In “Practical Aerodynamics,” Major B. Baden–Powell quotes the following; formulae for the pressure of the air on a moving plane when expressed as a function of the angle of attack α, taking the pressure as unity when α=90°.

I. Duchemin

II. Rayleigh (hydrodynamical)

IIa. Gerlach

III. Renard

IV. Soreau

V. Eiffel, 1907,

VI. Another formula frequently quoted is

and a more complicated formula, depending on the aspect ratio of the plane, has been proposed by Soreau.

The present investigation fills a gap in the applications of the formal methods of rigid dynamics to the study of the motions of aeroplanes.

The principal methods for the approximate solution of dynamical problems together with their aeronautical applications may perhaps be conveniently classified as follows:—

It will be seen that this order of development, which was contemplated by me long before the realisation of artificial flight, was that required at that time to answer the questions, whether an aeroplane could fly without upsetting, how it could be steered in a curve, and how it could be adapted for use in boisterous weather.

In the stability investigations which the late Captain Ferber published in the Revue d'Artillerie, the sustaining and other surfaces of an aeroplane were in certain cases taken to be represented, for dynamical purposes, by a system of three plane resisting laminæ fixed mutually at right angles. Unfortunately, however, such a system cannot in general be made equivalent to a collection of surfaces, such as those of an aeroplane, with the result that Captain Ferber's investigation failed to give the correct conditions of lateral stability. At the same time, Ferber's system of three orthogonal planes is so convenient, especially for forming a general idea of the effects of wind gusts on an aeroplane, that it is desirable to investigate conditions and limitations under which such a representation is valid. The desirability of a further investigation of the forces and couples acting on a system of resisting surfaces of a general character was foreshadowed in “Stability in Aviation,” and a more detailed discussion of the problem has now become necessary in order to prepare the way for further studies in the rigid dynamics of the motions of aeroplanes or of systems resembling them.

The solution of Laplace's well known formula giving the relation between the height above datum level and the pressure and temperature of the atmosphere, may be rapidly accomplished by means of a d'Ocagne alignment chart, an example of which is shown in Fig. i. Its use may be best explained by means of an example. Let the initial temperature (t0) and pressure (p0) be 20° C. and 29 inches of mercury, and at a certain height (z feet) suppose they have fallen 10 temperature (t) 0° C. and pressure (p) 20 inches. The ratio of the pressures and the mean temperature Place a straight edge across the chart so as to cut the right hand (R) scale at 1.45 and the left hand (T) scale at 10. The cut of the middle (z) scale gives the required altitude, namely, 10,000 feet.

Consider any form of aircraft provided with a small rudder plane placed behind its centre of gravity and at a certain distance l from it, which we may speak of as the length of the tail. When this tail is subjected to a given wind pressure, the moment of this pressure about the centre of gravity, tending to turn the machine round, is proportional to l. From this point of view a long tail is preferable to a short one.

On the other hand, