded the limit 



FLIGHT IN STILL AIR 75 

Op, and the elevator effect returned to its usual state and 
the machine actually started to descend. The pilot, unaware 
of the existence of this condition and brought to fly under 
it by certain circumstances (which, be it added, are purely 
hypothetical), would therefore regain normal flight by using 
his controls in the ordinary manner. 

Nevertheless, one is scarcely justified in applying to this 
second condition of horizontal flight the term " unstable " 
if employed in the sense ordinarily accepted in mechanics, 
for one may well believe that a pilot, aware of its 
existence, could perfectly well accomplish flight under 
this condition by reversing the usual operation of his 
elevator. 

Still, it would be a difficult proposition for machines 
normally flying at a low speed, since the speed of flight 
under the second condition (indicated by the point R 2 , fig. 
17) would be lower still. 

But in the case of fast machines the solution is obvious 
enough. For instance, according to Table II., the minimum 
speed of the aeroplane represented by the thin curve in 
fig. 17 is about 63 km. per hour, whereas in the early 
days of aviation the normal flying speed of aeroplanes 
was less. 

Now, note that by making an aeroplane fly under the 
second condition the angle of the planes would be quite 
considerable. In the case in question the angle would be 
in the neighbourhood of 15, which is about 10 in excess of 
the normal flying angle. 

The whole aeroplane would therefore be inclined at an 
angle equivalent to some ten degrees to the horizontal, with 
the result that the detrimental surface (which cannot be 
supposed constant for such large angles) would be increased, 
and with it the useful power required for flight. 

In practice, therefore, the power-plant would not enable 
the minimum speed Or 2 to be attained, and the second condi- 
tion of flight would take place at a higher speed and at a 
smaller angle of incidence. Still, it would be practicable 



76 FLIGHT WITHOUT FORMULA 

by working the elevator in the reverse sense to the 
usual.* 

Now let us just see how a pilot could make his aeroplane 
pass from normal flight to the second condition ; although, 
no doubt, in so doing we anticipate, for it is highly im- 
probable that any pilot hitherto has made such an attempt. 

When the aeroplane is flying horizontally and normally, 
the pilot would simply have to set his elevator to climb, and 
continue this manoeuvre until the flight-path had attained 
its greatest possible angle. The aeroplane would then return 
(and very quickly too, if practice is in accordance with 
theory) to horizontal flight, and now, flying very slowly, it 
would have attained to the second condition of flight. At 
this stage it would be flying at a large angle to the flight- 
path, very cabre, almost like a kite. 

The greater part of the useful power would be absorbed 
in overcoming the large resistance opposed to forward 
motion by the planes. It will now be readily seen that, 
under these conditions, any decrease in the angle of incidence 
would cause the machine to climb, since, while it would 
have but little effect on the lift of the planes, it would 
greatly reduce their drag. 

By the process outlined above, the aeroplane would 
successively assume every one of the series of speeds 
between the two speeds corresponding to normal and the 
second condition of flight (i.e. it would gradually pass from 
Or 1 to O 2 , fig. 17), though it would have to begin with 
climbing and descend afterwards. 

But we know that the pilot has a means of attaining 
these intermediary speeds while continuing to fly horizontally, 
namely, by throttling down his engine. This, at all events, 
is what he should do until the speed of the machine had 

* At present we are only dealing with the sustentation of the aeroplane. 
From the point of view of stability, which will be dealt with in subsequent 
chapters, it seems highly probable that the necessity of being able to fly at 
a small and at a large angle of incidence will lead to the employment of 
special constructional devices. 



FLIGHT IN STILL AIR 



77 



reached a certain point 01 (fig. 18) corresponding to that 
degree of throttling at which the power-plant curve (much 
flatter now by reason of the throttling-down process) only 
continues to touch the aeroplane curve at a single point L. 
Below this speed, if the pilot continues to increase the angle 
of incidence by using the elevator, horizontal flight cannot 
be maintained except by quickly opening the throttle. 

It would therefore seem feasible to pass from the normal 
to the second condition of flight, without rising or falling, 




FIG. 18. 

by the combined use of elevator and throttle. But up till 
now all this remains pure theory, for hitherto few pilots 
know how to vary their speed to any considerable extent, 
and probably not a single one has yet reduced this speed 
below the point 01 and ventured into the region of the 
second condition of flight, that wherein the elevator has to 
be operated in the inverse sense. 

The reason for this view is that the aeroplane, when 
its speed approaches the point 01, is flying without any 
margin, and consequently is then bound to descend. If 
therefore it obeys the impulse of descending given by the 
elevator, it no longer responds to the climbing manipulation. 



78 FLIGHT WITHOUT FORMULA 

As soon as the pilot perceives this,* he hastens to increase 
the speed of his machine again by reducing the angle of 
incidence and opening his throttle, whereas, in order to pass 
the critical point, he would in fact have to open the throttle 
but still continue to set his elevator to climb. 

The possibility of achieving several different speeds by 
the combined use of elevator and throttle forms the solution 
to the problem of wide speed variation. 

The greatest possible speed variation which any aeroplane 
is capable of attaining is measured by the difference between 
the normal and the second condition of flight. But, up to 
the present at any rate, the latter has not been reached, and 
the lowest speed of an aeroplane is that (indicated by 01, 
fig. 18) corresponding to flight at the " limit of capacity." 

This particular speed, not to be mistaken for one of the 
two essential conditions of flight, is usually very close to 
that corresponding to the economical angle of incidence 
(see Chapter II.). Hence the economical speed constitutes the 
lower limit of variation, which has probably never yet been 
attained. 

In the future, if the second condition of flight is achieved 
in practice, one will be able to fly at the lowest possible 
speed an aeroplane can attain. This conclusion may prove 
of considerable interest in the case of fast machines, for 
any reduction of speed, however slight, is then important. 

The highest speed is that of the normal flight of an aero- 
plane. In the example represented in fig. 17 this speed is 
23 m. per second, or about 83 km. per hour. Since the 
economical speed of the machine in question is about 66 
km. per hour, the absolute speed variation would be 17 km. 
per hour, or, relatively, about 20 per cent. This, however, 
is a maximum, since the economical speed, as we know, is 
never attained in practice. 

The above leads to the conclusion that the way to obtain 

* He is the more prone to do this owing to the fact that, with present 
methods of design and construction, stability decreases as the angle of 
incidence is increased. 



FLIGHT IN STILL AIR 



79 



a large speed variation is to increase the normal flying 
speed. 

In the previous example we assumed that the 50 h.p. 
motor turning at 1200 revolutions per minute was equipped 
with a propeller with a 0-7 pitch ratio, well adapted, 
whose characteristic qualities are given in Table VI. 

Now let us replace this propeller by another, equally well 




rStc] 



30 



adapted, but with a pitch ratio of 1-15. According to 
Table VI. the diameter of this propeller would be 1-98 m. 
and its pitch 2-28 m. The best speed corresponding to 
the new propeller would be 33 m. per second, and the 
maximum useful power developed at this speed 42 h.p. 

Now let the new power-plant curve (thick line) be super- 
posed on the previous aeroplane curve (see fig. 19). For 
the sake of comparison the previous power-plant curve is 
also reproduced in this diagram. 



80 FLIGHT WITHOUT FORMULA 

The advantage of the step is clear at a glance. In fact, 
the normal flying speed increases from Or l equivalent 
to 23 m. per second or 83 km. per hour to Or\ equivalent 
to 26 m. per second, or about 93 km. per hour. This in- 
creases the speed variation from 17 to 27 km. per hour, or 
from 20 to 29 per cent. 

Again, the maximum surplus power developed by the 
power-plant over and above that required merely for 
sustentation, amounting to about 7 h.p. with the former 
propeller, now becomes about 12 h.p. The quickest climb- 

7 x 75 
ing speed therefore grows from - =0-88 m. per second 

uOO 

12x75 

to = 1'5 m. per second. 

oOO 

Hence, by simply changing the propeller, one obtains the 
double result of increasing the normal flying speed of the 
aeroplane together with its climbing powers. Nor is the 
fact surprising, but merely emphasises our contention that 
since highly efficient propellers can be constructed, it will 
be just as well to use them. 

In order to gain an idea of the relative importance of 
increasing the pitch ratio when this ratio has already a 
certain value, we may superpose in a single diagram 
(fig. 20), on the aeroplane curve, all the power-plant curves 
representing the various propellers, well adapted, used with 
the same 50-h.p. motor turning at 1200 revolutions per 
minute, according to Table VI. 

Firstly, it will be evident that a pitch ratio of 0-5 would 
not enable the aeroplane in question to maintain horizontal 
flight, since the two curves that of the power-plant and of 
the aeroplane do not meet. In fact, the pitch ratio must 
be between 0-5 and 0-6 0-54, to be exact for the power- 
plant curve to touch the aeroplane curve at a single point. 
Horizontal flight would then be possible, but only at one 
speed and without a margin. 

But as soon as the pitch ratio increases, the normal flying 
speed and the climbing speed increase very rapidly. On 



FLIGHT IN STILL AIR 



81 



the other hand, once the pitch ratio amounts to 0-9, the 
advantage of increasing it still further, though this still 
exists, becomes negligible. Beyond 1-0 a further increase 
of pitch ratio (in the specific case in question) need not be 
considered. All of which are, of course, theoretical con- 
siderations, although they point to certain definite principles 
which cannot be ignored in practice a fact of which 















; 




I 












7 




1 










.1 






1 

fc 






4: 


^ 


V 

+- . 

0-8 


\jti 

-1-00 

I 


. 




$ 


^ 




7^-0-6 
**0-S4^ 


"0-7 ^ 
Y> 

,/ 








yjr 






^0-J 








/ 






Speed 


offtiyl 


tfanffl.j. 


erSecJ 




1 


1 


/ 

] 


J 2 
?IG. 20. 


2 


5 J 


J 


j 



constructors, as already remarked, are now becoming 
cognisant. 

At the same time, the reduction of the diameter necessi- 
tated by the use of propellers of great efficiency is not 
without its disadvantages, more especially in the case 
of monoplanes and tractor biplanes in which the propeller 
is situated in front. In these conditions, the propeller 
throws back on to the fuselage a column of air which be- 
comes the more considerable as the propeller diameter is 

6 



82 



FLIGHT WITHOUT FORMULA 



reduced, since practically only the portions of the blades 
near the tips produce effective work. 

It is on this ground that we may account for the fact 
that reduction in propeller diameter has not yet, up to a 
point, given the good results which theory led one to 
expect. 

But when the propeller is placed in rear of the machine 



70 



6C 



50 



4C 



30 



20 



20 




Sj teed oj flight 



Sec) 



30 



40 



35 
FIG. 21. 

The figures at the side of the curve indicate the lift. 



and the backward flowing air encounters no obstacle, there 
is every advantage in selecting a high pitch ratio, and we 
have already seen that M. Tatin, in consequence, on his 
Torpille fitted a propeller with a pitch exceeding the 
diameter.* 

* It may also be noticed that the need for reducing the diameter 
gradually disappears as the power of the motor increases, because the 
diameter of propellers well adapted to a motor increases with the power 
of the latter. 



FLIGHT IN STILL AIR 



83 



The use of propellers of high efficiency, therefore, obviously 
increases the speed variation obtainable with any particular 
aeroplane. 

The lower limit of this speed variation has already been 
seen to be the economical speed of the aeroplane. 

Now, it should be noted that, in designing high-speed 
machines, the use of planes of small camber and with a 
very heavy loading has the result of increasing the value 
of the economical speed. Thus, the Torpille, already 
referred to, appeared to be capable of attaining a speed of 
160 km. per hour ; * but its economical speed would have 
been about 28 m. per second or 100 km. per hour. Fig. 21 
shows, merely for the sake of comparison, the curve of an 
aeroplane of this type (weight, 450 kg. ; area, 12-50 sq. m. ; 
detrimental surface, 0-30 sq. m.) plotted from the following 
table. 



TABLE VII. 



+3 

a 


Speed Value. 


fei 


1 ;? 


C? c3 ^ 


1* 


11! 


1 






gi 13 .3^ 


H * 


** 't? . 


1 




1*1 


" 5fi gn^, 
f 8 X o 


^s 


i| 


ill 






fs 


m.p.s. 


km.p.h. 


"ill 


ir 


ill 


&i 


i*5 








Q 




3-x 




p<2~ 


1 


2 


3 


4 


5 


6 


7 


8 


o-oio 


60 


216 


0-0007 


31kg. 


87kg. 


118 kg. 


94 h.p. 


0-020 


42-4 


158 


0-0013 


29 


43 


72 


40 


0-030 


34-6 


125 


0-0020 


31 


29 


60 


28 


0-040 


30 


108 


0-0034 


38 


22 


60 


24 


0-050 


26-8 


97 


0-0055 


49 


18 


67 


24 


0-058 


24-9 


90 


0-0100 


77 


16 


93 


31 



* If we allow it a detrimental surface of 0'30 sq. metre, which is 
certainly not enough. 



84 



FLIGHT WITHOUT FORMULA 



The speed variation of such a machine would be 60 km. 
per hour =38 per cent. 

If it could fly in the second condition of flight, i.e. at 
90 km. per hour, the speed variation would be 70 km. per 
hour, or 44 per cent. 

In a machine o similar type, able to attain a speed of 
200 km. per hour (weight, 500 kg. ; area, 9 sq. m. ; detri- 













*f 




I 

^ 

5 








/ 


/ 




? 

^ 

3 


CO 
"5 

o 




o 

o , 


/ 






5 




Ll 





o ./ 


/ 










V6_ 


Jx 












































Speed c 


fflykl 


{ in Tn/x.rS 


y 



30 



35 



45 



50 



40 

FIG. 22. 

The figures by the side of the curve indicate the lift. 

mental surface, 0-03 sq. m.), whose characteristic curve is 
plotted in fig. 22, according to Table VIII., the economical 
speed would be 34 m. per second, or 125 km. per hour, 
giving a speed variation of 75 km. per hour, or 38 per cent. 
If it could attain the second condition of flight, i.e. 110 km. 
per hour, the variation would be 90 km. per hour, or 45 
per cent. 

Fortunately, as may be seen, the high-speed machine of 



FLIGHT IN STILL AIR 



85 



the future should possess a high degree of speed variation. 
And in the case of really high speeds even the smallest 
advantage in this respect becomes of great importance. It 
may well be that the necessity for achieving the greatest 

TABLE VIII. 



"S 


Speed Value. 


hi 


01 X '*-' 

C ~C5 di ^. 


iil 


|j 


||| 














56 




s.-s g 


PHjg 0<I< 


'So ^ 


rp 


fe3^ 




I 


j/J 


g,ix| 


ill 


! 


III 




m.p.s. 


km.p.h. 


$J! 


l^ 




l| 










p c2 








PS*- 


1 


2 


3 


4 


5 


6 


7 


8 


o-oio 


74-8 


270 


0-0007 


35kg. 


134 kg. 


169 kg. 


168 h.p. 


0-020 


53 


190 


0-0013 


33 


68 


101 


72 


0-030 


43-1 


155 


0-0020 


35 


45 


80 


46 


0-040 


37-4 


135 


0-0034 


43 


34 


77 


39 


0-050 


33-4 


120 


0-0055 


55 


27 


82 


37 


0-058 


31 


111 


o-oioo 


86 


24 


110 


46 



possible speed variation will induce pilots of the extra high 
speed machines of the future to attempt, for alighting, to 
fly at the second condition of flight.* In this they will 
only imitate a bird, which, when about to alight, places its 
wings at a coarse angle and tilts up its body. 

Fig. 20 further shows that when the pitch ratio is less 
than 0-8 the highest point of the power-plant curve lies to 
the left of the aeroplane curve. It only lies to the right of 
it when the pitch ratio is equal to or greater than 0-9. If 
the pitch ratio were 0-85, the highest point of the power- 
plant curve would just touch the aeroplane curve, and would 
hence correspond to normal flight. 

* Attention is, however, drawn to the remarks at the bottom of p. 74. 



86 . FLIGHT WITHOUT FORMULA 

In Chapter IV. it was shown that the highest point of 
the power-plant curve corresponds the propeller being 
supposedly well adapted to the motor to a rotational 
velocity of 1200 revolutions per minute, the normal number 
of revolutions at which it develops full power. If, there- 
fore, this highest point lies to the left of the aeroplane 
curve, the motor is turning at over 1200 revolutions per 
minute when the aeroplane is flying at normal speed. On 
the other hand, if the highest point lies to the right of the 
aeroplane curve, in normal flight the motor will be running 
at under 1200 revolutions per minute. 

In neither case will it develop full power. Moreover, 
there is danger in running the motor at too high a number 
of revolutions, particularly if it is of the rotary type. Only 
a propeller with a pitch ratio of 0-85 could enable the 
motor to develop its full power (in the special case in 
question). 

This immediately suggests the expedient of keeping the 
motor running at 1200 revolutions per minute while allow- 
ing the propeller to turn at the speed productive of its 
maximum efficiency through some system of gearing. 
Thus we are brought by a logical chain of reasoning to the 
geared-down propeller, a solution adopted in very happy 
fashion in the first successful aeroplane that of the brothers 
Wright. 

Let us suppose that an aeroplane whose curve is shown 
by the thin line in fig. 23 has a power-plant curve repre- 
sented by the thick line in the same figure, the propeller 
direct-driven, having a pitch ratio of 1-15, and hence possess- 
ing (according to Commandant Dorand's experiments) 84 
per cent, maximum efficiency. 

Evidently, however good this power-plant might be 
when considered by itself, it would be very badly adapted 
to the aeroplane in question, since, firstly, it would only 
enable the machine to obtain the low speed Or x ; and, 
secondly, the maximum surplus of useful power, the measure 
of an aeroplane's climbing properties, would fall to a very 



FLIGHT IN STILL AIR 



87 



low figure. Hence, the machine would only leave the 
ground with difficulty, and would fly without any margin. 
And all this simply and solely because the best speed, Om, 
suited to the power-plant would be too high for the aero- 
plane. 

Now let the direct-driven propeller be replaced by another 
of the same type, but of larger diameter, and geared down 
in such fashion that the best speed suited to this power- 
plant corresponds to the normal flying speed O' 15 of the 
aeroplane (see fig. 23). 




FIG. 23. 

The maximum useful power developed by this power- 
plant remains in theory the same as before, since the pro- 
peller, being of the same type, will still have a maximum 
efficiency of 84 per cent. The new power-plant curve will 
therefore be of the order shown by the dotted line in the 
figure. 

It is clear that by gearing down we first of all obtain an 
increase of the normal flying speed, and secondly, a very 
large increase in the maximum surplus of useful power 
that is, in the machine's climbing capacity. In practice, 
however, this is not a perfectly correct representation of 



88 FLIGHT WITHOUT FORMULA 

the case, since gearing down results in a direct loss of 
efficiency and an increase in weight. Whether or not to 
adopt gearing, therefore, remains a question to be decided 
on the particular merits of each case. Speaking very 
generally, it can be said that this device, which always 
introduces some complication, should be mainly adopted in 
relatively slow machines designed to carry a heavy load. 

In the case of high-speed machines it seems better to 
drive the propeller direct, though even here it may yet 
prove desirable to introduce gearing. 

This study of the power-plant may now be rounded off 
with a few remarks on static propeller tests, or bench tests. 
These consist hi measuring, with suitable apparatus, on the 
one hand, the thrust exerted by the propeller turning at a 
certain speed without forward motion, and, on the other, 
the power which has to be expended to obtain this result. 

Experiment has shown that a propeller of given diameter, 
driven by a given expenditure of power, exerts the greatest 
static thrust if its pitch ratio is in the neighbourhood of 
0-65.* On the other hand, we have seen that the highest 
thrust efficiency hi flight is obtained with propellers of a 
pitch ratio slightly greater than unity. Hence one should 
not conclude that a propeller would give a greater thrust 
hi flight simply from the fact that it does so on the bench. 
Thus, the propeller mounted on the Tatin Torpille, already 
referred to, which gave an excellent thrust hi flight, would 
probably have given a smaller thrust on the bench than a 
propeller with a smaller pitch. 

Consequently, a bench test is by no means a reliable 
indication of the thrust produced by a propeller hi flight. 
Besides, it is usually made not only with the propeller 
alone but with the complete power-plant, in which case the 
result is even more unreliable owing to the fact that the 
power developed by an internal combustion engine varies 
with its speed of rotation. 

For instance, suppose that a motor normally turning at 
* From Commandant Dorand's experiments. 



FLIGHT IN STILL AIR 89 

1200 revolutions per minute is fitted with a propeller of 
1-15 pitch ratio which, when tested on the bench by itself, 
already develops a smaller thrust than a propeller of 0-65 
pitch ratio ; the motor would then only turn at 900 revolu- 
tions per minute, whereas the propeller of 0-65 pitch ratio 
would let it turn at 1000 revolutions per minute, and hence 
give more power. The propeller with a high pitch ratio 
would therefore appear doubly inferior to the other, and 
this notwithstanding the fact that its thrust in flight would 
undoubtedly be greater. 

A propeller exerting the highest thrust in a bench test must 
not for that reason be regarded as the best. 



CHAPTER VI 
STABILITY IN STILL AIR 

LONGITUDINAL STABILITY 

AT the very outset of the first chapter it was laid down 
that the entire problem of aeroplane flight is not solved 
merely by obtaining from the " relative " air current which 
meets the wings, owing to their forward speed, sufficient 
lift to sustain the weight of the machine ; an aeroplane, in 
addition, must always encounter the relative air current in 
the same attitude, and must neither upset nor be thrown 
out of its path by a slight aerial disturbance. In other 
words, it is essential for an aeroplane to remain in equi- 
librium ; more, in stable equilibrium.* 

We may now proceed to study the equilibrium of an 
aeroplane in still air and the stability of this equilibrium. 

Since a knowledge of some of the main elementary 
principles of mechanics is essential to a proper understand- 
ing of the problems to be dealt with, these may be briefly 
outlined here. 

* The very fact that an aeroplane remains in flight presupposes, as 
we have seen, a first order of equilibrium, which has been termed the 
equilibrium of sustentation, which jointly results from the weight of the 
machine, the reaction of the air, and the propeller-thrust. The mainten- 
ance of this state of equilibrium, which is the first duty of the pilot, 
causes an aeroplane to move forward on a uniform and direct course. 

We are now dealing with a second order of equilibrium, that of the 
aeroplane on its flight-path. Both orders of equilibrium are, of course, 
closely interconnected, for if in flight the machine went on turning and 
rolling about in every way, its direction of flight could clearly not be 
maintained uniformly. 



STABILITY IN STILL AIR 



01 



The most important of these is that relating to the 
centre of gravity. 

If any body, such as an aeroplane, for instance (fig. 24), 
is suspended at any one point, and a perpendicular is 
drawn from the point of suspension, it will always pass, 
whatever the position of the body in question, through the 
same point G, termed the centre of gravity of the body. 

The effect of gravity on any body, in other words, the 




FIG. 24. 

force termed the weight of the body, therefore always 
passes through its centre of gravity, whatever position the 
body may assume. 

Another principle is also of the greatest importance in 
considering stability ; namely, the turning action of 
forces. 

When a force of magnitude F (fig. 25), exerted in the 
direction XX, tends to make a body turn about a fixed 
point G, its action is the stronger the greater the distance, 
Gx, between the point G and the line XX. In other words, 
the turning action of a force relatively to a point is the 
greater the farther away the force is from the point. 

Further, it will be readily understood that a force F', 
double the force F in magnitude but acting along a line 
YY separated from the fixed point G by a distance Gy, 



92 FLIGHT WITHOUT FORMULA 

which is just half of Gx, would have a turning force equal 
to F. In short, it is the well-known principle of the 
lever. 

The product of the magnitude of a force by the length 
of its lever arm from a point or axis therefore measures 
the turning action of the force. In mechanics this turning 
action is usually known as the moment or the couple. 

When, as in fig. 25, two turning forces are exerted in 
inverse direction about a single point or axis, and their 



X/ 




FIG. 25. 

turning moment or couple is equal, the forces are said to be 
in equilibrium about the point or axis in question. 

For a number of forces to be in equilibrium about a point 
or axis, the sum of the moments or couples of those acting 
in one direction must be equal to the sum of the couples of 
those acting in the opposite direction. 

It should be noted that in measuring the moment of a 
force, only its magnitude, its direction, and its lever arm 
are taken into account. The position of the point of its 
application is a matter of indifference. And with reason, 
for the point of application of a force cannot in any way 
influence the effect of the force ; if, for instance, an object 



STABILITY IN STILL AIR 93 

is pushed with a stick, it is immaterial which end of the 
stick is held in the hand, providing only that the force is 
exerted in the direction of the stick. 

Before venturing upon the problem of aeroplane stability 
a fundamental principle, derived from the ordinary theory 
of mechanics, must be laid down. 

FUNDAMENTAL PRINCIPLE. So far as the equilibrium 
of an aeroplane and the stability of its equilibrium are con- 
cerned, the aeroplane may be considered as being suspended 
from its centre of gravity and as encountering the relative 
wind produced by its own velocity. 

This principle is of the utmost importance and absolutely 
essential ; by ignoring it grave errors are bound to ensue, 
such, for instance, as the idea that an aeroplane behaves in 
flight as if it were in some fashion suspended from a certain 
vaguely-defined point termed the " centre of lift," usually 
considered as situated on the wings. An idea of this sort 
leads to the supposition that a great stabilising effect is 
produced by lowering the centre of gravity, which is thus 
likened to a kind of pendulum. 

Now, it will be seen hereafter that in certain cases the 
lowering of the centre of gravity may, in fact, produce a 
stabilising effect, but this for a very different reason. 

The " centre of lift " does not exist. Or, if it exists, it is 
coincident with the centre of gravity, which is the one and 
only centre of the aeroplane. 

The three phases of stability, which is understood to 
comprise equilibrium, to be considered are : 

Longitudinal stability. 

Lateral stability. 

Directional stability. 

First comes longitudinal stability, which will be dealt 
with in this chapter and the next. 

Every aeroplane has a plane of symmetry which remains 
vertical in normal flight. The centre of gravity lies in 
this plane. The axis drawn through the centre of gravity 
at right angles to the plane of symmetry may be termed 



94 FLIGHT WITHOUT FORMULA 

the pitching axis and the equilibrium of the aeroplane 
about its pitching axis is its longitudinal equilibrium. 

Hereafter, and until stated otherwise, it will be assumed 
that the direction of the propeller-thrust passes through 
the centre of gravity of the machine. Consequently, 
neither the propeller-thrust nor the weight of the aeroplane, 
which, of course, also passes through the centre of gravity, 
can have any effect on longitudinal equilibrium, for, hi 
accordance with the fundamental principle set out above, 
the moments exerted by these two forces about the pitch- 
ing axis are zero. 

Hence, in order that an aeroplane may remain in longi- 
tudinal equilibrium on its flight-path, that is, so that it 
may always meet the air at the same angle of incidence, 
all that is required is that the reaction of the air on the 
various parts of the aeroplane should be in equilibrium 
about its centre of gravity. 

Now, in normal flight all the reactions of the air must 
be forces situated in the plane of symmetry of the machine. 
These forces may be compounded into a single resultant 
(see Chapter II.), which, for the existence of longitudinal 
equilibrium, must pass through the centre of gravity. 

We may therefore state that : when an aeroplane is 
flying in equilibrium, the resultant of the reaction of 
the air on its various parts passes through the centre of 
gravity. 

This resultant will be called the total pressure. 

Let us take any aeroplane, maintained in a fixed position, 
such, for instance, that the chord of its main plane were at 
an angle of incidence of 10, and let us assume that a hori- 
zontal air current meets it at a certain speed. 

The air current will act upon the various parts of the 
aeroplane and the resultant of this action will be a total 
pressure of a direction shown by, say, P 10 (fig. 26). Without 
moving the aeroplane let us now alter the direction of the 
air current (blowing from left to right) so that it meets 
the planes at an ever-decreasing angle, passing successively 



STABILITY IN STILL AIR 



95 



from 10 to 8, 6, 4, etc. In each case the total pressure 
will take the directions indicated respectively by P 8 , P 6 , P 4 , 
etc. Let G be the centre of gravity of the aeroplane. 

Only one of the above resultants P 6 , for instance will 
pass through the centre of gravity. From this it may be 
deduced that equili- 
brium is only possible 
in flight when the 
main plane is at an 
angle of incidence of 
6. 

Thus, a perfectly 
rigid unalterable 
aeroplane could only 
in practice fly at a 
single angle of in- 
cidence. 

If the centre of 
gravity could be 
shifted by some means 
or other, to the posi- 
tion P 4 , for instance, 
the one angle of in- 
cidence at which the 
machine could fly 
would change to 4. 
But this method for 
varying the angle of 
incidence has not 
hitherto been success- 
fully applied in 
practice.* 

The same result, however, is obtained through an auxil- 
iary movable plane called the elevator. 

It is obvious that by altering the position of one of the 

* It will be seen hereafter that, if the method can be applied, it would 
have considerable advantages. 




FIG. 2fi. 



96 



FLIGHT WITHOUT FORMULA 



planes of the machine the sheaf of total pressures is altered. 
Thus, figs. 27 and 28 represent the total pressures in the 
case of one aeroplane after altering the position of the 




FIG. 27. 



elevator (the dotted outline indicating the main plane). 
If G is the centre of gravity, the normal angle of incidence 
passes from the original 4 to 2 by actuating the elevator. 



STABILITY IN STILL AIR 



97 



Therefore, as stated in Chapter I., by means of the 
elevator the position of longitudinal equilibrium of an 




FIG. 28. 

aeroplane, and hence its incidence, can be varied at 
will. 

The action of the elevator will be further considered in 
the next chapter. 

But the longitudinal equilibrium of an aeroplane must 

7 



98 



FLIGHT WITHOUT FORMULA 



also be stable ; in other words, if it should accidentally lose 

its position of equilibrium, the action of the forces arising 

through the air current from the very fact of the change 

in its position should cause 
it to regain this position 
instead of the reverse. 

If we examine once 
again the sheaf of total 
pressures we may be able 
to gain an idea of how 
this condition of affairs 
can be brought about. 

Returning again to fig. 
26, let us suppose that by 
an oscillation about its 
pitching axis the move- 
ment being counter-clock- 
wise the angle of the 
planes, which is normally 
6 since the total pressure 
P 6 passes through the 
centre of gravity, decreases 
to 4, the resultant of 
pressure on the aeroplane 
in its new position will 
have the direction P 4 ; 
hence this resultant will 
have, relatively to the 
pitching axis, a moment 
acting clockwise, which 

will therefore be a righting couple since it opposes the 

oscillation which called it into being. 

The same thing would come to pass if the oscillation was 

in the opposite direction. 

In this case, therefore, equilibrium is stable. 

On the other hand, if the sheaf of pressures was arranged 

as in fig. 29, the pressure P 4 would exert an upsetting 




FIG. 29. 



STABILITY IN STILL AIR 



99 



En te.rt.rty Ectgre. 



0-1 



0-2 



0-3 



0-4 



0-5 



couple relatively to the pitching axis, and equilibrium 
would be unstable. 

The stability or instability of longitudinal equilibrium 
therefore depends on the relative positions of the sheaf of 
total pressures and of the centre of gravity, and it may be laid 
down that when the line 
of normal pressure is in- 
tersected by those of the 
neighbouring total pres- 
sures at a point about 
the centre of gravity, 
equilibrium is stable, 
whereas it is unstable in 
the reverse case. 

Several experimenters, 
and among them notably 
M. Eiffel, have sought to 
determine by means of 
tests with scale models 
the position of the total 
pressures corresponding 
to ordinary angles of 
incidence. Hitherto M. 
Eiffel's researches have 
been confined to tests on 
model wings and not on 
complete machines, .but 
the latter are now being 
employed. Moreover, 
the results do not indi- v on , . ,,, 

FIG. 30. Angles t of the chord and the wind. 

cate the actual position 

and distribution of the pressure itself, but only the point 
at which its effect is applied to the plane, this point being 
known as the centre of pressure. 

The results of these tests have been plotted in two series 
of curves which give the position of the centre of pressure 
with a change in the angle of incidence. Figs. 30 and 31 



0-6 



0-7 



0-8 



0-9 



10 



Irt 



-40 -30 -20-IO 10 20 30 



40 



100 



FLIGHT WITHOUT FORMULAE 



reproduce, by way of indicating the system, the two series 
of curves relating to a Bleriot XI. wing. 

It has already been remarked that the point from which 
a force is applied is of no importance ; accordingly, a centre 



-30V 



-is* 




FIG. 31. 



of pressure is of value only in so far as it enables the direc- 
tion of the pressures themselves to be traced. 

By comparing the curve shown in fig. 31 with the polar 
curves already referred to in previous chapters, one obtains 



STABILITY IN STILL AIR 



101 



a means of reproducing both the position and the magnitude, 
relatively to the wing itself, of the pressures it receives at 
varying 




FIG. 32. Sheaf of pressures on a flat plane. 

Figs. 32, 33, and 34 show the sheaf of these pressures in 
the case, respectively, of : 
A flat plane. 

A slightly cambered plane (e.g. Maurice Farman). 
A heavily cambered plane (Bleriot XL). 
These diagrams, be it repeated, relate only to the plane 
by itself and not to complete machines. 

* A description of the method may be found in an article published 
by the author in La Technique Aeronautique (January 15, 1912). 



102 



FLIGHT WITHOUT FORMULAE 



Comparison of these three diagrams brings out straight 
away a most important difference between the flat and the 
two cambered planes. That relating to the flat plane, 
in fact, is similar in its arrangement to that shown in 






FIG. 33. Sheaf of pressures on a Maurice Farman plane. 

fig. 26, which served to illustrate a longitudinally stable 
aeroplane. 

The diagrams relating to cambered planes, on the other 
hand, are analogous, so far as the usual flying angles are 
concerned, to fig. 29, which depicted the case of a longi- 
tudinally unstable aeroplane. 

Thus we can state that, considered by itself, a flat plane 
is longitudinally stable, a cambered plane unstable (the 



STABILITY IN STILL AIR 



103 




A -HHffA- 

A [in / 




B 



FIG. 34. Sheaf of pressures on a Bleriot XI. plane. 



104 FLIGHT WITHOUT FORMULA 

latter statement, however, as will subsequently be seen, is 
not always absolutely correct). On the other hand, every 
one knows nowadays that flat planes are very inefficient, 
producing little lift with great drag. 

Hence the necessity for finding means to preserve the 
valuable lifting properties of the cambered plane while 
counteracting its inherent instability. The bird, inciden- 
tally, showed that it is possible to fly with cambered wings. 
And it was by adopting this example and improving upon 
it that the problem was solved, by providing the aeroplane 
with a tail. 

An auxiliary plane, of small area but placed at a con- 
siderable distance from the centre of gravity of the aero- 
plane, and therefore possessing a big lever arm relatively 
to the centre of gravity, receives from the air, when in 
flight the aeroplane comes to oscillate in either direction, 
a pressure tending to restore it to its original attitude. 
Since this pressure is exerted at the end of a long lever 
arm, the couples, which are always righting couples, are of 
considerably greater magnitude than the upsetting couples 
arising from the inherent instability of the cambered type 
itself. 

The adoption of this device has rendered it possible to 
utilise the great advantage possessed by cambered planes. 
Of course it is true that a machine with perfectly flat planes 
would be doubly stable, by virtue both of its main planes 
and of its tail, but to propel a machine of this type would 
mean an extravagant waste of power. 

Provided the tail is properly designed, there is nothing 
to fear even with an inherently unstable plane, and the 
full lifting properties of the camber are nevertheless 
retained. 

Subsequently it will be shown that the use of a tail 
entirely changes the nature of the sheaf of pressures, which, 
in an aeroplane provided with a tail, and even though its 
planes are cambered, assumes the stable form corresponding 
to a flat plane. 



STABILITY IN STILL AIR 105 

The aeroplane therefore really resolves itself into a main 
plane and a tail.* 

Assuming, once and for all, that the propeller-thrust 
passes through the centre of gravity, the longitudinal 
equilibrium of an aeroplane about the centre of gravity 
can be represented diagrammatically by one of the three 
figs., 35, 36, and 37. 

In fig. 35 the tail CD is normally subjected to no pressure 
and cuts the air with its forward edge. In this case, equi- 
librium exists if the pressure Q (in practice equal to the 
weight of the machine) on the main plane AB passes through 
the centre of gravity G. 

In fig. 36 the tail CD is a lifting tail, that is, normally 
it meets the air at a positive angle and therefore is sub- 
jected to a pressure q directed upwards. For equilibrium 
to be possible in this case the pressure Q on the main plane 
AB must pass in front of the centre of gravity G of the 
aeroplane, so that its couple about the point G is equal to 
the opposite couple q of the tail. 

The pressures Q and q must be inversely proportional to 
the length of their lever arms. When compounded they 
produce a resultant or total pressure equal to their sum 
(and to the weight of the aeroplane), which, as we know, 
would pass through the centre of gravity. 

Lastly, in fig. 37 the tail CD is struck by the air on its 
top surface and receives a downward pressure q. To obtain 
equilibrium the pressure Q on the main plane AB must 
pass behind the centre of gravity G, the couples exerted 
about this point by the pressures Q and q being, as before, 
equal and opposite. Once again, the pressures Q and q 
must be inversely proportional to the length of their lever 
arms. If compounded they would produce a resultant 
total pressure equal to their difference (and to the weight 

* In the case of a biplane both the planes will be considered as forming 
only a single plane, a proceeding which is quite permissible and could, 
if necessary, be easily justified. 



106 FLIGHT WITHOUT FORMULA 




G B CD 

FIG. 35. 




c D 



FIG. 36. 



P-Q-W /Q 




B 



V 

* 



FIG. 37. 



STABILITY IN STILL AIR 



107 



of the aeroplane), which would again pass through the 
centre of gravity. 

A fourth arrangement (fig. 38), and the first to be adopted 
in practice since the 1903 Wright and the 1906 Santos- 
Dumont machines were of this type is also possible. It 
has lately been made use of again hi machines of the 
" Canard " type (e.g. in the Voisin hydro-aeroplane), and 
consists in placing the tail, which must of course be a 
lifting tail, in front of the main plane. The conditions of 
equilibrium are the same as in fig. 36. 

In an aeroplane, to whichever type it belongs, the term 




FIG. 38. 

longitudinal dihedral, or Fee, is usually applied to the angle 
formed between the chords of the main and tail planes. 

Hitherto the relative positions of the main plane and 
the tail have been considered only from the point of view 
of equilibrium. We have now to consider the stability of 
this equilibrium. For this purpose we must return to the 
sheaf of pressures exerted, not on the main plane alone, 
but on the whole machine, that is, we have to consider the 
sheaf of total pressures. 

This is shown in fig. 39,* which relates to a Bleriot XI. 

* At the time when this treatise was first published, no experiments 
had been made to determine the actual sheaf of pressures as it exists in 
practice. The accompanying diagrams were drawn up on the basis of 
the composition of forces. 



108 



FLIGHT WITHOUT FORMULA 



Pf 



FIG. 39. Sheaf of total pressures on a complete Bleriot XI. monoplane. 



wing provided with a tail plane of one-tenth the area of 
the main plane, making relatively to the main plane a longi- 



STABILITY IN STILL AIR 109 

tudinal Vee or dihedral of 6, and placed at a distance 
behind the main plane equal to twice the chord of the 
latter. 

Let it be assumed that the normal angle of incidence of 
the machine is 6, which would be the case if its centre 
of gravity coincided with the pressure P 6 , at G 15 for in- 
stance. 

An idea of the longitudinal stability of the machine in 
these conditions may be guessed from calculating the couple 
caused by a small oscillation, such as 2. 

Since the normal incidence is 6, the length of the 
pressure P 6 is equivalent to the weight of the machine. 
By measuring with a rule the length of P 4 and P 8 , it will 
be found to be equal respectively to P 6 xO-74 and to 
P 6 Xl-23. The values of P 4 and P 8 therefore are the pro- 
ducts of the weight of the aeroplane multiplied by 0-74 
and 1-23 respectively. 

Further, the lever arms of these pressures will, on measure- 
ment, be found to be respectively 0-043 and 0-025 times 
the chord of the main plane. 

By multiplying and taking the mean of the results ob- 
tained, which only differ slightly, it will be found that an 
oscillation of 2 produces a couple equal to 0-031 times the 
weight of the aeroplane multiplied by its chord. 

This couple produced by an oscillation of 2 can obviously 
be compared to the couple which would be produced by an 
oscillation of 2 imparted to the arm of a pendulum or 
balance of a weight equal to that of the aeroplane. 

For these two couples to be equal, the pendulum arm 
must have a length of 0-88 of the chord, or, if the latter be 
2m., for instance, the arm would have to measure 1-76 m. 
Hence, the longitudinal stability of the machine under 
consideration could be compared to that of an imaginary 
pendulum consisting of a weight equal to that of the aero- 
plane placed at the end of a 1-76 m. arm. It is evident 
that the measure of stability possessed by such a pendulum 
is really considerable. 



110 FLIGHT WITHOUT FORMULA 

Having laid down this method of calculating the longi- 
tudinal stability of an aeroplane, fig. 39 may once again be 
considered. 

To begin with, it is evident that if the centre of gravity 
is lowered, though still remaining on the pressure line P 6 , 
the longitudinal stability of the machine will be increased 
since, the pressure lines being spaced further apart, then- 
lever arms will intersect. Therefore, under certain condi- 
tions, the lowering of the centre of gravity may increase 
longitudinal stability, though this has nothing whatsoever 
to do with a fictitious " centre of lift." Besides, in practice 
the centre of gravity can only be lowered to a very small 
extent, and the possible advantage derived therefrom is 
consequently slight, while, on the other hand, it entails 
disadvantages which will be dealt with hi the next 
chapter. 

Finally, the use of certain plane sections robs the lower- 
ing of the centre of gravity of any advantages which it 
may otherwise possess, a point which will be referred to in 
detail hereafter. 

Returning to fig. 39 the normal angle of incidence being 
6, and the non-lifting tail forming this same angle with the 
chord of the main plane, the tail plane will normally be 
parallel with the wind (see fig. 35). 

If the centre of gravity, instead of being at G l5 were at 
G 2 , on the pressure line P 8 , the tail would become a lifting 
tail (see fig. 36), having a normal angle of incidence of 2. 
Calculating as before, the length of the arm of the imaginary 
equivalent pendulum is found to be only 0-63 of the chord, 
or 1-26 m. if the chord measures 2 m. 

The aeroplane is therefore less stable than in the previous 
example. 

On the contrary, if the centre of gravity were situated at 
G 3 , corresponding to a normal incidence of 4, so that the 
tail is struck by the wind on its top surface at an angle of 
2 (in other words, is placed at a " negative " angle of 2, 
see fig. 37), the equivalent pendulum would have to have 



STABILITY IN STILL AIR 111 

an arm 3-50 m. long,* or about twice as long as when the 
normal incidence is 6. 

From this one would at first sight be tempted to conclude 
that the longitudinal stability of an aeroplane is the greater 
the smaller its normal flying angle, or, in other words, the 
higher its speed ; but, although this may be true in certain 
cases, it is not so in others. Thus, if the alteration in the 
angle of incidence were obtained by shifting the centre of 
gravity, the conclusion would be true, since the sheaf of 
total pressures would remain unaltered. 

But if the reduction of the angle is effected either by 
diminishing the longitudinal dihedral or, and this is really 
the same thing, by actuating the elevator, the conclusion no 
longer holds good, for the sheaf of total pressures does 
change, and in this case, as the following chapter will show, 
so far from increasing longitudinal stability, a reduction 
of the angle of incidence may diminish stability even to 
vanishing point. 

It should further be noted that the arrangement shown 
diagrammatically in fig. 37, which consists hi disposing 
the tail plane so that it meets the wind with its top surface 
in normal flight, is productive of better longitudinal stability 
than the use of a lifting tail.f This conclusion will be 
found to be borne out by fig. 40, showing the pressures 
exerted on the main plane by itself. 

By measuring the couples, it is clear that if the centre of 
gravity is situated at G 1? for instance, the plane is unstable, 
as we already knew ; but if the centre of gravity were 
placed far enough forward relatively to the pressures, at G 2 , 
for instance, a variation in the angle may set up righting 
couples even with a cambered plane. The couple resulting 
from a variation of this kind is the difference between the 

* Actually, the arm is longer if the oscillation is in the sense of a dive 
than in the case of stalling, which is quite in agreement with the con- 
clusions which will be set out later. 

t It will be seen later that this arrangement also seems to be excellent 
from the point of view of the behaviour of a machine in winds. 



112 FLIGHT WITHOUT FORMULAE 

PlS 



PIO 



A 



1 



FIG. 40. 



B 



STABILITY IN STILL AIR 



113 



couples of the pressure, before and after the oscillation, 
about the centre of gravity. 

Cambered planes in themselves may therefore be rendered 
stable by advancing the centre of gravity. 

This is not difficult to understand ; as a plane is further 
removed from the centre of gravity it begins to behave 



P. 5 



FIG. 41. Sheaf of total pressures on a Maurice Farman aeroplane. 

more and more like the usual tail plane. In these conditions 
the stability of an aeroplane becomes very good indeed, 
since it is assisted by main and tail planes alike. 

This explains why the tail-foremost arrangement (see 
fig. 38) can be stable, for in this arrangement the tail, 
situated in front, really performs the function of an " un- 
stabiliser," which is overcome by the inherent stability of 

8 



114 FLIGHT WITHOUT FORMULA 

the main plane owing to the fact that the latter is situated 
far behind the centre of gravity. 

Fig. 40 (which relates to the pressures on the main 
plane) further shows that if the centre of gravity is low 
enough, at G\, for instance, a Bleriot XI. wing would become 
stable from being inherently unstable. This is the reason 
for the stabilising influence of a low centre of gravity, 
which the examination of the sheaf of total pressures 
already revealed. 

For the sake of comparison, fig. 41 is reproduced, showing 
the sheaf of total pressures belonging to an aeroplane of 
the type previously considered, but with a Maurice Farman 
plane instead of a Bleriot XI. section. 

The pressure lines are almost parallel. 

Lowering the centre of gravity in a machine of this type 
would produce no appreciable advantage. 

It will be seen that the pressure lines draw ever closer 
together as the incidence increases, and become almost 
coincident near 90. This shows that if, by some means or 
other, flight could be achieved at these high angles which 
could only be done by gliding down on an almost vertical 
path, the machine remaining practically horizontal, which 
may be termed " parachute " flight, or, more colloquially, 
a " pancake " longitudinal stability would be precarious 
in the extreme, and that the machine would soon upset, 
probably sliding down on its tail. Parachute flight and 
" pancake " descents would therefore appear out of the 
question, failing the invention of special devices. 



CHAPTER VII 
STABILITY IN STILL AIR 

LONGITUDINAL STABILITY (cone 



IN the last chapter it was shown that the longitudinal 
stability of an aeroplane depends on the nature of the sheaf 
of total pressures exerted at various angles of incidence on 
the whole machine, and that stability could only exist if 
any variation of the incidence brought about a righting 
couple. 

But this is not all, for the righting couple set up by 
an oscillation may not be strong enough to prevent the 
oscillation from gradually increasing, by a process similar 
to that of a pendulum, until it is sufficient to upset the 
aeroplane. 

The whole question, indeed, is the relation between the 
effect of the tail and a mechanical factor, known as the 
moment of inertia, which measures in a way the sensitive- 
ness of the machine to a turning force or couple. 

A few explanations in regard to this point may here be 
useful. 

A body at rest cannot start to move of its own accord. 
A body in motion cannot itself modify its motion. 

When a body at rest starts to move, or when the motion 
of a body is modified, an extraneous cause or force must 
have intervened. 

Thus a body moving at a certain speed will continue to 
move in a straight line at this same speed unless some force 
intervenes to modify the speed or deflect the trajectory. 



116 



FLIGHT WITHOUT FORMULA 



The effect of a force on a body is smaller, the greater the 
inertia or the mass of the latter. 

Similarly, if a body is turning round a fixed axis, it will 
continue to turn at the same speed unless a couple exerted 
about this axis comes to modify this speed. 

This couple will have the smaller effect on the body, the 
more resistance the latter opposes to a turning action, that 
is, the more inertia of rotation it possesses. It is this 
inertia which is termed the moment of inertia of the body 
about its axis. The moment of inertia increases rapidly as 
the masses which constitute the body are spaced further 
apart, for, in calculating the moment of inertia, the dis- 
tances of the masses from the axis of rotation figure, not 




in simple proportion, but as their square. An example will 
make this principle, which enters into every problem con- 
cerning the oscillations of an aeroplane, more clear. 

At O, on the axis AB (fig. 42) of a turning handle a rod 
XX is placed, along which two equal masses MM can slide, 
their respective distances from the point O always remain- 
ing equal. Clearly, if the rod, balanced horizontally, were 
forced out of this position by a shock, the effect of this 
disturbing influence would be the smaller, the further the 
masses MM were situated from the point O, in other words, 
the greater the moment of inertia of the system. 

If the rod were drawn back to a horizontal position by 
means of a spring it would begin to oscillate ; these oscilla- 
tions will be slower the further apart the masses ; but, on 
the other hand, they will die away more slowly, for the 



STABILITY IN STILL AIR 117 

system would persist longer in its motion the greater its 
moment of inertia. 

These elementary principles of mechanics show that an 
aeroplane with a high moment of inertia about its pitching 
axis, that is, whose masses are spread over some distance 
longitudinally instead of being concentrated, will be more 
reluctant to oscillate, while its oscillations will be slow, thus 
giving the pilot time to correct them. On the other hand, 
they persist longer and have a tendency to increase if the 
tail plane is not sufficiently large. 

This relation between the stabilising effect of the tail 
and the moment of inertia in the longitudinal sense has 
already been referred to at the beginning of this chapter. 
It may be termed the condition of oscillatory stability. 

In practice most pilots prefer to fly sensitive machines 
responding to the slightest touch of the controls. Hence 
the majority of constructors aim at reducing the longi- 
tudinal moment of inertia by concentrating the masses. 

It should be added that the lowering of the centre of 
gravity increases the moment of inertia of an aeroplane 
and hence tends to set up oscillation, one of the disadvan- 
tages of a low centre of gravity which was referred to in 
the last chapter. 

By concentrating the masses the longitudinal oscillations 
of an aeroplane become quicker and, although not so easy 
to correct, present one great advantage arising from their 
greater rapidity. 

For, apart from its double stabilising function, the tail 
damps out oscillations, forms as it were a brake in this 
respect, and the more effectively the quicker the oscillations. 
The reason for this is simple enough. Just as rain, though 
falling vertically, leaves an oblique trace on the windows 
of a railway-carriage, the trace being more oblique the 
quicker the speed of travel, so the relative wind caused by 
the speed of the aeroplane strikes the tail plane at a 
greater or smaller angle when the tail oscillates than when 
it does not, and this with all the greater effect the quicker 



118 FLIGHT WITHOUT FORMULAE 

the oscillation. It is a question of component speeds 
similar to that which will be considered when we come to 
deal with the effect of wind on an aeroplane. 

The oscillation of the tail therefore sets up additional 
resistance, which has to be added to the righting couple due 
to the stability of the machine, as if the tail had to move 
through a viscous, sticky fluid, and this effect is the more 
intense the quicker the oscillation. It is a true brake effect. 

In this respect the concentration of the masses possesses 
a real practical advantage. 

According to the last chapter, an entirely rigid aeroplane, 
none of whose parts could be moved, could only fly at a 
single angle, that at which the reactions of the air on its 
various parts are in equilibrium about the centre of gravity. 
In order to enable flight to be made at varying angles the 
aeroplane must possess some movable part a controlling 
surface. 

Leaving aside for the moment the device of shifting the 
centre of gravity (never hitherto employed), the easiest 
method would be to vary the angle formed by the main 
plane and the tail, i.e. the longitudinal dihedral. 

The method was first adopted by the brothers Wright, 
and is even at the present time employed in several 
machines. Very powerful in its effect, the variations in 
the angle of the tail plane affect the angle of incidence by 
more than then' own amount, and this hi greater measure 
the bigger the angle of incidence. 

Figs. 43 and 44 represent two different positions of the 
sheaf of total pressures on an aeroplane with a Bleriot XI. 
plane, and a non-lifting tail of an area one-tenth that of 
the mam plane and situated in rear of it at a distance equal 
to twice the chord. In fig. 43 the tail plane forms an 
angle of 8 with the chord of the main plane ; in fig. 44 
this angle is only 6. 

If the centre of gravity is situated at G x , the normal 
angle of incidence passes from 4 in the first case to 2 in 
the second. This variation in the angle of incidence is 



STABILITY IN STILL AIR 



119 




FIG. 43. Sheaf of total pressures on a Bleriot XI. monoplane with a 
longitudinal V of 8. 



120 



FLIGHT WITHOUT FORMULA 



PlO 
P8 

PS 



Po 




ill 



FIG. 44. Sheaf of total pressures on a Bleriot XI. monoplane with a 
longitudinal V of 6. 



STABILITY IN STILL AIR 121 

therefore integrally the same as that of the angle of the 
tail plane. 

If the centre of gravity is at G 2 , the normal angle of 
incidence would pass from 6 to 3|, and would therefore 
vary by 2| for a variation in the angle of the tail of 
only 2. 

Lastly, if the centre of gravity is at G 3 , the normal angle 
of incidence would pass from 8 to 5, a variation equal to 
one and a half times that of the angle of the tail. 

A comparison of figs. 43 and 44 further shows that the 
lines of total pressure are spaced further apart the greater 
the longitudinal dihedral. Now, other things being equal, 
the farther apart the lines of pressure the greater the longi- 
tudinal stability of an aeroplane. Hence the value of the 
longitudinal dihedral is most important from the point of 
view of stability. 

If the tail plane (non-lifting) is normally parallel to the 
relative wind, the longitudinal dihedral is equal to the 
normal angle of incidence. But if a lifting tail is employed, 
the longitudinal dihedral must necessarily be smaller than 
the angle of incidence (this is clearly shown in fig. 36). 
If the normal angle of incidence is small, as in the case of 
large biplanes and high-speed machines, the longitudinal 
dihedral is very small indeed and stability may reach a 
vanishing point. 

But if, in normal flight, the tail plane meets the wind 
with its upper surface (i.e. flies at a negative angle), the 
longitudinal dihedral, however small the normal angle of 
incidence, will always be sufficient to maintain an excellent 
degree of stability. This conclusion may be compared 
with that put forward in the previous chapter in regard 
to the advantage of causing the tail to fly at a negative 
angle. 

The foregoing shows that the reduction of the angle of 
incidence by means of a movable tail plane i.e. by alter- 
ing the longitudinal dihedral has the disadvantage that 
every alteration in the position of the tail plane brings 



122 FLIGHT WITHOUT FORMULAE 

about a variation in the condition of stability of the 
aeroplane. 

By plotting the sheaf of total pressures corresponding 
to very small values of the longitudinal dihedral, it would 
soon be seen that if the latter is too small, equilibrium 
may become unstable. 

A machine with a movable tail and normally possessing 
but little stability such, for instance, as a machine whose 
tail lifts too much may lose all stability if the angle of 
incidence is reduced for the purpose of returning to earth. 
This effect is particularly liable to ensue when, at the 
moment of starting a glide, the pilot reduces his incidence, 
as is the general custom. 

Losing longitudinal stability, the machine tends to 
pursue a flight-path which, instead of remaining straight, 
curls downwards towards the ground, and at the same time 
the speed no longer remains uniform and is accelerated. 

The glide becomes ever steeper. The machine dives, 
and frequently the efforts made by the pilot to right it by 
bringing the movable tail back into a stabilising position 
are ineffectual by reason of the fact that the tail becomes 
subject, at the constantly accelerating speed, to pressures 
which render the operation of the control more and more 
difficult. 

In the author's opinion, the use of a movable tail is 
dangerous, since the whole longitudinal equilibrium depends 
on the working of a movable control surface which may 
be brought into a fatal position by an error of judgment, 
or even by too ample a movement on the part of the pilot. 

For, apart from the case just dealt with, should the 
movable tail happen to take up that position in which 
the one angle of incidence making for stability is that 
corresponding to zero lift, i.e. when the main plane meets 
the wind along its " imaginary chord " (see Chapter I.), 
longitudinal equilibrium would disappear and the machine 
would dive headlong. 

In this respect, therefore, the movements of a movable 



STABILITY IN STILL AIR 123 

tail should be limited so that it could never be made to 
assume the dangerous attitude corresponding to the rupture 
or instability of the equilibrium. 

A better method is to have the tail plane fixed and rigid, 
and, hi order to obtain the variations in the angle of in- 
cidence required in practical flight, to make use of an 
auxiliary surface known as the elevator. 

Take a simple example, that of the aeroplane diagram- 
matically shown in fig. 45, possessing a non-lifting tail 




C D 

FIG. 45. 



plane CD, normally meeting the wind edge-on, to which is 
added a small auxiliary plane DE, constituting the elevator, 
capable of turning about the axis D. 

So long as this elevator remains, like the fixed tail, 
parallel to the flight-path, the equilibrium of the aeroplane 
will remain undisturbed. But if the elevator is made to 
assume the position DE (fig. 46), the relative wind strikes 
its upper surface and tends to depress it. Hence the 
incidence of the main plane will be increased until the 
couple of the pressure Q exerted about the centre of gravity, 
and the couple of the pressure q' exerted on the elevator, 
together become equal to the opposite moment of the 
pressure q on the fixed tail. 

Again, if the elevator is made to assume the position 



124 



FLIGHT WITHOUT FORMULAE 



DE 2 (fig. 47), the incidence decreases until a fresh condition 
of equilibrium is re-established. 

Each position of the elevator therefore corresponds to 





FIG. 46. 

one single angle of incidence ; hence the elevator can be 
used to alter the incidence according to the requirements 
of the moment. 




FIG. 47. 



It will be obvious that the effectiveness of an elevator 
depends on its dimensions relatively to those of the fixed 
tail, and, further, that if small enough it would be incapable, 
even in its most active position, to reduce the angle of 



STABILITY IN STILL AIR 125 

incidence to such an extent as to break the longitudinal 
equilibrium of the aeroplane. 

This, in the author's opinion, is the only manner in 
which the elevator should be employed, for the danger of 
increasing the elevator relatively to the fixed tail to the 
point even of suppressing the latter altogether has already 
been referred to above. 

In the position of longitudinal equilibrium corresponding 
to normal flight, the elevator, in a well-designed and well- 
tuned machine, should be neutral (see fig. 45). It follows 
that all the remarks already made with reference to the 
important effect on stability of the value of the longitudinal 
dihedral apply with equal force when the movable tail 
has been replaced by a fixed tail plane and an elevator. 

The extent of the longitudinal dihedral depends on the 
design of the machine, and more especially on the position 
of the centre of gravity relatively to the planes, and on 
its normal angle of incidence, which, again, is governed by 
various factors, and in chief by the motive power. 

The process of tuning-up, just referred to, consists prin- 
cipally in adjusting by means of experiment the position of 
the fixed tail so that normally the elevator remains neutral. 
Tuning-up is effected by the pilot ; in the end it amounts to 
a permanent alteration of the longitudinal dihedral ; where- 
fore attention must be drawn to the need for caution in 
effecting it. 

There are certain pilots who prefer to maintain the 
longitudinal dihedral rather greater than actually necessary 
(i.e. with the arms of the V close together), with the con- 
sequence that their machines normally fly with the elevator 
slightly placed in the position for coming down, or meeting 
the wind with its upper surface. In the case of machines 
with tails lifting rather too much, the practice is one to 
be recommended, for machines of this description are 
dangerous even when possessing a fixed tail, since if the 
elevator is moved into the position for descent the longi- 
tudinal dihedral is still diminished, though in a lesser 



126 FLIGHT WITHOUT FORMULA 

degree, and if it were already very small, stability would 
disappear and a dive ensue. 

Therefore the tuning-up process referred to has this 
advantage in the case of an aeroplane with a fixed tail 
exerting too much lift, that it reduces the amplitude of 
dangerous positions of the elevator and increases the 
amplitude of its righting positions. 

If the size of the elevator is reduced, with the object of 
preventing loss of longitudinal equilibrium or stability, 
to such a pitch as to cause fear that it would no longer 
suffice to increase the angle of incidence to the degree 
required for climbing, an elevator can be designed which 
would act much more strongly for increasing the angle 
than for reducing it, by making it concave upwards if 
situated in the tail, or concave downwards if placed in 
front of the machine. 

For it may be placed either behind or in front, and 
analogous diagrams to those given in figs. 46 and 47 would 
show that its effect is precisely the same in either case. 

But it should also be noted that if an elevator normally 
possessing no angle of incidence is moved so as to produce 
a certain variation in the angle of incidence of the main 
plane, of 2, for instance, the angle through which it must 
be moved will be smaller in the case of a front elevator 
than in that of a rear elevator, the difference between the 
two values of the elevator angle being double (i.e. 4 in the 
above case) that of the variation in the angle of incidence 
(assuming, of course, that front and rear elevators are of 
equal area and have the same lever arm). 

This is easily accounted for by the fact that a variation 
in the angle of incidence, which inclines the whole machine, 
is added to the angular displacement of a front elevator, 
whereas it must be deducted from that of the rear 
elevator. 

Thus, if we assume that the elevator must be placed at 
an angle of 10 to cause a variation in the incidence of 2, 
the elevator need only be moved through 8 if placed in 



STABILITY IN STILL AIR 127 

front, whereas it would have to be moved through 12 if 
placed in rear. 

A front elevator, therefore, is stronger in its action than 
a rear elevator. But it is also more violent, as it meets the 
wind first, which may tend to exaggerated manoeuvres. 
Finally, referring to the remarks in the previous chapters 
regarding the " tail-first " arrangement, the longitudinal 
stability of an aeroplane is diminished to a certain degree 
when the elevator is situated in front. These are no doubt 
the reasons that have led constructors to an ever-increasing 
extent to give up the front elevator.* 

All these facts plainly go to show, as already stated, that 
stability does not necessarily increase with speed. Aero- 
planes subject to a sudden precipitate diving tendency only 
succumb to it when their incidence decreases to a large 
extent and their speed exceeds a certain limit, sometimes 
known as the critical speed, at which longitudinal stability, 
far from increasing, actually disappears altogether. The 
term critical speed is not, however, likely to survive long, 
if only because it refers to a fault of existing machines 
which, let us hope, will disappear in the future. And it 
would disappear all the more rapidly if the variations in 
the angle of incidence required in practical flight could be 
brought about, not by a movable plane turning about a 
horizontal axis, but by shifting the position of the centre of 
gravity relatively to the planes, which could be done by 
displacing heavy masses (such as the engine and passengers' 
seats, for example) on board or, also, by shifting the planes 
themselves. 

In this case, as we have seen, the variations of the in- 
cidence would have no effect on the longitudinal dihedral, 
so that the sheaf of total pressures would not change, and 
then it would be true that stability increased with the 
speed. Then, also, there would be no critical speed. 

* The placing of the propeller in front and the production of tractor 
machines though, in the author's opinion, an unfortunate arrangement 
has also formed a contributory cause. 



128 FLIGHT WITHOUT FORMULA 

As stated previously, the horizontal flight of an aeroplane 
is a perpetual state of equilibrium maintained by con- 
stantly actuating the elevator. The idea of controlling 
this automatically is nearly as old as the aeroplane itself. 
But, as this question of automatic stability chiefly arises 
through the presence of aerial disturbances and gusts, its 
discussion will be reserved for the final chapter, which 
deals with the effects of wind on an aeroplane. 

Hitherto it has been assumed that the propeller-thrust 
passes through the centre of gravity, and therefore has 
no effect on longitudinal equilibrium. The angle of inci- 
dence corresponding to a given position of the elevator 
therefore remains the same in horizontal, climbing, or 
gliding flight. 

But if the propeller-thrust does not pass through the 
centre of gravity, it will exert at this point a couple which, 
according to its direction, would tend either to increase or 
diminish the incidence which the aeroplane would take up 
as a glider (assuming that the elevator had not been moved). 
In that case any variation in the propeller- thrust, more 
particularly if it ceased altogether either by engine failure 
or through the pilot switching off, would alter the angle of 
incidence. 

Thus if the thrust passed below the centre of gravity 
the stopping of the engine would cause the angle of in- 
cidence to diminish, and thus produce a tendency to dive. 
On the other hand, if the thrust is above the centre of 
gravity, the stopping of the engine would increase the 
angle of incidence, and therefore tend to make the machine 
stall. 

Practical experience with present-day aeroplanes teaches 
that in case of engine stoppage it is better to decrease the 
angle of incidence than to leave it unchanged, and, above 
all, than to increase it. 

The reason for this is that the transition from horizontal 
flight to gliding flight is not instantaneous as is often 
thought from purely theoretical considerations. An aero- 



STABILITY IN STILL AIR 129 

plane moving horizontally tends, through its inertia, to 
maintain this direction. Since there is now no longer any 
propeller-thrust to balance the head-resistance of the 
machine, it loses speed, which is to be avoided at all 
costs by reason of the ensuing dive. Therefore a pilot 
reduces his angle of incidence in order to diminish the 
drag of the aeroplane, and hence to maintain speed as far 
as possible. 

This action usually produces the desired effect, as the 
normal angle of incidence of most aeroplanes is greater 
than their optimum angle ; but this would not be the case if 
the optimum angle, or a still smaller angle, constituted the 
normal flying angle. 

The reduction of the angle of incidence at the moment 
the engine stops has the additional effect of producing the 
flattest gliding angle, which, as has already been shown, 
corresponds to the use of the optimum angle. On the 
other hand, stability increases through the reduction of 
the incidence (which is here equivalent to an increase 
in speed) so long as this does not reduce the longitudinal 
dihedral. 

Bearing these various considerations in mind, it would 
seem preferable, in contradiction to a very general view 
which at one time the author shared, to make the propeller- 
thrust pass below rather than above the centre of gravity, 
at any rate in the case of machines normally flying at a 
fairly large angle of incidence. 

As a general rule the propeller-thrust passes approxi- 
mately through the centre of gravity, and this, perhaps, is 
the best solution of all. 

Since the direction of the propeller-thrust is under con- 
sideration, it may be as well to note that this direction need 
not necessarily be that of the flight-path of the aeroplane. 
Take the case where the thrust passes through the centre 
of gravity ; it will be readily understood that if the direction 
of the thrust is altered this cannot have any effect on 
longitudinal equilibrium. Hence there is no theoretical 

9 



130 FLIGHT WITHOUT FORMULAE 

reason why an aeroplane with an inclined propeller shaft 
should not fly horizontally. 

The only effect on the flight of an aeroplane by tilting 
the propeller shaft up at an angle would be to reduce the 
speed, because the thrust doing its share in lifting, the planes 
need only exert a correspondingly smaller amount of lift. 
Therefore the lifting of the propeller shaft virtually amounts 
to diminishing the weight of the aeroplane, thereby, other 
things being equal, reducing the speed. 

If the thrust became vertical, the planes could be dispensed 
with, horizontal speed would disappear, and the aeroplane 
would become a helicopter. 

It can easily be shown that the most advantageous 
direction to give to the propeller-thrust is that wherein the 
shaft is slightly inclined upwards, as is done hi the case of 
certain machines, though in others the thrust is normally 
horizontal. 

To wind up these remarks on longitudinal stability, we 
will describe various types of little paper gliders which 
will afford in practical fashion some interesting information 
concerning certain aspects of longitudinal equilibrium and 
of gliding flight. The results, of course, are only approxi- 
mate in the widest sense, since such paper gliders are very 
erratic as they do not preserve their shape for any length 
of time. 

Experiments with these little paper models are most 
instructive and are to be highly recommended to every 
reader ; however childish they may at first appear, they 
will not be waste of time. By experimenting oneself with 
such miniature flying-machines one can learn many valuable 
lessons in regard to points of detail, only a few of which 
can here be set out. To make these little models it is best 
to use the hardest obtainable paper, though it must not be 
heavy ; Bristol-board will serve the purpose. Even better 
is thin sheet aluminium about one- tenth of a millimetre in 
thickness, but in this case the dimensions given hereafter 
should be slightly increased. 



STABILITY IN STILL AIR 131 

TYPE I. 

An ordinary rectangular piece of paper, in length about 
twice the breadth (12 cm. by 6 cm., for instance), folded 
longitudinally down the centre (see fig. 48) so as to form a 
very open angle (the function of this, which affects lateral 
stability, will be explained in the next chapter). 

Reference to fig. 32, Chapter VI., will show that for a 
single flat plane to assume one of the ordinary angles of 
incidence (roughly, from 2 to 10), its centre of gravity 
must be situated at a distance of from one-third to one- 
quarter of the fore-and-aft dimension of the plane from the 
forward edge. This is easily obtained by attaching to 




FIG. 48. Perspective. 

the paper a few paper clips or fasteners, fixed near one of 
the ends of the central fold at a slight distance from the 
edge (about | cm.). 

If the ballasted paper is held horizontally by its rear end 
and is thrown gently forward, it will behave in one of the 
three following ways : 

(a) The paper inclines itself gently and glides down 
regularly without longitudinal oscillations. 

This is the most favourable case, for at the first attempt 
the ballast has been placed in the position where the corre- 
sponding single angle of incidence was one of the usual 
angles. Practice therefore confirms theory, which taught 
that a single flat plane is longitudinally stable. 

(b) The paper dips forward and dives. 

The centre of gravity is too far forward and in front 
of the forward limit of the centre of pressure. To obtain 



132 FLIGHT WITHOUT FORMULA 

a regular glide the ballast must be moved slightly toward 
the rear. In effecting this, it will probably be moved too 
far back and the paper will in that case behave in the 
opposite manner, which is about to be described. 

(c) The paper at first inclines itself, but, after a dive 
whose proportions vary with several factors, and chiefly 
with the force with which the model has been thrown, it 
rears up, slows down, and starts another dive bigger than 
the first, and thus continues its descent to the ground, stall- 
ing and diving in succession (see fig. 49). 




FIG. 49. 



As a matter of fact, the dive following the first stalling 
may be final and become vertical if during the accompany- 
ing oscillation the paper should meet the air edge-on, so 
that actually it has no angle of incidence, for such a glider 
if dropped vertically, leading edge down, has no occasion 
to right itself and continues to fall like any solid body. 

The above experiment is quite instructive. It corresponds 
to the case where the single angle of incidence at which 
flight is possible, owing to the centre of gravity being too 
far back, is greater than the usual angles of incidence. 

As it begins its descent the sheet of paper, having been 
thrown forward horizontally, has a small angle of incidence, 
and hence tends to acquire the fairly high speed correspond- 
ing to this small angle. But the pressure of the air, passing 



STABILITY IN STILL AIR 133 

in front of the centre of gravity, produces a stalling couple 
which increases the incidence. Owing to its inertia, the 
paper will tend to maintain its speed, which has now be- 
come higher than that corresponding to its large angle 
of incidence, and so the pressure of the air becomes greater 
than the weight, on account of which the flight-path becomes 
horizontal again and even rises. 

The same thing, in fact, always happens if for some 
reason or other a glider or an aeroplane should attain to 
a higher speed than that corresponding to the incidence 
given it by the elevator, and also if the angle of the planes 
is suddenly increased. This rising flight-path by an in- 
crease in the angle of incidence is constantly followed by 
birds, and especially by birds of prey such as the falcon, 
which uses it to seize its prey from underneath. 

Pilots also use it in flattening out after a steep dive or 
vol pique, though the manoeuvre is distinctly dangerous, 
since it may produce in the machine reactions of inertia 
which may cause the failure of certain parts of the 
structure. 

Returning to the ballasted sheet of paper : as the flight- 
path rises, the glider loses speed ; in fact, it may stop 
altogether. It is then in the same condition as if it were 
released without being thrown forward, and falls in a 
steep dive which, as already stated, may prove final. 

There are many variants of the three phenomena described. 

Thus, the stalling movement may become accentuated to 
such an extent as to cause the sheet of paper to turn right 
over and " loop the loop." * Again, the paper may start to 
glide down backwards and do a " tail-slide." 

These variants depend mainly on how far back the 
centre of gravity is situated, that is, on the value of the 
single angle of incidence at which the sheet can fly. If 

* It is interesting to note that this and many of the following 
mano3uvres are precisely those practised by Pegoud and his imitators, 
although the above was written long before they were attempted in 
practice. TKANSLATQR. 



134 FLIGHT WITHOUT FORMULA 

this angle is only slightly greater than the usual angles of 
incidence, the stability of the glider which is less, of 
course, at large angles than at small ones will still be 
sufficient to prevent the effect of inertia of oscillation from 
bringing it into a position where it is liable to dive, to turn 
over on its back, or slide backwards. It will therefore 
follow a sinuous flight-path consisting of successive stalling 
and diving, but will not actually upset. 

But if the centre of gravity is brought further back and 
the angle of incidence corresponding to this position is 
much greater than the usual angles of incidence, the stabilis- 
ing couples no longer suffice to overcome the effects of 
inertia to turning forces, the condition of stability in 
oscillation is no longer fulfilled, and the glider behaves in 
one of the ways already described. 

It should, however, be pointed out that a rectangular 
sheet of paper has a far larger moment of inertia in respect 
to pitching than a glider generally conforming, as our next 
models will do, to the shape of an aeroplane. 

To prevent these occurrences from taking place, all that 
is required is to bring the ballast further forward and to 
adjust the incidence by cutting off thin strips from the 
forward edge. By these means it is possible eventually 
to obtain a regular gliding path without longitudinal 
oscillations. 

If thin strips of paper are thus cut off with sufficient 
care,* the various properties of gliding flight set forth in 
Chapter II. can be very easily followed. 

It will be seen that by gradually reducing the angle 
of incidence by cutting back the forward edge, the glide 
becomes both longer and faster. Next, when the angle 
has become smaller than the optimum angle of this embryo 
glider, the length of the glide diminishes, the path becomes 
steeper, and the glider tends to dive. 

Towards the end the process of adjustment becomes 

* In case the ballast should be in the way, the paper can be cut away 
diagonally and equally on either side, as shown in fig. 50. 



STABILITY IN STILL AIR 135 

exceptionally delicate, for since the optimum angle of a 
model of this nature is very small indeed, by reason of the 
fact that its detrimental surface is almost zero relatively 
to its lifting area, the slightest shifting of the centre of 
gravity is enough to cause a large variation in the gliding 
angle and to upset longitudinal stability. 

Now let us suppose that, the ballast being so placed that 
the glider tends to dive, we proceed to rectify by cutting 
away pieces of the trailing edge as in fig. 50. If the outer 
rear tips thus symmetrically formed are bent upwards, 




FIG. 50. 

the glider will no longer tend to dive and will assume a 
position of equilibrium. 

By bending these outer tips through various degrees, and 
also, if necessary, bending up the inner portion of the 
trailing edge, all the various forms of gliding flight can be 
reproduced which were previously obtained by shifting the 
ballast and cutting back the forward edge.* 

But to whatever degree the tips may be bent up, hence- 
forward the stalling movement will not be followed by a 
dive, nor will the glider loop the loop or do a tail-slide. 

This is due to the fact that instead of being constituted 
by a'single flat plane, the glider now possesses a tail, which 
gives it much better longitudinal stability. The effects of 

* The rear tips may not be bent exactly equally on either side, with 
the result that the glider may tend to swerve to left or right. To counter- 
act this, the tip on the side towards which the paper swerves should be 
bent up a little more. 



136 FLIGHT WITHOUT FORMULA 

inertia are now overcome by the stabilising moments arising 
from the tail. Moreover, a glider of this description when 
dropped vertically rights itself. It can no longer dive 
headlong. 

If the tips are bent back to their original horizontal 
position, it is evident that the sheet of paper will dive once 
more, and to an even greater extent if the tips were bent 
down instead of up. This plainly shows the danger of 
allowing the elevator to constitute the solitary tail plane, 
for, unless its movement is limited, it could cause equili- 
brium to be lost. 

TYPE II. 

1. Fold a sheet of paper in two, and from the folded 
paper cut out the shape shown in fig. 51. 

2. Fold back the wings and the tail plane along the 
dotted lines. The wings should make a slight lateral V or 
dihedral. 

3. Ballast the model somewhere about the point L the 
exact spot must be found by experiment with one or 
more paper fasteners. 

This model approaches more nearly to the usual shape of 
an aeroplane. By finding the correct position for the 
ballast, so that the centre of gravity is situated on the 
total pressure line corresponding approximately to the 
optimum angle, this little glider can be made to perform 
some very pretty glides.* 

The ballast may be brought further forward or additional 
paper fasteners may be affixed without making the model 
dive headlong. 

It will dive, and on this account may be brought to fall 
headlong if the height above the ground is only slight ; but 
if there is room enough it will recover and, though coming 
down steeply, will not fall headlong. It is still gliding, 

* Should it tend to swerve to either side, bend up slightly the rear 
tip of the wing on the opposite side of that towards which the aeroplane 
tends to turn. 



STABILITY IN STILL AIR 137 

since during its descent the air still exerts a certain amount 
of lift. Longitudinal equilibrium is not upset, and if the 
glider does not lose its proper shape on account of its high 
speed, it cannot fall headlong, whatever the excess of load 
carried, by reason of the fact that the main and tail planes 
are placed at an angle to one another. 

The reduction in the angle of incidence by bringing the 
centre of gravity further forward therefore maintains 
stability, and even increases it as the speed grows. And 
this because the longitudinal dihedral has not been touched. 

By shifting the ballast toward the rear, the model will 





FIG. 51. 

also follow a steep downward path, but this time the angle 
of incidence is large, the speed slow, and therefore the glider 
remains almost horizontal and " pancakes." This shows 
conclusively that the same gliding path can be followed at 
two different normal angles of incidence and at two different 
speeds. 

By still shifting the ballast farther back, the model may 
be made to glide as if it belonged to the tail-foremost or 
" Canard " type (cf. the third model described hereafter). 
Flight at large angles of incidence is now possible and 
will not cause the model to overturn as in the case of the 
single sheet of paper, as the moment of pitching inertia 
is much feebler than in the former case. The stability of 



138 FLIGHT WITHOUT FORMULA 

oscillation is therefore still adequate at large angles of 
incidence. 

Now let us shift the ballast back again so that the glide 
becomes normal once more ; at the rear of the tail plane, 
bend down either the whole or half the trailing edge to 
the extent of 2 mm. This will give us an elevator, while 
the fixed tail is retained. 

By moving this elevator the conditions of gliding flight 
can obviously be modified ; for instance, if the outer halves 
of the rear edge are evenly bent down to an angle of some 
45 that is, to have their greatest effect in reducing the 
angle of incidence the glider will extend the length of 
its flight and travel faster (see fig. 52). 




FIG. 52. Perspective. 

But it will still be impossible by the operation of the 
elevator to make the model fall headlong. The fixed tail 
will prevent this, and will overcome the action of the 
elevator because the latter is small in extent. Hence, an 
elevator small enough relatively to the tail plane cannot make 
an aeroplane dive headlong. 

If the whole of the trailing edge is bent down it might 
possibly cause longitudinal equilibrium to be upset and 
make the glider dive. And should this not prove to be 
the case, it could be done without fail by increasing the 
depth of the elevator. 

The experiment shows that the size of the elevator 
should not be too large ; it should merely be sufficient to 
cause the alterations of the angle of incidence required for 
ordinary flight and should never be able to upset stability. 



STABILITY IN STILL AIR 139 

TYPE III. 

1. Cut out from a sheet of paper folded in two a piece 
shaped as in fig. 53. 

2. Fold back the wings along the dotted lines. 

3. Fold the wing- tips upwards along the outer dotted 
line. 

This tail-first glider will be stable without ballast and 




FIG. 53. 

glides very prettily on account of its lightness. It will be 
referred to again in connection with directional stability.* 

TYPE IV. 

1. Cut out from a sheet of paper folded in two the shape 
shown in fig. 54. 

2. Cut away from the outer edge of the fold two portions 
about 1 mm. deep, and of the length shown at AB and CD. 

3. Inside the fold fix with glue 

(a) At AB a strip of cardboard or cut from a visiting 
card ; 5 cm. long, 1 cm. broad. The inner end 
of the strip is shown by the dotted line at AB. 

(6) At CD glue a similar strip as shown. 

* If it tends to swerve, slightly bend the whole of the front tail in 
the opposite sense. 



140 



FLIGHT WITHOUT FORMULAE 



4. Fold back the wings and the tail plane along the 
dotted lines. 





ci L 


n fi 

_i 4-^ ^\ 


y 


A^ J B c L 1 

^7 a 



FIG. 54. 




FIG. 55. Perspective. 

5. Ballast the model with a paper clip placed at the end of 
the strip AB, and with another in the neighbourhood of L. 



STABILITY IN STILL AIR 141 

The exact positions are to be found by experiment, and it 
may therefore be as well to turn the cardboard about its 
glued end before the glue has set. 

If this glider is thrown upwards towards the sky, it will 
right itself and glide away in the attitude shown in fig. 55. 
Now the centre of gravity of a glider of this kind lies some- 
where about G. 

On the other hand, the point sometimes termed the 
" centre of lift " is situated on the plane at the spot which, 
in equilibrium, is on the perpendicular from the centre of 
gravity and shown at S. This point S lies below the centre 
of gravity. 

Now, if an aeroplane ought to be considered as suspended 
in space from a so-called " centre of lift," its centre of 
gravity could not, perforce, be anywhere but below this 
" centre of lift." 

In the case just mentioned the opposite took place, which 
shows very clearly that this idea of a " centre of lift " is 
erroneous. 

An aeroplane has one centre only, its centre of gravity. 



CHAPTER VIII 
STABILITY IN STILL AIR 

LATERAL STABILITY 

FOR the complete solution of the problem of aviation the 
aeroplane must possess, in addition to stable longitudinal 
equilibrium, stable lateral equilibrium or, more briefly, 
lateral stability. 

The fundamental principle laid down hi Chapter VI. is 
equally applicable to lateral equilibrium.* 

But hi the case of longitudinal equilibrium the move- 
ments that had to be considered hi respect of stability 
could be simply reduced to turning movements about a 
single axis, the pitching axis. The matter becomes exceed- 
ingly complicated hi the case of lateral equilibrium, for 
the turning movements can take place about an infinite 
number of axes passing through the centre of gravity and 
situated in the symmetrical plane of the machine. 

For instance, assume that the aeroplane diagrammatically 
shown hi fig. 56 were moving horizontally and that the 
path of the centre of gravity G were along GX. If the 
machine were to turn through a certain angle about the 
path GX, clearly no change would take place in the manner 
in which the air struck any part of the machine,f and 
no turning moment would arise tending to bring the machine 

* From the point of view of equilibrium and stability, the aeroplane 
may be regarded as if it were suspended from its centre of gravity, and 
were thus struck by the relative wind created by its own speed. 

t Assuming, of course, that the turning movement does not alter the 
path of the centre of gravity. 



STABILITY IN STILL AIR 143 

back to its former position or to cause it to depart there- 
from still further. 

It can therefore be stated that the lateral equilibrium 
of an aeroplane is neutral about an axis coincident with 
the path of the centre of gravity. 

But when we come to consider turning movements about 
other axes such as GXj or GX 2 which do not coincide with 
the path of the centre of gravity, it is evident that such 
movements will have the effect of causing the aeroplane to 
meet the air dissymmetrically, and consequently to set up 
lateral moments tending to increase or diminish the tilt of 
the machine that is, upsetting or righting couples. 

Before going further it is readily evident that, the axis 



X 



FIG. 56. 



GX being neutral, axes such as GXj and GX 2 , lying on 
opposite sides of GX, will have a different effect, and that 
a turning movement begun about one series of axes will 
encounter a resistance due to the dissymmetrical reaction of 
the air which it creates, while any turning movement begun 
about the other series of axes, again owing to the dis- 
symmetrical reaction of the air, will go on increasing until 
the machine overturns. 

The former series will be known as the stable axes, the 
latter as the unstable axes. The neutral axis is that co- 
inciding with the path of the centre of gravity. Further, 
the term raised axis will be used to denote an axis with 
its forward extremity raised like GXj and lowered axis 
for that which, like GX 2 , has its forward extremity 
lowered. 



144 FLIGHT WITHOUT FORMULA 

The shape of the aeroplane determines which axis is 
unstable. 

In many aeroplanes, and in monoplanes in particular,* 
the forward edges of the wings do not form an exact straight 
line, but a dihedral angle or V opening upwards. 

We shall also have to examine though the arrangement 
hi question has never to the author's knowledge been 
adopted in practice the case of the machine with wings 
forming an inverted dihedral or A-t Lastly, the forward 
edges of the two wings may form a straight line, and such 
wings will hereafter be described as straight wings. 

In an aeroplane with straight wings, a turning movement 
imparted about an axis situated in the symmetrical plane 
of the machine increases the angle of incidence if the axis 





FIG. 57. 

is a lowered axis, and diminishes the angle if the axis is a 
raised axis. This can easily be proved geometrically, and 
can be shown very simply by the following experiment. 

Make a diagonal cut in a cork, as shown in fig. 57 (front 
and side views). In this cut insert the middle of one of 
the longer sides of a visiting-card, and thrust a knitting- 
needle or the blade of a knife into the centre of the cork 
on the side where the card projects. Now place this con- 

* In the case of large-span biplanes the flexing on the planes in flight 
forces them into a curve which in its effects is equivalent, for purposes 
of lateral stability, to a lateral dihedral. 

t The " Tubavion " shown at the 1912 Salon is stated to have flown 
with wings thus disposed. 

TRANSLATOB'S NOTE. The same device was adopted by Cody in his 
earlier machines, and in the " June Bug," the first machine designed by 
Glenn Curtiss. 



STABILITY IN STILL AIR 



145 



trivance in the position shown in fig. 58, with the needle 
horizontal and at eye-level. If the needle is rotated slowly, 
the card will always appear to have the same breadth 
whatever its position. 

If this visiting-card is taken to represent the straight 




FIG. 58. 

wings of an aeroplane struck by the wind represented by 
the line of sight, this shows that a turning movement about 
the neutral axis of an aeroplane with straight wings 
produces no change in the angle of incidence, as already 
known. 




FIG. 59. 

But if the needle is inserted in the position shown in 
fig. 59, it will be found that by rotating the needle without 
altering its position, the breadth of the card will appear to 
increase, thus showing, retaining the same illustration, that 
when the axis of rotation of a machine with straight wings 
is a lowered axis, the incidence increases as the result of 
the turning movement. 

10 



146 



FLIGHT WITHOUT FORMULA 



This effect is the more pronounced the smaller the angle 
of incidence. 

But if the needle is inserted as shown in fig. 60, the 
breadth of the card when the needle is rotated will appear 
to diminish. 

If the needle is parallel to the card, a turn of the needle 
through 90 brings the card edge-on to the line of sight. 

Lastly, if the needle and the card are in converging 
positions, a slight turn of the needle brings the card edge-on, 
and beyond that its upper surface alone is in view. 

From this we may conclude that if the axis of rotation 
of an aeroplane with straight wings is a raised axis, the 
angle of incidence diminishes as the result of a turning 




FIG. 



movement, and if the axis is raised to a sufficient degree, 
the angle of incidence may become zero and even negative. 

This effect is the more pronounced the larger the angle of 
incidence. 

It should be noted that in neither case is the action 
dissymmetrical and that both wings are always equally 
affected. In other words, should a machine with straight 
wings turn about an axis lying within its plane of symmetry, 
no righting or upsetting couple is produced by the turning 
movement. 

On the other hand, if the eye looks down vertically upon 
the cork from above, it will be seen that a turning move- 
ment about a lowered axis has the effect of causing the 
rising wing to advance, while in the case of a raised axis 
a turning movement causes it to recede (fig. 61). Now, by 



STABILITY IN STILL AIR 



147 



advancing a Aving, the centre of pressure is slightly shifted ; 
this may produce a couple tending to raise the advancing 
wing. 

Should the advancing wing be the lower one, which 
corresponds to the case of a raised axis, this couple is a 
righting couple. In the reverse case it is an upsetting 
couple. 

In this respect, for aeroplanes with straight wings a 
raised axis is stable, a lowered axis unstable. 

This effect in itself is very slight, but it represents the 
nature of the lateral equilibrium of an aeroplane with 




Raised Axis. 



FIG. 61. 



straight wings ; for if it were not present, a machine with 
straight wings would be hi neutral equilibrium and possess 
no stability. 

But as soon as the wings form a lateral dihedral, whether 
upwards or downwards, this effect practically disappears 
and becomes negligible. This is the case next to be 
examined. 

Let us suppose, to begin with, that the wings form an 
upward lateral dihedral, or open V- Each of the wings 
may be considered in the light of one-half of a set of straight 
wings which has begun to turn about the axis represented 
by the apex of the V> the movement of each whig being 



148 



FLIGHT WITHOUT FORMULAE 




in the opposite direction, i.e. while one is falling the other 

is rising. 

The considerations set forth above show that a turning 

movement about a raised axis causes the incidence of the 

rising wing to diminish 
while that of the fall- 
ing wing increases ; the 
contrary takes place in 
the case of a lowered 
axis. 

This is easily demon- 
strated by tilting up- 

FI<J. 62. Stable. Lateral V and raised axis. Ward the two halves of 

the visiting-card used in 

the previous experiment. If the contrivance is looked at 
as before, so that the axis of the cork is horizontal and 
on a level with the eye, it will be found that any rotation 
about the needle, when this is directed upwards, causes the 
rising wing to appear to 
diminish in surface while 
the falling wing in- 
creases (see fig. 62). 
But if the needle points 
downwards, the opposite 
takes place (fig. 63). 

In the first case, there- 
fore, the turning move- 
ment produces a righting 
couple, in the second case 
an upsetting couple. 

This effect is the more 
pronounced the larger 
the angle of incidence. 

Therefore in the case of wings forming a lateral dihedral, 
a raised axis is stable, a lowered axis unstable, and the 
more so the greater the angle of incidence. 

This effect is added to the secondary effect already referred 




FIG. 63. Unstable. Lateral V with lowered 
axis. 



STABILITY IN STILL AIR 



149 




to in the case of straight wings ; but as soon as the dihedral 
is appreciable, the former effect becomes by far the stronger. 
Now consider the case of wings forming an inverted 
dihedral or A- The same line of reasoning shows (see figs. 
64 and 65) that : 

In the case of wings forming an inverted lateral dihedral 
a raised axis is un- 
stable, a lowered axis 
stable, and this the more 
so the smaller the angle 
of incidence. 

In this case the 
secondary effect acts 

in opposition, but it Fic " ^-^fsed wif 6 1 Awith 
becomes negligible as 
soon as the inverted dihedral is appreciable. 

These various effects are increasingly great, it will 
be readily understood, as the span is increased in size, 
for the upsetting or righting couples have lever arms 
directly proportional to the span. Besides, but quite 
apart from the value of the incidence in a given case, 

it is clear that the 
righting couples are 
greater the higher the 
speed of flight, since 
they are proportional 
to the square of the 
speed. Broadly speak- 
ing, therefore, though 
with certain reserva- 
tions into which we need not here enter in detail, it may 
be stated that the higher the flying speed the greater is 
lateral stability. 

Although the stability or instability of any axis depends 
chiefly on the main planes, other parts of the aeroplane can 
affect it to a certain extent, hence their effect should be 
taken into account as well. 




150 FLIGHT WITHOUT FORMULAE 

The tail plane, which is usually straight, only affects 
lateral stability to an inappreciable extent. 

But it should be noted, as already stated, that any turning 
movement about an axis other than the neutral axis will 
affect the incidence at which the tail plane meets the air ; 
and, since such a turning movement also affects, as already 
known, the incidence of the main plane, this dual effect 
must needs disturb the longitudinal equilibrium of the 
machine. Hence, we arrive at the general proposition that 
rolling begets pitching. 

As regards the remaining parts of the aeroplane fuselage, 
chassis, vertical surfaces, etc. they experience from the 
relative wind, when the aeroplane turns about an axis 
in the median plane, certain reactions which may be dis- 
symmetric and would thus affect the equilibrium of the 
machine on its flight-path. More particularly when the 
parts in question are excentric relatively to the turning 
axis can they influence though usually only to a small 
extent lateral equilibrium. 

For the sake of convenience and in a manner similar to 
that previously adopted in the case of the detrimental 
surface, the effects of all these parts may be concentrated 
and assumed to be replaced by the effect of a single fictitious 
vertical surface, which may be termed the keel surface, 
which would, as it were, be incorporated in the symmetrical 
plane of the machine. 

Certain parts of the aeroplane, such as the vertical rudder, 
the sides of a covered-in fuselage, vertical fins, form actual 
parts of the keel surface. 

Evidently, according to whether the pressure exerted on 
the keel surface, by reason of a turning movement about 
a given axis, passes to one side or the other of this axis, the 
couple set up will be either a righting or upsetting couple. 

It is easily shown that a keel surface which is raised 
relatively to the axis of rotation can be compared, pro- 
portions remaining the same, to a plane with an upward 
dihedral, or V, and that a keel surface which is low relatively 



STABILITY IN STILL AIR 



151 



to the axis of rotation to a plane with a downward dihedral 
or A- 

For this purpose, the cork, visiting-card, and needle 
previously employed may be discarded in favour of a 
visiting-card fixed flag- wise to a knitting-needle. It is 
clear, as shown in fig. 66, that when the axis of rotation 
is raised, a high keel surface renders this axis stable and a 
low keel surface renders it unstable, while the reverse is the 
case if the axis of rotation is lowered (see fig. 67). 




Unstable. Raised axis and low kt 
FIG. 66. 

But this effect, as previously explained, is of small 
importance as compared with that due to the shape of the 
main plane ; for, while the pressures on the keel surface are 
never far removed from the axis of rotation, the differential 
variations in the pressure exerted on the two wings of a 
plane folded into a dihedral have, relatively to the axis, a 
lever arm equal to half the span of the wing, and accord- 
ingly these variations are considerable. 

The effect of the dihedral of the main plane is therefore 
not equivalent in magnitude to that of the keel surface 
formed by the projected dihedral (fig. 68). The dihedral 



152 



FLIGHT WITHOUT FORMULA 



has a much greater effect on lateral stability than a similar 
keel surface. 

We now know the position of the stable and the un- 
stable axes of rotation according to the particular struc- 




Unstable. Lowered axis and high keel. 
\ 




Stable. Lowered axis and low keel. 
FIG. 67. 

ture of the aeroplane, and we have found that the same 
machine can be stable laterally for one axis of rotation, 
and unstable for another. 

This is scarcely reassuring and inevitably leads to the 




FIG. 68. 

question : About which axis can an aeroplane, flying freely 
in space, be brought to turn ? 

In the first place, the position of the axes obviously 
depends on the causes which can bring about the turning 
movement. But these causes are known : so far as lateral 



STABILITY IN STILL AIR 



153 



equilibrium is concerned, they can only consist in excess of 
pressure on one wing or on the keel surface. 

Here, then, we have one important element of the ques- 
tion already settled. Nevertheless, the problem cannot 
be solved in its entirety without having recourse to ordinary 
mechanics and calculations, though the results thus obtained 
may well be called into question, since the calculations have 
to be based on hypotheses which are not always certain in 
the present state of aerodynamical knowledge. 

Without attempting to examine this difficult problem in 
all its details, we may never- 
theless remark that in its 
solution the most important 
part is played by the dis- 
tribution of the masses 
constituting the aeroplane 
or, in other words, by its 
structure considered from 
the point of view of inertia. 

Let us take a long iron 
rod AB (fig. 69), ballasted 
with a mass M, and sus- 
pend it from its centre of 
gravity G ; add a small 
pair of very light wings in the neighbourhood of the centre 
of gravity. 

If, with a pair of bellows, pressure is created beneath one 
of the wings, the device will start to oscillate laterally, and 
these oscillations will obviously take place about the axis 
of the iron bar. If this is placed in the position shown in 
fig. 69, the axis of rotation will be a raised axis ; if in the 
position illustrated in fig. 70, it will be a lowered axis. 

Now every aeroplane, and every long body in fact, has a 
certain axis passing through the centre of gravity, about 
which axis we can assume the masses to be distributed, as 
in the case of the present device they are about the axis of 
the iron bar. 




FIG. 69. 



154 



FLIGHT WITHOUT FORMULA 




Lateral oscillations tend to take place about this axis, 
which may be termed the rolling axis. The term, it is 
true, is not absolutely accurate, and lateral oscillations do 

not take place mathe- 
matically about this 
axis ; but at the same 
time, as further investi- 
gations would show, the 
true rolling axes only 
differ from it to a very 
slight extent, and are 
always slightly more 

.-,, raised than the rolling 

M ^rj{ FlG ' 70 ' axis. This brings us to the 

moment of rolling inertia. 

In Chapter VII. was defined the moment of inertia of 
a body about any axis ; in the examination of longitudinal 
stability the moment of inertia of an aeroplane about its 
pitching axis was considered as the moment of pitching 
inertia. But in the present case we have only to deal 
with the moment of inertia of an aeroplane about its rolling 
axis that is, its moment of rolling inertia. 

As a matter of fact, the true axis of lateral oscillations 
coincides more closely with the rolling axis as, on the one 
hand, the incidence of the main plane is nearer to the lest 
incidence (see Chapter III.) and the corresponding drag-to- 
lift ratio is smaller, and, on the other hand, as the ratio 
between the moment of rolling inertia and the moment 
of pitching inertia is smaller. 

Owing to the fact that this latter ratio is very small in 
the diagrammatic case just considered, the lateral oscilla- 
tions of this device take place almost exactly about the 
rolling axis, i.e. about the axis of the iron bar.* 

* The moment of rolling inertia is very slight, since those parts which 
are at any distance from the rolling axis, i.e. the wings, are very light, 
while the moment of pitching inertia is great, owing to the length and 
the weight of the iron bar. 



STABILITY IN STILL AIR 155 

From all this it is clear that, according to the position of 
the aeroplane in flight, its natural axis of lateral oscillation, 
or approximately its rolling axis, will be either a raised or 
a lowered axis. For an aeroplane to possess lateral stability, 
its natural axes of oscillation must obviously be stable axes. 

Thus, if the wings of an aeroplane form an upward 
dihedral or y, or if the machine has a high keel surface, its 
natural axes of oscillation must be raised axes, if lateral 
stability is to be ensured. This condition is complied with 
if the rolling axis of the aeroplane is itself a raised axis, 
and even when the rolling axis is slightly lowered, since 
the natural axes of oscillation are relatively slightly raised. 

It is also clear that the stability will be better the greater 
the angle of incidence. 

On the other hand, if the main planes form a downward 
dihedral or A> the machine will be unstable laterally if the 
rolling axis is a raised one or even if it is only slightly 
lowered. But the aeroplane can be made stable if its 
rolling axis is lowered to a sufficient extent, and the more 
so the smaller the angle of incidence. 

This conclusion is distinctly interesting since it is directly 
at variance with the views held by the late Captain Ferber, 
whose great scientific attainments lent him all the force of 
authority, to the effect that an upward dihedral was essential 
to lateral stability. 

But it is even more important by reason of the fact 
which will be duly discussed in the final chapter, already 
noted by Ferber himself, that whereas the upward dihedral 
or V is disadvantageous in disturbed air, the downward 
dihedral has distinct advantages in this respect. 

On the whole, however, Ferber's view is correct at present, 
since in the majority of aeroplanes of to-day the rolling 
axis is practically identical with the trajectory of the 
centre of gravity or only very slightly lowered. But in 
an aeroplane with a rolling axis lowered to an appreciable 
extent, the upward dihedral might be highly injurious from 
the point of view of lateral stability, whereas the inverted 



156 FLIGHT WITHOUT FORMULA 

dihedral or A would, contrary to general opinion, be 
eminently stable. 

How is this arrangement to be carried out in practice ? 

The rolling axis is a line which passes through the centre 
of gravity and lies close to the masses situated at the end 
of the fuselage, such as the tail plane and controlling 
surfaces. When the centre of gravity is normal, this line 
consequently lies along the axis of the fuselage. But if 
the centre of gravity is situated low relatively to the wings, 
the rolling axis is also lowered. The same would occur if 
the machine was so arranged as to fly with its tail high, so 
that the axis of its fuselage would form an angle, distinctly 
greater than the normal incidence, with the chord of the 
main plane. 

On the other hand, a low centre of gravity, if unduly 
exaggerated, presents certain disadvantages. 

The best method of obtaining a rolling axis such that 
the inverted dihedral of the main plane produces lateral 
stability would seem to be by combining both devices, i.e. 
by slightly lowering the centre of gravity and raising the 
tail in flight. 

This conclusion was formed by the author several years 
ago ; and in 1909, somewhat fearful of running counter to 
the authoritative views of Captain Ferber, the opinion was 
sought of the eminent engineer, M. Rodolphe Soreau, another 
recognised authority, in regard to the position of the axis 
about which an aeroplane's natural lateral oscillations take 
place. In 1910 in a previous work,* the author first enun- 
ciated in definite form the proposition that an aeroplane 
with its main planes arranged to form an inverted dihedral 
could, under given conditions, remain stable laterally. 
Since then the point has been dealt with in an article in 
La Technique Aeronautique and in a paper read before the 
Academic des Sciences.f 

* The Mechanics of the Aeroplane (Longmans, Green & Co.). 
f La Technique Aeronautique, December 15, 1910 ; Comptes Rendus, 
May 15, 1911. 



STABILITY IN STILL AIR 157 

Summing up : 

( 1 ) In aeroplanes of the shape hitherto generally employed 

a straight plane produces no lateral stability, apart 
from the very slight stabilising effect produced by 
the secondary cause, already referred to. 

(2) In such aeroplanes a dihedral angle of the wings or 

the use of a high keel surface produces lateral 
stability, and this in an increasing degree as the 
angle of incidence is greater. 

(3) If the centre of gravity of the aeroplane is low, or if 

its tail hi normal flight is high (or if both these 
features are incorporated in the machine), an 
inverted dihedral or A of the wings with a low 
keel surface may produce lateral stability, and this 
to an increasing extent the smaller the angle of 
incidence. 

Lateral stability, therefore, depends on several different 
parts of the structure, but it can never attain the same 
magnitude as longitudinal stability, which is easily explained. 
For, whereas in the case of longitudinal stability any 
angular displacement in the sense of diving or stalling 
affects to its full extent the angle of the main and the 
tail planes, as regards lateral stability a great angular 
displacement in the sense of rolling is required to pro- 
duce even a slight difference in the incidence of the two 
wings. 

The righting couples are therefore much smaller in 
the lateral than the longitudinal sense for any given 
oscillations. If, as hi Chapter VI., the degree of lateral 
stability of an aeroplane is represented by the length of 
a pendulum arm, it will be found that even with the 
most stable machines this length is hardly in excess of 
0'5 m., while attaining 2-5 and even 3 m. in the case of 
longitudinal stability. 

As with longitudinal stability, so here again there exists 
a condition of stability of oscillation that is, a definite 



158 FLIGHT WITHOUT FORMULA 

proportion must exist between the stabilising effect of 
the shape of the machine and the value of its moment of 
rolling inertia, so that the lateral oscillations can never 
increase to the point of making the aeroplane turn turtle. 
For this reason, since lateral stability is relatively small, 
the moment of rolling inertia should not be too great. 
On the other hand, an increase in span, which increases 
this moment of inertia, also gives the stabilising effect 
a long lever arm. Hence, a middle course had best be 
adopted. 

Aeroplanes with a large rolling inertia oscillate slowly, 
so that there is time to correct the oscillations, but these 
tend to persist. 

So far as the wind is concerned, it would appear an 
advantage to concentrate the masses, thus keeping the 
moment of inertia small (see Chapter X.). 

A low centre of gravity, as already shown, increases to a 
considerable extent the moment of inertia both to pitching 
and to rolling. Hence, if unduly low, it may set up lateral 
oscillations, which constitute the disadvantage previously 
referred to. If, therefore, a low centre of gravity is resorted 
to with the object of inclining the rolling axis to permit the 
use of wings with an inverted dihedral or A, care 'should 
be taken that it be not too low, and it would seem in every 
respect preferable to obtain the same result by raising 
the tail.* 

Aeroplanes with little rolling inertia oscillate more 
quickly than the others. If this is slightly disadvantageous 
since these oscillations cannot be so easily corrected, quick 
oscillations, on the other hand, possess the advantage of 
being accompanied by a damping effect similar to that 
existing in the case of longitudinal oscillations and referred 
to above. For, if a plane oscillates laterally, the wind 
strikes it at either a greater or smaller angle than when 
it is motionless, and this becomes the more marked the 
quicker the oscillation. 

* Such a tail should obviously offer as little resistance as possible. 



STABILITY IN STILL AIR 159 

The small degree of lateral stability possessed by aero- 
planes, especially of those with straight planes, would, 
generally speaking, usually not suffice to prevent the 
upsetting of the machine owing to atmospheric disturb- 
ances, the more so since, as Chapter X. will show, the very 
shapes and arrangements which produce lateral stability 
may at times interfere with the flying qualities of the 
machine in disturbed air. 

Hence it is necessary to give the pilot a means of power- 
ful control over lateral balance in order to counteract the 
effects of air disturbances. 

This means consists in warping, which was probably 
first conceived by Mouillard, and first carried out in practice 
by the brothers Wright. Other devices, such as ailerons, 
have since been brought out, the object in each case being 
to produce, differentially or not, an excess of pressure on 
one wing. 

The pilot therefore controls the lateral balance of his 
machine, and this has to be constantly corrected and 
maintained by him. 

Naturally, the idea of providing some automatic device 
to replace this controlling action by the pilot has arisen, 
but this question will be left for discussion in the last 
chapter, which deals with the effects of wind on the 
aeroplane. 

The rotation of a single propeller causes a reaction in 
an aeroplane tending to tilt it laterally to some extent. 
This could easily be corrected by slightlj 7 overloading the 
wing that shows the tendency to rise ; but in this event the 
reverse effect would take place when the engine stopped, 
either unintentionally or through the pilot's action when 
about to begin a glide that is, at the very moment when 
longitudinal balance is already disturbed. 

Lateral balance is bound to be disturbed in some degree 
owing to the propeller ceasing to revolve, but it would seem 
preferable that at the moment when this occurs both wings 
should be evenly loaded. 



160 FLIGHT WITHOUT FORMULAE 

For this reason constructors generally leave it to the 
pilot to correct the effect referred to by means of the warp 
(which term includes all the different devices producing 
lateral stability). Probably the effect is responsible for 
the tendency which most aeroplanes possess of turning 
more easily in one direction than the other.* 

* Another effect due to the rotation of the propeller, the gyroscopic 
effect, will be briefly considered in the following chapter. 



CHAPTER IX 
STABILITY IN STILL AIB 

LATERAL STABILITY (concluded) DIRECTIONAL 
STABILITY TURNING 

OUR examination of lateral stability may well be brought 
to a conclusion by considering the interesting lessons that 
may be learned from experiments with little paper gliders. 
First, we will take some of those gliders which have been 
described in previous chapters and examine them in regard 
to lateral stability. 

Type 1 (see Chapter VII.). This, it will be remembered, 
is a simple rectangular piece of paper. It has already been 
explained that it was necessary to bend it so as to form a 
lateral V- 

This arrangement is essential for obtaining lateral stability 
with this particular glider, since its rolling axis, which 
corresponds approximately with the fold along the centre, 
is a raised axis, for the reason that the path followed by 
the centre of gravity must be at a lesser angle than this 
central fold, in order to give the gliders an angle of incidence. 
Practice here will be seen to confirm theory. 

Cut out a rectangular sheet of very stiff paper and, without 
folding it, load it with ballast as shown in Chapter VII. 

During the process of finding, by experiment, the correct 
position for the ballast, it will be found that the flight of 
such a glider is accompanied by considerable lateral 
oscillations. More, these oscillations are both lateral and 
directional ; in other words, the path followed by the 

161 



162 FLIGHT WITHOUT FORMULA 

centre of gravity is a sinuous one, and the glider not only 
tilts up on to one wing and the other in succession, but 
each time it tends to change its course and swerve round 
towards the lower whig, and thus it is virtually always 
skidding or yawing sideways. 

In this way it appears to oscillate not about an axis 
passing through the centre of gravity, but about a higher 
axis. 

The reason for this, which will be entered into more fully 
in connection with directional stability, is the extremely 
small keel surface of such a glider. This might at first 
appear to conflict with the fundamental principle,* but the 
anomaly is simply due to the fact that the lateral oscillations 
which, as always, do indeed occur about an axis passing 
through the centre of gravity, are combined with the zig- 
zag movement due to the small keel surface, which is 
moreover the outcome of the oscillations. 

The tendency to roll is the result of the very slight 
lateral stability of a straight plane, which possesses practi- 
cally no keel surface, and this tendency is counteracted by 
nothing but the small secondary damping effect referred 
to in Chapter VIII. 

Oscillatory stability is therefore almost absent, and the 
first rolling movement would increase until the glider was 
overturned, but for the fact that, the air pressure being no 
longer directed vertically upwards, the path followed by 
the centre of gravity is deflected sideways and the glider 
tends to turn bodily towards its down-tilted side. 

The glider promptly obeys this tendency, for its mass is 
feeble while it possesses practically no keel surface offer- 
ing lateral resistance ; hence the centre of pressure moves 
towards the side in which movement is taking place and 
thus creates a righting couple. Consequently, in a measure, 
the yawing saves the glider from overturning. 

This tendency to yaw which is displayed by machines with 

* That an aeroplane should be considered as being suspended from 
its centre of gravity (see p. 93). 



STABILITY IN STILL AIR 163 

straight wings and a small keel surface is to be observed, 
for it would seem to furnish the reason for the side-slips to 
which aeroplanes devoid of a lateral dihedral are prone. 

Now, if the sheet of paper is slightly folded upwards 
from the centre, these various movements decrease, and, 
finally, if folded up still further, disappear altogether. 
The dihedral angle increases lateral stability and oscillatory 
stability, while the considerable keel surface which the 
glider now possesses stops all tendency to yaw. 

Since the stabilising effect of the dihedral in the example 
chosen is due to the rolling axis being a raised axis, it is 
to be expected that, when launched upside down, the 
glider will prove to be laterally unstable. This, in fact, is 
what occurs if the dihedral is pronounced enough. The 
glider immediately turns right side up. 

If the glider has no dihedral, or only a slight one, and if 
the span is reduced (in this case either the ballast must be 
moved back or and this is preferable the wing-tips must 
be turned up aft, for the weight of the ballast is now dis- 
proportionate to the weight of the paper so that the centre 
of gravity has moved forward), it may be observed that 
the lateral oscillations become quicker, which is due to the 
moment of rolling inertia having diminished. 

Type, 2 (see Chapter VII.). This model represents the 
normal shape of an aeroplane (fig. 51). It has already 
been explained that it should be given a slight dihedral, 
which, in any case, it will tend to assume of its own accord 
owing to the combined forces of gravity and air pressure. 

This glider will be found to possess good lateral stability, 
its rolling axis coinciding approximately with the central 
fold, so that it is a raised axis. 

Oscillatory stability is good and rolling almost absent. 
If the glider is bent into a downward dihedral or A (this 
fold should be somewhat emphasised in view of the tendency 
of the glider to assume an ordinary V of its own accord), 
it will overturn, which is quite in accord with theory, as 
the natural rolling axis is a raised axis. 



164 FLIGHT WITHOUT FORMULA 

Type 3 (see Chapter VII.). This is the " Canard " or tail- 
first type (fig. 53). The wing-tips being bent vertically 
upward, form a high keel surface, while, as in the previous 
case, the glider naturally tends to assume a V- 

Hence it is laterally stable, and the rolling axis, coincid- 
ing with the central fold, is a raised axis. 

There are no appreciable oscillations. If it is bent 
downward into a f\, and this to a pronounced extent owing 
to the stabilising effect of the fins, or if these latter are 
folded down, the glider overturns. 

These three types, therefore, are in accordance with the 
principle laid down by Captain Ferber, to the effect that 
a lateral V is necessary for lateral stability. 

Examination of the following two models, on the other 
hand, shows them to be in contradiction with this principle, 
and bears out the author's contention that lateral stability 
may be obtained in an aeroplane possessing a downward 
dihedral or A- 

Type 5* 1. Cut out from a sheet of paper folded in two 
the outline shown in fig. 71. 

2. Cut away, hi the dimensions shown, the folded edge 
at AB and CD. 

3. Glue to the inside of the fold : (a) at AB, a small 
strip of cardboard cut from a visiting-card (5 to 6 cm. 
long and 1 cm. broad), (b) at CD a rectangular piece of 
paper (4 cm. by 15 mm.). 

4. Fold back the wings and the tail plane along the thick 
dotted lines. The wings should be folded so as to form a 
lateral A- (If any difficulty is experienced in maintaining 
this shape, a thin strip of cardboard, 4 to 5 mm. wide, 
may be glued along the forward edge.) 

5. Affix the ballast, consisting of one or more paper 
fasteners, to the end of the paper strip in front. (The 
correct position must be found by experiment, and it 
may be useful, for this purpose, to adjust the forward 

* This number was given so as not to break the numerical sequence. 
Type 4 was dealt with in Chapter VII. 



STABILITY IN STILL AIR 



165 



strip of cardboard to the correct position before the glue 
is quite dry.) 




FIG. 71. 

Type 6.\. Cut out from a'sheet'of paper folded in two 
the outline shown in fig. 72. 



Fold along this line. 




FIG. 72. 

2. Fold back the wings and the tail plane along the 
thick dotted lines. The wings are folded so as to form an 



166 FLIGHT WITHOUT FORMULAE 

inverted dihedral or A> the edge of the fold being upper- 
most in this model. 

3. Glue to the inside of the fold : (a) in front and so as 
to form a continuation of the fold, a strip of cardboard cut 
from a visiting-card, and measuring 5 or 6 cm. by 1 cm. ; 
(b) in the rear and at right angles to the fold, another strip 
of cardboard measuring 4 cm. by 15 mm. 

4. Affix ballast in the shape of two or three paper clips 
at the extremity of the foremost cardboard strip. 

These models belonging to types 5 and 6 have to be 
adjusted with great care and will probably turn over at 
the first attempt, until balance is perfect, but this need not 
discourage further attempts. 

If they display a marked tendency to side-slip or yaw 
or to turn to one side, the trailing edge of the opposite 
whig-tip should be slightly turned up, until balance is 
obtained. 

But if the model rolls in too pronounced a fashion, 
such oscillations may be caused to disappear either by 
shifting the ballast or even by turning up or down the 
rear edge of the tail plane. In some cases the same result 
may be obtained simply by emphasising the A of the 
wings. 

Once they are properly adjusted, these gliders assume on 
their flight-path the attitudes shown respectively in figs. 
73 and 74. These are the attitudes imposed by the laws 
of equilibrium ; and if the gliders are thrown skyward 
anyhow, they will always resume these positions, provided 
they are at a sufficient height above the ground. 

Model 5 displays a slight tendency to roll, but model 6 
follows its proper flight-path, which can be made perfectly 
straight, in quite a remarkable manner. 

This is all in accordance with the theory put forward in 
Chapter VIII. Because the rolling axis is a lowered axis 
since in model 5 the centre of gravity is situated very low 
and in model 6 the tail is very high the inverted dihedral 
or A of the wings produces stable lateral equilibrium. 



STABILITY IN STILL AIR 



167 



On the other hand, in model 5 the centre of gravity is 
exceptionally low ; hence the moment of rolling inertia is 




FIG. 73. Perspective. 




PIG. 74. Perspective. 

great, so that oscillatory stability is not quite perfect and 
there remains a tendency to swing laterally, whereas in 



168 PLIGHT WITHOUT FORMULA 

model 6 this defect is absent, since the centre of gravity is 
only slightly lowered. 

The latter arrangement is therefore the one to be adopted 
in designing a full-size aeroplane of this type. 

In both cases, by increasing the A the tendency to 
oscillate is reduced, owing to the fact that, up to a certain 
limit, the value of the righting couples is hereby increased 
likewise. The same is true of a decrease in the angle of 
incidence, effected either by displacing the ballast farther 
forward or by adjusting the tail, because, as has been 
shown previously, any decrease in the incidence augments 
lateral stability if the wings are placed at a A- 

Our theory would be even more conclusively proved 
correct, if, when the wings were turned up into a V> the 
glider overturned. 

As a matter of fact this does occur sometimes, especially 
with model 6, but not always, and with its wings so arranged 
the model may still retain a certain amount of lateral 
stability. 

This apparent conflict of practice and theory may be 
explained by the fact that, by turning up the wings of 
such a model the centre of gravity is raised, since the 
wings constitute an important part of the weight of these 
little gliders ; consequently the rolling axis is also raised, 
and since, as previously stated, lateral oscillation occurs 
not precisely about the rolling axis but about a higher axis 
still, the true rolling axis may prove to be a stable axis 
for V-shaped wings. This is borne out further by the 
fact that in many cases, and especially with model 6, this 
does not occur and that the glider overturns. 

With the kind assistance of M. Eiffel, the author carried 
out hi the Eiffel laboratory a series of tests with a scale 
model of greater size and so designed that its wings could 
be altered to form either an upward or a downward dihedral, 
and these tests appear to be conclusive. 

The model, perfectly stable when its wings formed a 
A > showed a strong tendency to overturn when the wings 



STABILITY IN STILL AIR 



169 



formed a V- (The raising of the centre of gravity caused 
by upturning the wings was neutralised by lowering the 
ballast to a corresponding extent.) 

But even if this further proof were absent, it would 
nevertheless remain true and the fact is most important, 
as will be shown in Chapter X. that it seems possible to 
build aeroplanes, with wings forming an inverted dihedral 
angle, which in spite of this are laterally stable. 

DIRECTIONAL STABILITY 

An aeroplane must possess more than longitudinal and 
lateral stability ; it must maintain its direction of flight, 
must always fly head to the relative wind, and must not 
swing round owing to a slight disturbance from without. 
This is expressed by the term directional stability. 

In other words, an aeroplane should 
behave, in the wind set up by its own 
speed through the air, like a good * 

weathercock. 

In fig. 75, let AB represent, looking 
downwards, a weathercock, turning about \ / 

the vertical axis shown at 0, the direction 
of the wind being shown by the arrow. 

From our knowledge of the distribu- 
tion of pressure on a flat plane (fig. 32, 
Chapter VI.), it is clear that if the axis 
O is situated behind the limit point of 
the centre of pressure, the weathercock, 
in order to be in equilibrium, would have 
to be at an angle with the wind such 
that the corresponding pressure passed 
through the point 0. Hence, the weathercock would 
assume the position A'B' or A*B". 

It would be a bad weathercock because it formed an 
angle with the wind. A good weathercock always lies 
absolutely parallel with the wind, which thus always meets 
it head-on. 



A" A 



A' 



B 



B 



FIG. 75. Plan. 



170 PLIGHT WITHOUT FORMULA 

Therefore, in a good flat weathercock the axis of rotation 
must be situated in front of the limit point of the centre 
of pressure, i.e. in the first fourth from front to rear. 

In so far as its direction in the air is concerned, an aero- 
plane behaves in exactly the same way as the weather- 
cock which we have termed its keel surface, the axis of 
rotation being approximately a vertical axis passing through 
the centre of gravity. 

It can therefore be stated that for an aeroplane to possess 
directional or weathercock stability, the limit point of the 
centre of pressure on its keel surface, when it meets the air 
at even smaller angles, must lie behind the centre of gravity. 

Directional equilibrium is thus obviously stable, since 
any change of direction sets up a righting couple, because 
the pressure on the keel surface always passes behind the 
centre of gravity.* 

Directional stability is usually maintained by the means 
already provided to secure lateral stability, the rear portion 
of the fuselage, which is often covered in with fabric, con- 
stituting the rear part of the keel surface. Moreover, this 
is further increased by the presence of a vertical rudder 
still further aft. 

But there are certain machines in which special means 
have to be taken to secure directional stability the tail- 
first or " Canard " machine is of this type. 

In Chapters VI. and VII. it was stated that the fact that 
this type of machine has its tail plane in front tends to 
longitudinal instability, which is only overcome by the 
unusually high stabilising efficiency of the main planes, 

* Reference to Chapters VI. and VII. will show that longitudinal 
equilibrium is also, in effect, weathercock equilibrium. But in this 
respect, the planes must always form an angle with the relative wind, 
which constitutes the angle of incidence and produces the lift. In 
regard to longitudinal stability, the aeroplane should therefore be a bad 
weathercock. Further, it will be shown in Chapter X. that, in con- 
sidering the effect of the wind on an aeroplane, two classes of bad weather- 
cocks have to be distinguished, and that an aeroplane should be, if the 
term be allowed to pass, a " good variety of bad weathercocks." 



STABILITY IN STILL AIR 171 

which are of relatively great size and situated at a con- 
siderable distance behind the centre of gravity. 

The same is true in regard to directional stability, and 
the existence in the forward part of the machine of a long 
fuselage, comparable to a weathercock turned the wrong 
way round, would speedily cause the aeroplane to turn 
completely round if it were not provided with considerable 
keel surface behind the centre of gravity. The necessity 
for this arrangement will readily appear if, in the little 
paper glider No. 3, already described, the vertical fins at 
the wing-tips are removed. The glider will then turn 
about itself without having any fixed flight-path.* 

In Chapter VIII. it was shown that lateral stability 
is affected by raising or lowering the vertical keel surrace. 
But even if it is neither high nor low, and though it may 
appear to affect only directional stability, every bit of keel 
surface plays an important part in lateral stability. For 
these two varieties of stability are not absolutely distinct. 
Both, in fact, relate to the rotation of the aeroplane about 
axes situated in the plane of symmetry. 

When these axes are close to the flight-path of the centre 
of gravity, only lateral stability comes in question ; but 
when they are more nearly vertical, the rotary movement 
about them belongs to directional stability. 

Nevertheless, any turning movement about any axis 
other than that formed by the path of the centre of gravity 
plays its part in both lateral and directional stability, and 
it is only in so far as it affects the one more than the other 
that it is classified as belonging to lateral or directional 
stability. The line of cleavage between these two varieties 
of stability is by no means clear. 

From this it follows, in the author's opinion, that the 
means for obtaining lateral stability gain considerably in 
effectiveness if they also produce directional stability. If 

* To obtain good directional stability, those paper models with a 
ballasted strip of cardboard in front were all provided with a vertical 
fin in the rear. 



172 FLIGHT WITHOUT FORMULA 

aeroplanes with wings forming a A are ey er built, they 
should be provided with a considerable amount of keel 
surface aft (placed low rather than high). 

In conclusion, it may be said that, of the three varieties 
of stability, directional stability is at the present time the 
most perfect, which is to be accounted for on the ground 
that the pressure on the keel surface must always pass 
behind the centre of gravity, whence arise strong righting 
couples. 

In the order of their effectiveness at the present day, 
the three classes of stability can therefore be arranged as 
follows : 

Directional stability. 

Longitudinal stability. 

Lateral stability. 

By careful observation of the oscillations of an aeroplane 
the truth of this statement will be borne out. Every 
aeroplane betrays some tendency to roll ; at times it also 
tends to pitch, but it hardly ever swerves from side to side 
on its flight-path, zigzag fashion. 

TURNING 

The vertical flight-path of an aeroplane is controlled by 
the elevator ; but the pilot must also be able to change his 
direction and to execute turning movements to right and 
left. 

A few points of elementary mechanics may here be 
usefully recalled. 

If a body is freely abandoned to its own devices after 
having been launched at a certain speed (omitting from 
consideration the action of gravity), it continues by reason 
of its inertia to advance in a straight line at its original 
speed, and an outside force is required in order to modify 
this speed or to alter the direction followed by the body. 

A body following a curved path therefore only does so 
through the action of an outside force. 

If the body follows a circular path, the force which pre- 



STABILITY IN STILL AIR 173 

vents it from getting further away from the centre of the 
circle, although its inertia seeks to propel it in a straight 
line, to move away at a tangent, is termed centripetal force. 

For instance, if a stone attached to the end of a string 
is whirled round, it describes a circle instead of following 
a straight line only because the string resists and exerts 
on it a centripetal force. If this force is stopped and the 
string is let go, the stone will fly off at a tangent. 

On the other hand, a body, in this case the stone, always 
tends to fly off ; it thus reacts, exerting in its turn on the 
cause which maintains it in a circular path in this case, 
on the string a force termed centrifugal force, which, in 
accordance with the well-known principle of mechanics 
concerning the equality of action and reaction, is exactly 
equal and opposite to the centripetal force which causes it. 

In the example chosen, the value of the centripetal and 
centrifugal forces (the same in both cases) could be 
measured by attaching a spring balance to the string. It 
would be found that, as is easily shown in theory, this 
value is proportional to the square of the speed of rotation 
and inversely proportional to the radius of the circle 
described. 

From this it is clear that in order to curve the flight- 
path of an aeroplane, that is, to make it turn, it is necessary 
to exert upon it by some means or other a centripetal 
force directed from the side hi which the turn is to be 
made. This can be done by creating, through movable 
controlling surfaces, a certain lack of symmetry in the 
shape of the aeroplane which will result in a correspond- 
ing lack of symmetry in the reactions of the air upon it. 

The most obvious proceeding is to provide the aeroplane 
with the same device by which ships are steered and to 
equip it with a rudder. But, just as a ship without a 
keel responds only in a slight measure to the action of a 
rudder, so an aeroplane offering little lateral resistance 
that is, having but little keel surface only responds to the 
rudder in a minor degree. 



174 



FLIGHT WITHOUT FORMULAE 



In order to make this clear, we will take the case of an 
aeroplane entirely devoid of keel surface, though this is 
an impossibility on a par with the case of an aeroplane 
wholly devoid of detrimental surface, since the structure 
of an aeroplane must perforce always offer some lateral 
resistance, even though the constructor has tried to reduce 
this to vanishing-point. 

However, let us assume that such an aeroplane, having 



Q 




/t 



FIG. 76. 



its centre of gravity at G (fig. 76), is provided with a 
rudder CD. 

If the rudder is moved to the position CD', the aero- 
plane will turn about its centre of gravity until the rudder 
lies parallel with the wind. But there will not be exerted 
on the centre of gravity any unsymmetrical reaction, any 
centripetal force capable of curving the flight-path. 

The aeroplane will therefore still proceed in a straight 
line, and the only effect of the displacement of the rudder 



STABILITY IN STILL AIR 



175 



will be to make the aeroplane advance crabwise, without 
any tendency to turn on its flight-path. 

But if the machine is equipped with a keel surface AB 
(fig. 77), directional equilibrium necessitates that this keel 
surface should present an angle to the wind, and become 
thereby subjected to a pressure Q, whose couple relatively 




Fro. 77. Plan. 



to the centre of gravity is equal and opposite to the pressure 
q exerted 011 the displaced rudder CD'. Since Q is con- 
siderably greater than q, there is exerted on the centre of 
gravity, as the result of their simultaneous effect, a resultant 
pressure approximately equal to their difference (which 
could be found by compounding the forces), and this forms 
a centripetal reaction capable of curving the flight-path 
that is, of making the machine turn. 



176 



FLIGHT WITHOUT FORMULA 



It should be observed that the nearer the keel surface 
is to the centre of gravity the greater is the centripetal 
force set up by the action of the rudder. Similarly, the 
intensity of this force also depends on the extent of the 
keel surface. And lastly, since the centripetal force has 
a value equal to the difference between the pressures Q 
and q, it becomes greater the smaller the latter pressure. 
Hence there is an advantage in using a small rudder, which 

must, in consequence, 
have a long lever arm 
in order to balance the 
effect of the keel surface. 
A turn might also be 
effected by lowering a 
flap CD, as shown in 
fig. 78 at the extremity 
of one wing, this flap 
constituting a brake. 
In this case, too, a keel 
surface is essential and 
equilibrium would exist 
if the couples set up by 
the pressures Q and q, 
exerted on the keel sur- 
face and on the brake 
respectively, were equal. 
A centripetal reaction, 
the resultant of these pressures, would act on the centre 
of gravity and bring about a turn. 

There remains a third and last means of making an 
aeroplane perform a turn, and this requires no keel surface. 
This consists in causing the aeroplane to assume a permanent 
lateral tilt. 

The pressure exerted on the plane (which is roughly 
equal to the weight of the machine) is tilted with the aero- 
plane and has a component p (fig. 79) which assumes the 
part of centripetal force, and makes the machine turn. 




FIG. 78. Plan. 



STABILITY IN STILL AIR 



177 



The machine can be tilted in various ways for instance, 
by overloading one of the wings. But the more usual 
method is that of the warp, which has already been referred 
to as the pilot's means of maintaining lateral balance. 

By increasing the incidence, or its equivalent the lift, of 
one wing-tip and decreasing that of the other, the former 
wing is raised and the latter lowered, so that the machine 
is tilted in the manner required to make a turn. 

But in warping, the wing with increased lift also has 
an increased drag or head-resistance, while the reverse 
takes place with the other wing. 

This secondary effect 
is analogous with that of 
the air brake just con- 
sidered and is exerted in 
the opposite way to that 
required to perform the 
turn. It is usually smaller 
than the main effect of the 
warp, but still interferes 
with its efficacity. On 
the other hand, in some 
aeroplanes it may gain FIG. 79. Front elevation, 

the upper hand, as in the noteworthy case of the Wright 
machines. 

In order to overcome this defect, the brothers Wright 
produced, through the means of the rudder (which played 
no other part), a couple opposed to the braking effect, 
which left its entire efficiency to the differential pressure 
variation exerted on the wings by the action of the warp. 
Further, the warp and rudder could be so interconnected 
as to act simultaneously by the movement of a single 
lever (this constituted the main principle of the Wright 
patents). 

This detrimental secondary effect could, it would appear, 
be easily overcome by using a plane with wing-tips uptilted 
in the rear as at BC in fig. 80. 

12 




Centripetal Force. 



178 FLIGHT WITHOUT FORMULA 

By depressing the trailing edge BC of the wings, which 
are purposely made flexible, the lift is increased and the 
drag diminished at the wing-tip. By turning up the trailing 
edge the lift is decreased and the drag increased. Both 
effects therefore combine to assist in making the turn 
instead of impeding it. Instead, finally, of adopting this 
particular warping method, the same result could be obtained 
by using negative-angle ailerons. 

It should be noted and the fact is of importance both 
so far as turning and lateral balance are concerned that 
the effect of the warp is definitely limited. It is known 
that beyond a certain incidence (usually in the neighbour- 
hood of 15 to 20) the lift of a plane diminishes while the 
drag increases rapidly. 

If the warp is therefore used to an exaggerated extent, 
the detrimental secondary effect referred to above comes 

into play, with the result 

"""'""'- -^.^^^ ., that its effect is the reverse 
" -Q G of the usual one. This may 

prove a source of danger, 
FIG. 80. -Profile. and j t might be weU ^ 

certain machines, if not to limit the warp absolutely, at 
any rate to provide some means of warning the pilot that 
he is approaching the danger-point. 

Since the rudder sets up a couple tending to counteract 
this secondary effect, it should be resorted to in case an 
undue degree of warp causes a reverse action to the one 
intended. 

The banking of the planes which, as already seen, 
may provoke a turn, always results from it ; for, as the 
aeroplane swings round, the outer wing travels faster than 
the inner wing, so that the pressure on the one differs 
from that on the other, with the result that the outer one 
is raised. 

Therefore, if the centripetal force which causes the 
turn does not originate from the intentional banking of 
the planes, this banking which results from the turning 



STABILITY IN STILL AIR 179 

movement produces the necessary force to balance the 
centrifugal force set up by the circular motion of the 
machine. 

It follows that the amount of the bank during a turn 
depends on those factors which determine the amount of 
centrifugal force. Hence, the bank is steeper the faster the 
flying speed (being proportional to the square of the speed), 
and the sharper the turn. It may therefore be dangerous 
to turn too sharply at high speeds. 

Equilibrium between centripetal and centrifugal force is 
important simply in so far as it concerns the movement of 
the aeroplane along its curved path, or, in other words, the 
movement of its centre of gravity. But, in addition, the 
machine itself should be in equilibrium about its centre of 
gravity that is, the couples exerted upon it by the air in 
its dissymmetrical position during the turn must exactly 
balance one another. 

This position of equilibrium during a turn evidently 
depends on various factors, among which are the means 
whereby the turn has been produced and the distribution 
of the masses of the machine. 

For instance, if the turn is caused by banking it might 
be thought that so long as the cause remained, the bank 
would continue to grow more and more steep. But usually 
this is not the case, for if the aeroplane possesses any 
natural stability, the bank will itself set up a righting 
couple balancing the couple which produced the bank. 

The value of this righting couple depends, of course, on 
the shape of the aeroplane and especially on the position 
of its rolling axis. If the machine has little natural 
stability, the pilot may have to use his controls in order to 
limit the bank, as otherwise the machine would bank ever 
more steeply and the turn become ever sharper until the 
aeroplane fell.* 

* Pilots have often mentioned an impression of being drawn towards 
the centre when turning sharply. 



180 FLIGHT WITHOUT FORMULA 

As a rule, the warp is not used for producing a turn, for 
the majority of machines possess sufficient keel surface to 
answer the rudder perfectly. 

Often the rudder aids the warp in maintaining lateral 
balance : for instance, by turning to the left a downward tilt 
of the right wing may be overcome. 

Possibly in future the warp will become even less im- 
portant, so that this device, which is generally thought 
to have been imitated from birds (which have no vertical 
rudder), may eventually vanish altogether.* The Paulhan- 
Tatin " Torpille," referred to in previous chapters, had no 
warp, neither had the old Voisin biplane, one of the first 
aeroplanes that ever flew. This was due to the fact that 
in both cases the keel surface (a pronounced curved dihedral 
in the " Torpille," and curtains in the Voisin) was sufficient 
to render the rudder highly effective. 

It is to be noted that, whatever the cause of the turn, the 
dissymmetrical attitude adopted as a result by the aeroplane 
simultaneously causes the drag to increase while the lift 
decreases owing to the bank. At the same time, the angle 
of incidence alters, since any alteration in lateral balance 
brings about an alteration in longitudinal balance, for rolling 
produces pitching. 

For these reasons an aeroplane descends during a turn. 

The pilot feels that he is losing air-speed and puts the 
elevator down. Theory, on the other hand, would appear to 
teach that he ought to climb. But, as already stated, this 
apparent divergence is due to the fact that theory applies 
chiefly to a machine in normal flight. When an aeroplane 
changes its flight and passes from one position to another, 
effects of inertia may arise during the transition stage 
which may vitiate purely theoretical conclusions, and in 

* Although the author has carefully studied the flight of large soaring 
and gliding birds in a wind, he has never found them to warp their wing- 
tips to a perceptible extent to obtain lateral balance, while, on the other 
hand, probably for this very purpose, they continually twist their tails 
to right and left. 



STABILITY IN STILL AIR 181 

such a case theory must give way to practice. In any 
event, practice need not necessarily remain the same should 
the shape of the aeroplane undergo considerable alterations 
and, more especially, if in future the lift coefficient becomes 
very small.* 

In conclusion, something remains to be said of the 
gyroscopic effect of the propeller. Any body turning about 
a symmetrical axis tends, for reasons of inertia, to preserve 
its original movement of rotation. 

The direction of the axis about which turning takes 
place remains fixed in space, and, in order to alter it, a force 
must be applied to it, which must be the greater the higher 
the speed of rotation, the greater the movement of inertia, 
and the sharper the effort to alter it. 

But now arises the curious fact that if it is sought to 
move the axis in a given direction, it will actually move in 
a direction at right angles to this. This characteristic of 
rotating bodies may be observed in the case of gyroscopic 
tops, which only remain in equilibrium and only adopt a 
slow conical motion when their axis becomes inclined 
towards the end of their spinning, for this very reason. 

Now a propeller which has a high moment of inertia, 
especially if of large diameter, and turning at a great speed, 
constitutes a powerful gyroscope (which is further increased 
if the motor is of the rotary type). 

It follows that any sudden action tending to modify the 
direction of flight results in a movement at right angles to 
that desired. Thus, a sudden swerve to one side may pro- 

* It may be added that at very high speeds an aeroplane during a 
sharp turn actually rises instead of coming down, but this is due to 
quite a different cause. At the moment of turning, when already banked 
and the rudder is brought into play, the machine for a fraction of time, 
owing to its inertia, slides outward and upward on its planes. This 
effect was particularly noticeable during the Gordon-Bennett race in 
1913, when, long before the turning-point was reached, the aeroplanes 
were gradually banked over, until at the last moment a sudden move- 
ment of the rudder bar sent them skimming round, the while shooting 
sharply upward and outward. TRANSLATOR. 



182 FLIGHT WITHOUT FORMULA 

duce a tendency either to dive or to stall, according to which 
side the swerve is made and to the direction of rotation of 
the propeller. 

Accidents have sometimes been ascribed to this gyroscopic 
effect, but its importance would appear to have been greatly 
exaggerated, and so long as the controls are not moved 
very sharply, it remains almost inappreciable. 



CHAPTER X 
THE EFFECT OF WIND ON AEROPLANES 

EVERY previous chapter related to the flight of an aero- 
plane in perfectly still air. To round off our treatise, the 
behaviour of the aeroplane must be examined in disturbed 
air in other words, we now have to deal with the effect of 
wind on an aeroplane. 

The atmosphere is never absolutely at rest ; there is 
always a certain amount of wind. The two ever-present 
characteristic features of a wind are its direction and its 
speed. No wind is ever regular. Both its velocity and its 
direction constantly vary and, save in a hurricane, these 
variations do not depart from the mean beyond certain 
limits. Hence, the wind as it exists in Nature may be 
regarded as a normal wind, as if it had a mean speed and 
direction, with variations therefrom. 

These variations may be in themselves irregular or 
regular up to a point. Near the ground the wind follows 
the contour of the earth, encounters obstacles, and flows 
past them in eddies ; hence it is perforce irregular, like the 
flow of a stream along the banks. 

Eddies are formed in the air, as in water : valleys, forests, 
damp meadows where humidity is present all these 
produce in the air that lies above them descending currents, 
sometimes called " holes in the air " ; while hills and bare 
ground radiating the sun's heat produce rising currents 
of air. 

These effects are only felt up to a certain height in the 
atmosphere, and the higher one flies the more regular 



184 FLIGHT WITHOUT FORMULA 

becomes the wind. In the upper reaches the wind seems 
to pulsate and to undulate in waves comparable to the 
waves of the sea. 

The regular mean wind which reigns there may there- 
fore be considered as possessing atmospheric pulsations, 
propagated at a speed differing from the speed of the wind 
itself, comparable to the ripples produced by throwing a 
stone in flowing water ripples which move at a speed 
differing from that of the current itself. 

This comparison of a regular wind with a flowing stream 
enables the effect of such a wind on an aeroplane to be 
studied in a very simple manner. 

For the last time we will refer to that elementary principle 
of mechanics applicable to any body moving through a 
medium which itself is in motion the principle of the 
composition of speeds. 

A speed, just as a force, may be represented by an arrow 
of a length proportional to the speed and pointing in the 
direction of movement. 

For example, let us suppose that a boat is moving 
through calm water at a speed represented by the arrow 
OA (fig. 81). 

Now, if instead of being still, the water were flowing at 
a speed represented by the arrow OB, the ship, although 
still heading in the same direction, would have a real speed 
and direction represented by the arrow 00. This speed is 
the resultant of the speeds OA and OB, and this composition 
of speeds, it will be seen, is simply effected by drawing the 
parallelogram. 

The ship will appear still to be following the course OA, 
which will be its apparent course, while in fact following 
the real course 00. 

Instead of a ship through flowing water, let us now take 
the case of an airship or aeroplane moving through a current 
of air or regular wind. Such a craft, while driven for- 
ward through the air by its own motive power at the speed 
it would attain if the air were perfectly calm, is at the 



THE EFFECT OF WIND ON AEROPLANES 185 

same time drawn along by the wind together with 'the 
surrounding air, of which it forms, as it were, a part, and 
this without the pilot being able to perceive this motion, 
unless he looks at some fixed landmark on the ground. 

An aeroplane may be likened to a fly in a railway carriage, 
which is unable to perceive, and remains unaffected by, 
the speed at which the train is moving. 

In a free spherical balloon drifting before a regular wind 
not a breath of air is perceptible. On board an aeroplane 




B 



FIG. 81. 

or airship only the relative wind is felt which is created by 
the speed of flight, no matter whether in still air or in wind. 

In a side-wind, in order to attain to a given spot, a pilot 
does not steer straight for his objective, but allows for the 
drift, like a boatman crossing a swift-flowing river. 

When the direction of the wind coincides with the path 
of flight the speeds are either added to or subtracted from 
one another ; for instance, an aeroplane with a flying speed 
of 80 km. per hour in a calm will only have a real speed of 
50 km. per hour against a 30-km. per hour wind, but will 
attain 110 km. per hour when flying before it. 

In order to be dirigible, an aircraft must have a speed 



186 FLIGHT WITHOUT FORMULA 

greater than that of the wind. In practice an aeroplane 
virtually never flies in a wind of greater velocity than its 
own flying speed, and hence is always dirigible. 

The wind further affects the gliding path of an aeroplane. 
For example, if an aeroplane with a normal gliding path 
OA in a calm (fig. 82) comes down against the wind, its 
real gliding path will be OC 15 which is steeper than OA, 
while with the wind behind it will be flatter, as shown by 
OC 2 . The arrows OC^ and OC 2 represent the resultant 
speeds of the gliding speed OA in calm air and of the speeds 
of the wind OB X and OB 2 . 

But in all these different gliding paths, the gliding angle 




of the aeroplane remains the same, since the apparent gliding 
path relatively to the wind always remains the same. 

If the speed of the wind is equal to that of the aeroplane, 
the machine, still preserving its normal gliding angle, 
would come down vertically and would alight gently on 
the earth without rolling forward. 

Birds often soar in this manner without any perceptible 
forward movement, but, apart perhaps from the brothers 
Wright during the course of their gliding experiments in 
1911, no aeroplane pilot would appear to have attempted 
the feat hitherto.* 

* This statement is no longer correct. Many pilots have undoubtedly 
flown in winds equal and even superior to their own flying speed. 
Moreover, this vertical descent is sometimes made intentionally with 
such machines as the Maurice Farman, the engine being stopped and 
the aeroplane being purposely stalled until forward motion appears 
to cease and the machine seems to float motionless in the air. 
TBANSLATOB. 



THE EFFECT OF WIND ON AEROPLANES 187 

A regular wind may be a rising current. In this case, 
if sufficiently strong, it may render the gliding path hori- 
zontal. Thus, if an aeroplane in calm air glides at a speed 
OA (fig. 83), which has a horizontal component equal 
to 15 m. per second, and follows a descending path of 1 
in 6, a regular ascending current with a speed OB X or OB 2 , 
with a vertical component equal to 2-5 m. per sec., would 
enable an aeroplane to glide horizontally. 

The existence of such ascending currents is sometimes 
taken in order to explain the soaring flight practised by 
certain species of large birds over the great spaces of the 
ocean or the desert. But it is difficult to accept this as 
the only explanation of this wonderful mode of flight, 




which often extends for hours at a time, and would pre- 
suppose the permanency of such rising currents. Another 
explanation will be given hereafter. 

We may now examine the effects on an aeroplane of 
irregularities in the wind. 

Any disturbance in the air may at any time be character- 
ised by the modification in speed and direction of the wind ; 
such modifications could be measured by means of a very 
sensitive anemometer mounted on a universal joint. 

The first effect of a disturbance is to tend to impart its 
own momentary speed and direction to anything borne 
by the air which it affects. Very light objects, feathers, 
tissue-paper, etc., immediately yield to a gust. 

If an aeroplane were devoid of mass, and therefore of 
inertia, it would behave in the same way ; it would instantly 
assume the new speed and direction of the wind and would 



188 FLIGHT WITHOUT FORMULA 

promptly obey its every whim. In this case the pilot 
would be unable to perceive, except by looking at the 
ground, any gusts or their effect ; for him it would be the 
same as though he were flying in a regular wind. 

But all aircraft possess considerable mass, and therefore 
do not immediately obey the modifications resulting from 
a wind gust in which they are flying. The disturbance 
therefore exerts upon it, during a variable period, a certain 
action, also variable, which can be likened to that which 
would be experienced if the movements of the aeroplane 
were restrained. This action, which may be termed the 
relative action of a disturbance, modifies both in speed and 
in direction the relative wind which the aeroplane normally 
encounters, and these modifications can be felt by the 
pilot and measured by an anemometer. 

For the sake of simplicity, let us suppose that a wind of 
a certain definite value is quite instantaneously succeeded 
by a wind of another value, the wind being regular in each 
case. A craft without mass would forthwith conform to 
the new wind. The primary gust effect would be complete, 
its relative action would be zero. 

For any craft possessing mass the primary gust effect 
would at first be zero and the relative action at a maximum ; 
but, as the machine gradually yields to the gust, the rela- 
tive action grows smaller and finally vanishes altogether 
when the aeroplane has completely conformed to the new 
wind. The greater the inertia of the machine, the longer 
will be the transition period. 

Still keeping to our hypothesis of an instantaneous 
change of condition, an anemometer fixed in space and 
another carried on the aeroplane might for one brief instant 
record the same indications ; but while those of the fixed 
anemometer would be constant, the other instrument 
would sooner or later, according to the aeroplane's inertia, 
return to its original indications. 

If it is remembered that gusts, even the most violent, 
are never perfectly instantaneous, it seems probable that 



THE EFFECT OF WIND ON AEROPLANES 189 

the relative action of a gust on an aeroplane is never so 
intense as it would be were the machine fixed in space, 
and that it dies away the more quickly the lighter the 
aeroplane. 

But the pilot of a machine in flight does not perceive 
this relative action in the same way that he would if the 
machine were immovable for instance, if the aeroplane 
were struck by a gust coming from the right at right angles, 
the pilot of a stationary aeroplane would only feel the gust 
on his right cheek, while in flight he would only perceive 
the existence of a gust by the fact that the relative wind 
was just a little stronger on his right cheek than on the 
left. It is simply a question of the composition of 



We have distinguished a primary gust effect and a rela- 
tive effect. The results of each may now be examined. 

The primary effect modifies in magnitude and in direction 
the real speed of the aeroplane, which yields the more 
slowly the greater its mass and inertia. 

Now, instead of consisting, as our hypothesis required, 
of an instantaneous succession of two winds of different 
value, a gust is a more or less gradual and wavelike modi- 
fication of the mean speed of the wind, lasting usually not 
more than a few seconds. 

Hence, if the aeroplane's inertia be sufficient, the cause 
may cease before the gust has exerted its primary effect on 
the aeroplane, the whole energies of the gust being absorbed 
in producing the relative effect. 

The direction of flight and the real speed of the aeroplane, 
provided it has enough inertia, may consequently be only 
slightly altered by the gust which would pass like a wave 
past a floating body. This is why, whereas a toy balloon 
is tossed by every little gust, a great passenger balloon sails 
majestically on its way without being affected in the 
slightest degree. 

Why, therefore, should this not be the case with an 
aeroplane which has a mass not differing widely from that 



190 FLIGHT WITHOUT FORMULA 

of a balloon ? The cause must be sought for in the relative 
effect of the gust. This relative effect is only slight in the 
case of a balloon which is based on static support according 
to the Archimedean law ; but it affects the very essence of 
the equilibrium of an aeroplane based on the dynamic 
principle of sustentation by its speed and incidence. 

Any variation in the speed or direction of the relative 
wind, therefore, usually affects the value of the pressures 
on the various planes, and consequently further affects its 
attitude hi the air which is determined by a perpetual 
equilibrium. 

The effects produced by the relative action of a gust may 
be divided into two classes : the displacement effect and the 
rotary effect. 

The displacement effect is that produced by the relative 
action of the gust on the machine as a whole, and seen 
in the modification of the path followed before by the 
centre of gravity and the speed at which it moved until 
then. 

The displacement effect must not be confused with the 
primary gust effect previously referred to. 

For instance, if an aeroplane in horizontal flight is struck 
head-on by a horizontal gust, the primary gust effect takes 
the shape of a reduction in the real flying speed, which 
reduction is the greater the smaller the inertia of the 
machine. But this will not alter the horizontal nature of 
the flight-path. 

On the other hand, the displacement effect produced by 
the gust will result in raising the whole machine which, 
owing to its inertia and in increasing measure as its inertia 
is greater, experiences an increase in the speed of the 
relative wind, with the result that the lift on the planes 
also increases. 

The rotary effect is that produced by the relative action 
of the gust on the equilibrium of the aeroplane about its 
centre of gravity. This is due to the fact that the modi- 
fications in the relative wind destroy the harmony between 



THE EFFECT OF WIND ON AEROPLANES 191 

the pressures on the various parts of the aeroplane, which 
balanced one another and thus maintained the machine in 
stable equilibrium. 

Certain rotary effects are due to the fact that no gust is 
instantaneous, but always moves at a speed which, however 
great, is still limited. A gust may therefore first strike 
one part of the aeroplane and produce a first rupture of 
equilibrium ; then, continuing, it may strike the opposite 
side which may already have been shifted out of position, 
and affect this in turn either in the sense of restoring equili- 
brium or the reverse. 

The displacement and rotary effects due to a gust will 
now be successively examined, beginning with those which 
affect equilibrium of sustentation and longitudinal equili- 
brium, these being closely interconnected. For the time 
being, therefore, we will only deal with gusts moving in 
the plane of symmetry of the aeroplane that is, with 
straight gusts, which affect the speed and the angle at 
which the relative wind meets the planes. 

First, let us examine the displacement effect. It will 
result in a modification in the lift of the planes. The lift, 
normally equal to the weight of the machine, has for its 
value the lift coefficient of the planes multiplied by their 
area and the square of the speed. If the lift coefficient 
remains constant, and the relative wind increases as a 
result of the gust, the lift of the planes increases ; if the 
speed of the wind diminishes, so does the lift. 

It is readily seen that in the case of small variations in 
the speed, the variations in the lift are increasingly large, 
the greater the weight of the machine and the lower its 
normal flying speed. These variations depend neither on 
the wing area nor on the value of the lift coefficient. 

For instance, if an aeroplane weighing 400 kg. and flying 
at 20 m. per second or 72 km. per hour, experienced, as the 
result of a gust from the rear, a decrease in the relative 
speed of 2 m. per second, the lift will decrease by 76 kg. If 
it weighed 600 kg. instead of 400, its normal flying speed 



192 FLIGHT WITHOUT FORMULA 

being still 20 m. per second, the same decrease in the speed 

600 
would bring about a reduction in the lift of 76 x = 114 kg. 

proportional to the weight. 

If, weighing 400 kg., its normal speed were 30 m. per 
second instead of 20, the same decrease of 2 m. per second in 
the speed would produce a reduction in the lift of only 
52 kg. instead of 76 as before. 

These results remain true irrespectively of the plane 
area and the lift coefficient.* 

Now, suppose that, the speed of the relative wind re- 
maining constant, the angle at which it meets the aero- 
plane changes ; the value of the angle of incidence of the 
planes is thereby modified and with it the lift coefficient. 
The lift therefore also varies in this case, and a simple cal- 
culation shows that these variations are the greater the 
greater the weight and the smaller the lift coefficient. 

For example, a machine weighing 400 kg. and possessing 
a lift coefficient of 0-05, will, if this lift coefficient is re- 
duced by 0-005 which is equivalent to lessening the 
angle of incidence by one degree experience a loss of lift of 
about 40 kg. If the weight were 600 kg., the loss of lift 
would be 60 kg. 

If it weighed 400 kg. and the normal lift coefficient were 
0-025 instead of 0-05, the loss of lift resulting from a re- 

* The method of calculation is quite simple. 

Example. it the weight is 400 kg. and the speed 20 m. per second 
the square of the latter being 400, the product of the plane area and 
the lift coefficient remains 1 whether the area be 20 sq. m. and the lift 
coefficient 0'05, or the area 25 sq. m. and the lift coefficient 0'04, or 
whatever be the combination. This being so, if the speed decreases to 
18 m. per second, the square of which is 324, it is clear that the lift is 
reduced from 400 to 324 kg.,- and consequently there is a reduction in 
the lift of 76 kg. as stated. 

If the normal speed were 30 m. per second, the product of the area and 

the lift coefficient would be OArk = 0'444; the decrease in the speed to 
900 

28 m. per second (the square of which is 784) would give the lift a value 
of 0-444 x 784 = 348 kg. The loss of lif t, therefore, would be only 52 kg. 



THE EFFECT OF WIND ON AEROPLANES 193 

duction of the lift coefficient by 0-005 would be 80 kg. 
instead of 40 kg. 

These results hold good irrespectively of the area and 
the speed. 

Finally, if both the speed and the angle of wind vary 
at one and the same time, both results are added to one 
another. 

From this it may be deduced that for an aeroplane to 
experience the least possible loss of lift owing to an atmo- 
spheric disturbance, it should be light, fly at a high speed, 
and possess a big lift coefficient. 

These two latter conditions are not so contradictory as 
might be supposed ; and if considered together, further 
confirm the view expressed in Chapter III., as the result 
of totally different considerations, that an increase in 
the speed of aeroplanes should be sought for rather in 
the reduction of their area than of their lift coefficient. 
Apart from the question of weight, which will be dealt 
with further on, this may be one of the reasons why, as 
a general rule, monoplanes behave better in a wind than 
biplanes.* 

The relative action of a gust moving in the plane of 
symmetry of an aeroplane, results, as we have just seen, 
in a modification of the lift of the planes. This modifica- 
tion produces the displacement effect. 

Suppose, for instance, that an aeroplane flying hori- 
zontally at a definite speed suddenly were to lose the 
whole of its lift ; it would become comparable to a 
projectile launched horizontally, and, while retaining a 
certain forward speed, would fall. If the air in no way 
resisted its fall, this would take place at the rate of any 
body falling freely in a vacuum ; that is, after one second 
it would have fallen about 5 m., at the end of 2 seconds 
20 m., etc. 

Its trajectory would be a curve bending ever more steeply 

* Responsibility for this statement, in which I do not concur, rests 
entirely in the author. TRANSLATOR. 

13 



194 FLIGHT WITHOUT FORMULA 

towards the earth. Naturally this curve would be flatter 
the higher the flying speed of the aeroplane. 

Actually the air opposes, in the vertical sense, con- 
siderable resistance to the fall of a machine provided with 
planes, so that an aeroplane would not fall so fast as men- 
tioned above. 

Moreover, as a gust is not instantaneous and only lasts 
a short while, the flight-path straightens out again fairl}- 
quickly as soon as the lift returns, and this the more quickly 
the smaller the mass of the aeroplane. 

This modification of the flight-path constitutes the dis- 
placement effect due to the gust. 

The pilot only feels, in the case under consideration, the 
sensation of a vertical fall though actually this move- 
ment is progressive. According to pilots' accounts these 
vertical falls are considerable, from which one judges that 
either the duration of the gusts is fairly long or that the 
planes may, under given conditions, lose more than their 
total lift.* 

This displacement effect is devoid of danger, when it is 
not excessive, if it is in the sense of raising the machine. 
When it is considerable, the pilot corrects it by reducing 
his incidence by means of the elevator. 

On the other hand, if it tends to make the aeroplane 
fall, it may be dangerous if occurring near the ground ; it 
is here, moreover, that there always exists a source of 
danger, for eddies are more frequent than higher up in 
the atmosphere. 

Besides, pilots always fear a loss of lift or, what is often 
the equivalent, a loss of air speed, for, apart altogether 

* The discovery made during the inquiry into certain accidents that 
the upper stay-wires of monoplanes have broken in the air, would at first 
sight appear to confirm the view that their wings may at times be struck 
by the wind on their upper surface. 

Nevertheless this view should be treated with caution, for the break- 
age of the overhead stay-wires could be attributed equally well to the 
effects of inertia produced when, at the end of a dive, the pilot flattens 
out too abruptly. 



THE EFFECT OF WIND ON AEROPLANES 195 

from the ensuing fall, the aeroplane then flies in a con- 
dition where the ordinary laws normally determining the 
equilibrium and stability of an aeroplane no longer apply. 
This stability may become most precarious, and this is 
apparent to the pilot by the fact that the controls no longer 
respond. The only remedy is to regain air speed, which 
is effected by diving.* 

Usually, therefore, the correction of displacement effects 
due to gusts consists in diving. Nevertheless, if a head 
gust slanting downward forced the aeroplane down, the 
pilot would naturally have to elevate. In this case there 
would be no loss of air speed, and the loss of lift would be 
due to the reduction of the relative incidence. 

Let us now turn to the rotary effects of atmospheric 
disturbances acting in the plane of symmetry of the aero- 
plane. A machine with a fixed elevator can only fly at a 
single angle of incidence. Therefore, if the relative wind 
which normally strikes an aeroplane changes its inclination 
by reason of a gust, the machine will of its own accord 
seek to resume, relatively to the new direction of the re- 
lative wind, the only angle of incidence at which it flies 
in longitudinal equilibrium. 

The same thing will happen if the displacement effect 
already referred to should modify the trajectory of the 
centre of gravity ; the latter will always tend to adhere to 
its flight-path. 

The rotary effect resulting will take place all the quicker, 
and will die away all the more rapidly, as the longitudinal 
moment of inertia of the machine is smaller. Thus, in 
the case, already considered, of an aeroplane losing air speed 
and falling, it may do this bodily, without any appreciable 
dive, if its moment of inertia is big ; whereas, if bow and 
tail are lightly loaded, it yields to the gust and dives in a 
more or less pronounced fashion. 

* Air-speed indicators, consisting of some form of delicate anemo- 
meter, constantly record the relative speed and enable the pilot to 
operate his controls in good time. 



196 PLIGHT WITHOUT FORMULA 

This latter quality would appear to be the better one 
of the two, since, in the case under consideration, the pilot 
always has to dive to re-establish equilibrium. Hence, in 
this respect, an aeroplane should have as small a longi- 
tudinal moment of inertia as possible. 

Another rotary effect may arise through a cause already 
referred to if the disturbance does not reach the main plane 
and the tail simultaneously. In this case there is exerted 
on the first surface struck, if considered independently from 
the rest, a modification in the magnitude and the position 
of the pressure, which in turn brings about a modification 
in the couple which it normally exerted about the centre 
of gravity. 

If the couple due to the main plane takes the upper hand, 
the machine tends to stall ; if the reverse takes place, it 
tends to dive. A stalling aeroplane always loses some of 
its air speed ; moreover, if the gust strikes it head-on the 
machine is still further exposed, being stalled, to its dis- 
turbing effect. As has already been shown, the correcting 
movement for the majority of cases of displacement effect 
consists not hi stalling but in diving. 

For these various reasons, and excepting always the case 
of a downward current forcing the machine down, the 
rotary effect of a gust should cause the aeroplane to dive 
of its own accord. 

In this respect, the manner in which fore-and-aft balance 
is maintained is most important. If the tail is a lifting tail 
(see fig. 36, Chapter VI.), the pressure normally exerted on 
the main plane passes in front of the centre of gravity. 
This being so, the action of a gust striking the main plane 
first, would produce as its rotary effect a stalling move- 
ment, except only if the gust had a pronounced downward 
tendency, in which case the stalling movement is the 
right one. 

A gust from the rear, striking the tail first, decreases its 
lift and also provokes stalling. In every case, therefore, 
the rotary effect of the gust is detrimental to stability. 



THE EFFECT OF WIND ON AEROPLANES 197 

A lifting tail which, as seen in Chapter VI., is the most 
defective in regard to lateral stability in still air, is con- 
sequently equally unfavourable in disturbed air. 

On the other hand, if the tail is normally placed at a 
negative angle (see fig. 42, Chapter VI.), the normal pressure 
on the main plane passes behind the centre of gravity. 
The action of a head gust, unless pointing downward to a 
considerable extent, in this case produces as its rotary effect 
a diving movement, and the same is true of a gust from 
behind which diminishes the downward pressure normally 
exerted on the tail plane. If the gust is a downward one 
to a marked extent, it will tend to stall the machine, which, 
again, is as it should be. In every case the rotary effect of 
the gust is favourable. 

The use of a negative tail plane, which has already been 
seen to be excellent in regard to stability hi still air, is 
therefore equally beneficial in disturbed air. Nor should 
this cause surprise. 

Previously it was shown that the presence of a plane 
normally acting in front of the centre of gravity was 
productive of longitudinal instability, since it really acted 
as a reversed and overhung weathercock. It is quite clear 
that if a gust strikes such a plane first, it will tend, being 
a bad weathercock, to be displaced still further and thereby 
become still more exposed to the disturbing action of 
the gust. 

On the other hand, if both the main plane and the tail 
act behind the centre of gravity, where they combine to 
procure for the machine an excellent degree of longitudinal 
stability in still air, they will constitute a good weather- 
cock which will always float in a head gust so that the 
upsetting action vanishes,* and the aeroplane itself absorbs 
the gust. In so far as gusts from behind are concerned, 

* Earlier, it was stated (see p. 170) that longitudinally an aeroplane 
must necessarily always be a bad weathercock, but some distinction of 
quality still remains and, so far as the effects of wind are concerned, an 
aeroplane should belong to a " good variety of bad weathercocks." 



198 FLIGHT WITHOUT FORMULA 

this arrangement is again productive of good stability 
since the rotary effect due to the gust brings about the 
very manoeuvre which the pilot would have otherwise to 
perform in order to correct the displacement effect. 

These rotary effects have an intensity and duration 
depending on the moment of longitudinal inertia of the 
machine. The science of mechanics proves that a definite 
amount of disturbing energy applied to aeroplanes possess- 
ing the same degree of longitudinal stability * gives them 
an identical angular displacement irrespective of their 
moment of inertia. The latter only affects the duration 
of the displacement. The greater the moment of inertia, 
the slower does the oscillation come about. 

Nevertheless, it should be remembered that a force, 
however great, can only put forth an amount of energy 
proportional to the displacement produced.! 

Hence, if the gust is only a brief one, the disturbing 
energy applied to the aeroplane and the ensuing angular 
displacement will be all the smaller the more reluctantly 
the aeroplane yields to the gust. Wherefore, there is a 
distinct advantage to be derived from increasing the 
longitudinal moment of inertia. 

But, if the gust lasts some considerable time, this ad- 
vantage disappears and the great moment of inertia has 
the effect of prolonging the disturbing impulse. Besides, 
it may happen that two gusts follow one another at a 
brief interval and that the second, which would encounter 
an aeroplane with little inertia already re-established in 
a position of equilibrium, would strike a machine heavily 
loaded fore and aft before it had recovered, or even when 
it was still under the influence of the first gust. 

* In Chapter VI. it was shown that the longitudinal stability of an 
aeroplane can be represented by the length of a pendulum arm weighted 
at the end with the weight of the aeroplane. 

f If a pony is harnessed to a heavy wagon, it will be unable to move 
it ; its force will be wasted, since it will produce no energy. But if it 
is harnessed to, a light cart, its force, though smaller than that put forth 
in the former case, will produce useful energy. 



THE EFFECT OF WIND ON AEROPLANES 199 

Moreover, for the same reason, the first machine would 
more readily answer its controls and would respond more 
perfectly to the wishes of most pilots, who desire, above 
all, a controllable aeroplane. 

It should be noted that, in so far as rotary effects are 
concerned, it is desirable that gusts should clear an aeroplane 
as quickly as possible, and, for this reason, it should be 
fairly short fore and aft, after the example of birds who 
fly particularly well. 

The negative-angle tail complies well with this require- 
ment and also compensates the lessening of the lever arm of 
the tail plane which ensues through its important increase 
in stability due to the increase in the longitudinal V- 

Moreover, by bringing the main and tail planes closer 
together, the longitudinal moment of inertia is reduced, 
whereby the machine is rendered more responsive to its 
controls. 

For these reasons, the author is of opinion that the 
present type of aeroplane with its tail far outstretched 
will give way to a machine at once much shorter, more 
compact, and easier to control.* 

Summarising our conclusions, we find that : 

(1) In regard to the relative action of gusts, which are 
the main cause of loss of equilibrium, an aeroplane should 
be as light as possible, so as to be able to yield in the greatest 
possible measure to the displacement effect of gusts, which 
reduces their relative effect. This conclusion is clearly 
open to question, and may be opposed by the illustration 
that large ships have less to fear from a storm than small 
boats. But the comparison is not exact, for the simple 
reason that boats are supported by static means, whereas 
aeroplanes are upheld in the air dynamically. 

* Not that it will be possible to suppress the tail entirely, as some 
have attempted to do. Oscillatory stability (see Chapter VII.) would 
suffer if this were done, and the braking effect would disappear. Besides, 
Nature would have made tailless birds, could these have dispensed with 
their tails. 



200 FLIGHT WITHOUT FORMULA 

(2) Regarding its behaviour in a wind, an aeroplane 
should : 

(a) possess high speed, with the proviso that its speed 
should not be obtained by reducing its lift co- 
efficient, so that any increase in speed should 
be achieved rather by reducing the area than 
the lift coefficient ; 
(6) be naturally stable longitudinally ; 

(c) have a small longitudinal moment of inertia ; 

(d) be short in the fore-and-aft dimension ; 

(e) be so designed that any initial displacement due 

to a gust causes it to turn head to the gust 
instead of exposing it still further to its dis- 
turbing effect. 

The negative tail arrangement seems to answer the 
most perfectly to (6), (c), (d), and (e). 

It has often been stated that those provisions ensuring 
stability in still air were harmful to stability in disturbed 
air. If this were true, the future of aviation would indeed 
be black. Fortunately it is erroneous, even though practice 
has borne it out hitherto with few exceptions. 

It has been attempted, as in the case of the brothers 
Wright, to overcome this difficulty by only providing the 
minimum degree of stability essential to the correct be- 
haviour of a machine in still air, leaving the pilot to make 
the necessary corrections to counteract the disturbing 
effects of the wind by giving him exceptionally powerful 
means of control. 

The slight degree of natural stability possessed by such 
an aeroplane renders it most responsive to its controls a 
feature agreeable to the majority of pilots. On the other 
hand, by actuating the control a pilot may unduly modify, 
even to a dangerous extent, the normal state of equilibrium. 
More especially is this true of longitudinal equilibrium, 
for here, as has been shown, a slight degree of stability 
may change into actual instability for instance, by putting 
the elevator down too far. This is due (as explained in 



THE EFFECT OF WIND ON AEROPLANES 201 

Chapters VI. and VII.) to the fact that the sheaf of total 
pressures of the aeroplane is thereby altered, with the result 
that the longitudinal V is diminished, and consequently 
the diminution of the angle of incidence, instead of increas- 
ing stability, as in the case of advancing the centre of 
gravity, would bring it down to vanishing-point. 

Aeroplanes which display a tendency towards uncontrol- 
lable dives, are simply momentarily unstable longitudinally 
and refuse to answer the pilot's controls because, owing 
to their acceleration, their dive soon becomes a headlong 
fall, so that the precarious degree of stability which they 
possessed in normal flight has disappeared. In such a case 
it would be incorrect to say that an increase of speed 
augments stability, for, on the contrary, when the speed 
passes a certain limit termed the " critical speed " (in the 
author's opinion, this term is not correct, since a well- 
designed aeroplane should have no critical speed), all stability 
vanishes. 

An aeroplane should always be so designed as to be 
naturally stable in still air, and at the same time every 
effort should be made to arrange its structure so as to 
render it stable also in disturbed air. 

It has already been shown that it seems possible, in 
regard to longitudinal stability, to achieve this result with- 
out sacrificing controllability, which would appear to be 
dependent, above all, on a small moment of inertia. 

Whether the negative tail arrangement, previously re- 
ferred to, or some other similar device should prove the 
better in the long run, this for the time being is the right 
road along which to make endeavours and to try to reduce 
to the lowest possible degree the intervention of the pilot 
in controlling the stability of an aeroplane. The whole 
future of aviation is bound up in the solution of this problem. 
An aeroplane should be able to fly in the worst weather 
without demanding from its pilot an incessant, tiring, and 
often dangerous struggle against the elements. Not until 
this is achieved will aviation cease to be the sport of the 



202 FLIGHT WITHOUT FORMULA 

few and become a speedy and practical, and above all, safe, 
means of locomotion. 

It has ere now been sought to reduce the necessity for 
constant control on the part of the pilot by rendering 
aeroplanes automatically stable. The problem is an un- 
usually complex one, for automatic stability devices are 
required to correct not only the effects of gusts that come 
from without, but faults that arise from within the aero- 
plane itself, such as a loss of power, motor failure, mis- 
takes in piloting, etc. 

This being so, if a device of this nature fulfils one part' 
of its required functions, almost inevitably it will fail in 
others, and this is the rock against which all attempts so 
far made have been shattered. Not that the difficulty 
cannot be overcome, but it is undoubtedly a grave one. 

Hitherto such attempts at solution as have been made 
have usually related to longitudinal stability. Among such 
devices may be mentioned the ingenious invention of 
M. Doutre, who utilised the effects of inertia exerted on 
weights to actuate, at any change of air speed, the elevator 
through the intermediary of a servo-motor. 

Even now some lessons may be drawn from previous 
attempts. More especially would it seem desirable to 
prevent the effects of gusts rather than to correct them 
once they have been produced. The use of " antenna " or 
" feelers " that is, of some kind of organ instantaneously 
yielding to aerial disturbances and thus preparing, through 
the intermediary of the requisite controls, an aeroplane to 
meet the gust would seem preferable to organs which only 
right it once it has assumed an inclined position after 
having been struck by the gust. 

Important results, in this respect, also appear to have 
been obtained by M. Moreau, who seems to have succeeded 
in applying the principle of the pendulum to produce a 
self-righting device. 

In addition it has been sought to ensure automatic 
stability by constantly maintaining the air speed of an 



THE EFFECT OF WIND ON AEROPLANES 203 

aeroplane. But it has already been shown that this is 
inadequate in certain circumstances, more especially if the 
aeroplane has a small lift coefficient, which is the case with 
machines of large wing area, and the lift often decreases 
to a far greater extent as the result of a decrease hi the 
relative incidence than in the speed. Hence, not only the 
relative speed, but the relative incidence should be preserved. 

In regard to the effects of wind alone, therefore, the 
problem is already complicated enough ; but it becomes 
even more complex if disturbances due to the machine 
itself are taken into consideration. 

Without the slightest wish to deny the great importance 
of the problem, the author, nevertheless, reiterates his 
opinion that the first necessity is to so design the structure 
of an aeroplane as to render it immune from dangers through 
wind. Later, an automatic stability device could be 
added in order to correct in just proportion the effects of 
gusts and further to correct disturbances due to the machine 
itself. If this were done, the functions of automatic stability 
devices would be greatly simplified. 

In addition, it should not be forgotten that by adding to 
an aeroplane further moving organs which are consequently 
subject to lagging and even to breakdowns, an element of 
danger is created. In any event, any such device must 
perforce constitute a complication.* 

Until now only those gusts have been considered which 
blow in the plane of symmetry of the aeroplane straight 
gusts which only affect the flying and longitudinal equili- 

* Virtually, this stricture, while perfectly correct in itself, only applies 
to such extraneous stability devices as those of Doutre and Moreau, and 
not to the automatic stability inherent in the forms of the aeroplane 
itself which has been produced by J. W. Dunne. This latter possesses 
automatic stability in both senses, and in principle is based on the auto- 
matic maintenance of air speed without the pilot's intervention. In 
this respect it undoubtedly constitutes one of the greatest advances yet 
made in aviation, though opinions may well differ on the point whether 
it is desirable to rob the pilot of control in order to confide it to automatic 
mechanism. TRANSLATOR. 



204 



FLIGHT WITHOUT FORMULA 



brium of the machine. Let us now examine the effect 
of side-gusts. By doing so, we shall have considered 
the effect of almost every variety of aerial disturbance, 
which can in most cases be resolved into an action directed 
in the plane of symmetry of the aeroplane and into one 
acting laterally. 

In this case again we distinguish a primary effect and a 
relative effect. 

If the aeroplane had no inertia, it would immediately be 




FIG. 84. 

carried away by the gust together with the mass of sup- 
porting air, and this movement would not be perceptible 
to the pilot except by observing the ground beneath. But 
this is a purely hypothetical case, and the gust exerts 
a relative action on the machine, which is the more pro- 
nounced the greater the mass of the latter. 

This action is perceptible by a modification in the 
direction and speed of the relative wind. For, if the 
aeroplane were flying in still air, thereby encountering a 
relative wind GA (fig. 84), and were struck by a lateral 
gust whose action is represented by the speed GB, the 
relative speed of the machine becomes GC. Both the 



THE EFFECT OF WIND ON AEROPLANES 205 

magnitude and direction of the speed have, consequently, 
altered. 

The fact of the relative speed varying in magnitude 
shows that, apart from effects due to its dissymmetrical 
position, the relative action of a side-gust must exert on 
the flying and longitudinal equilibrium an influence similar 
to that produced by the straight gusts already considered. 

A lateral gust, therefore, can cause an aeroplane to rise or 
fall at the same tune that it disturbs its longitudinal equili- 
brium. But for the sake of simplicity this part of the 
effects of side-gusts may be ignored, and only those effects 
need be taken into account which modify the direction 
of flight, and lateral and directional stability. 

First, the displacement effect due to the relative action of 
a side-gust consists in creating a centripetal force tending 
to curve the flight-path and to produce a turn in the direction 
opposite to that from which the gust comes. 

Among the rotary effects, as in the case of straight gusts, 
that particular one should first be distinguished which 
causes an aeroplane, in regard to directional equilibrium, 
to adhere to its flight-path or, in other words, to behave 
like a good weathercock. 

If the flight -path curves, as the result of the displacement 
effect of a gust, in the opposite direction to that from which 
the gust comes, the rotary effect which will tend to make 
the aeroplane adhere to its new flight-path will cause it to 
be exposed still further to the disturbing effect of the gust. 
It will turn away from the wind. So far as this point is 
concerned, it would seem desirable that an aeroplane should 
take up its new flight-path as slowly as possible. 

But a second rotary effect causes the aeroplane to assume 
the new direction of the relative wind, like a good weather- 
cock, and this is an advantage, since, by heading into the 
wind, the lateral disturbing effect of the gust is damped out. 

Of these two rotary effects the second is probably the 
first to occur and to remain the more intense. 

In order to reduce the first rotary effect, the lateral 



206 FLIGHT WITHOUT FORMULA 

resistance of the aeroplane that is, its keel surface should 
not exceed certain proportions. Moreover, the directional 
stability should also be reduced to a minimum from this 
point of view ; the second and more important rotary effect, 
on the other hand, points to an increase in directional 
stability as desirable. 

Both theories have their friends and foes, and here again 
the view has been advanced that the aeroplane should be 
given only that measure of stability which is strictly 
necessary in order to prevent it from yielding too easily 
to the rotary effects of gusts and to render it easily 
controllable. Such a reduction in directional stability 
is not so detrimental as a diminution of longitudinal 
stability, since it in no way affects the cardinal principles 
of sustentation. 

Nevertheless, in the author's opinion a definite degree of 
directional stability is desirable, since this would also 
produce some amount of lateral stability which is always 
somewhat defective. In any case, usually the structure of 
the aeroplane and the rudder in the rear suffice for the 
purpose. 

There remain the most important rotary effects due to 
side-gusts those which affect lateral stability. 

Any modification in the direction of the relative wind 
results in a lateral displacement of the normal pressure on 
the main planes, which causes a couple tending to tilt the 
aeroplane sideways. If that wing which is struck by the 
gust rises, the aeroplane will turn into the opposite direction, 
thus turning away from the wind, and thereby, as already 
seen, exposes itself still further to the disturbing effect of 
the gust. 

But if the wing struck by the gust falls, the aeroplane 
swings round, heading into the wind, which damps out the 
disturbing effect. These movements are intensified by 
reason of the gust not striking both wings at once. 

According to the principle already cited, the initial 
displacement due to a gust should cause an aeroplane to 



THE EFFECT OF WIND ON AEROPLANES 207 

turn into the wind instead of causing it to become exposed 
to the disturbing influence still further, which renders the 
second rotary effect the more favourable. 

If the wings are straight and, still more, if they have a 
lateral dihedral or V, the first effect is produced. Hence 
a lateral dihedral seems unfavourable in disturbed air. 
Besides, it is fast disappearing, and pilots of such machines 
are obliged to counteract the effects of gusts by lowering 
the wing struck first that is, of momentarily suppressing, 
as far as is in their power, the lateral dihedral, while swinging 
round into the wind. 

On the other hand, if the wings have an invertefl dihedral 
or A> the rotary effect of a side-gust will be the second 
and desirable effect ; the aeroplane will turn into the 
wind of its own accord, which will cause the disturbing 
effect to disappear. 

Captain Ferber from the very first pointed out this fact 
and remarked that sea-birds only succeeded in gliding in 
a gale because they placed their wings so as to form an 
inverted dihedral angle. But he also thought that these 
birds could only assume this attitude, believed by him to 
be unstable, by constant balancing. In Chapters VIII. and 
IX. it was shown that it is possible, by lower ing the rolling 
axis of an aeroplane in front (by lowering the centre of 
gravity, or better, by raising the tail), to build machines 
with wings forming a downward dihedral and nevertheless 
stable in still air.* 

In regard to lateral stability, as with longitudinal, the 
natural stability of an aeroplane and good behaviour in a 
wind are, contrary to general opinion, in no wise incom- 
patible, and both these important qualities can be obtained 
in one and the same machine by a suitable arrangement of 
its parts. 

* As previously mentioned, the " Tubavion " monoplane has flown 
with its wings so arranged, and the pilot is stated to have noted a great 
improvement in its behaviour in winds. This machine had a low centre 
of gravity and a high tail. 



208 FLIGHT WITHOUT FORMULA 

Attempts to produce automatic lateral stabilisers have 
hitherto not given very good results.* 

So far as the moment of rolling inertia is concerned, 
previous considerations point to the desirability of reducing 
this as much as possible by the concentration of masses. 
The machine is thus rendered easily controllable, and the 
rapidity of its oscillations guards against the danger aris- 
ing from too quick a succession of two gusts. This is 
of exceptional importance from the point of view of lateral 
stability, which we know to be the least effective of all or, 
at any rate, the most difficult to obtain in any marked 



Summarising these conclusions, it may be stated, that 
for good behaviour in winds, an aeroplane should : 

(1) be light, thus yielding more readily to the primary 

effect of gusts, whereby it is not so much affected by 
their relative action ; only if this relative action 
could be wholly eliminated would an increase in the 
weight become an advantage ; 

(2) fly normally at high speed, provided that an increase 

in speed be not obtained by unduly reducing the lift 
coefficient ; 

(3) be naturally stable both longitudinally and laterally ; 

(4) have a small moment of inertia and its masses con- 

centrated ; 

(5) head into the wind instead of turning away from it. 
The fulfilment of the last condition is the most likely to 

produce the best results in regard to the behaviour of an 
aeroplane in a wind, and this has been shown to be in 
no way incompatible with excellent stability in still air 
and adequate controllability. The arrangement proposed 
by the author a negative-angle tail and a downward 

* This is hardly correct so far as the Dunne aeroplane is concerned, 
which is automatically stable in a wind. This machine, it should be 
noted, has in effect a downward dihedral and a comparatively low centre 
of gravity, coupled with a relatively high tail which is constituted by the 
wing-tips. TRANSLATOR. 



THE EFFECT OF WIND ON AEROPLANES 209 

dihedral* is not perhaps that which careful experiment 
methodically pursued would finally cause to be adopted ; 
but at any rate it provides a good starting-point. 

What is required first of all is to so design the structure 
itself of the aeroplane as to render it immune to danger 
from gusts. The future of aviation depends upon this to 
a large extent, and it is for this reason that attention has 
been drawn to it with such insistence in these pages, for 
in this respect much, if not almost all, remains to be done. 

Afterwards, may come the study of movable organs 
producing automatic stability, and in all probability this 
study will have been greatly simplified if the first essential 
condition has been complied with. 

Who knows whether one day we shall not learn how to 
impress into our service, like the birds, that very internal 
work of the wind which now constitutes a source of danger 
and difficulty ? Some species of birds appear to know 
the secret of how to utilise the external energy of the 
movements of the atmosphere and to remain aloft hi the 
air for hours at a time without expending the slightest 
muscular effort. 

It is certain that for this purpose they make use of ascend- 
ing currents, but it is difficult to believe that these currents 
are sufficiently permanent to explain the mode of soaring 
flight alluded to. 

More probable is it that birds which practise soaring flight 
be it noted that they are all large birds, and consequently 
possessing considerable inertia meeting a head gust, give 
their wings a large angle of incidence and thus rise upon 
the gust, and then glide down at a very flat angle in the 
ensuing lull. 

Even in our latitudes certain big birds of prey, such as 
the buzzard, rise up into the air continuously, without any 
motion of then- wings, but always circling, when the wind 

* This arrangement was first proposed by the author in a paper con- 
tributed to the Academic des Sciences on March 25, 1911 (Comples Bendus, 
voL clii. p. 1295). 

14 



210 FLIGHT WITHOUT FORMULAE 

is strong enough. This circling appears essential, and may 
possibly be explained on the supposition that the circling 
speed is in some way connected with the rhythmic wave- 
like pulsations of the atmosphere in such a fashion that 
these pulsations, whether increasing or diminishing, are 
always met by the bird as increasing pulsations, and on 
this account it circles. 

It appears in no way impossible that we should one day 
be able to imitate the birds and to remain, without expend- 
ing power, in the air on such days when the intensity of 
atmospheric movements, an inexhaustible supply of power, 
is sufficient for the purpose. 

One thing is to be remembered : wind, and probably 
irregular wind, is absolutely essential to enable such flight 
to be possible ; it would be an idle dream to hope to over- 
come the never-failing force of gravity without calling 
into play some external forces of energy, and on those 
days when this energy could not be derived from the wind, 
it would have to be supplied by the motor. 

But in any event this stage has not yet been reached, 
and before we attempt to harness the movements of the 
atmosphere they must no longer give cause for fear. To 
this end the aerial engineer must direct all his efforts for 
the present. 

The really high-speed aeroplane forms one solution, even 
though probably not the best, since such machines must 
always remain dangerous in proximity to the surface of 
the earth.* 

Without a doubt, a more perfect solution awaits us some- 
where, and the future will surely bring it forth into the 
light. On that day the aeroplane will become a practical 
means of locomotion. 

Let the wish that this day may come soon conclude this 

* Slowing up preparatory to alighting forms no solution to the difficulty, 
since the machine would lose those very advantages, conferred by its 
high speed, precisely at the moment when these were most needed, in the 
disturbed lower air. 



THE EFFECT OF WIND ON AEROPLANES 211 

work. Every effort has been made to render the chapters 
that have gone before as simple and as attractive as the 
subject, often it is to be feared somewhat dry, permitted. 

Not a single formula has been resorted to, and if the 
author has succeeded in his task of rendering the under- 
standing of his work possible with the simple aid of such 
knowledge as is acquired at school, this is mainly due to 
the distinguished research work which has lately furnished 
aeronautical science with a mass of valuable facts : to the 
work of M. Eiffel, to which reference has so often been 
made in the foregoing pages. 

No more fitting conclusion to these chapters could there- 
fore be devised than this slight tribute to the indefatigable 
zeal and the distinguished labours of this great scientific 
worker who has rendered this book possible. 



Printed by T. and A. CONSTABLE, Printers to His Majesty 
at the Edinburgh University Press, Scotland 



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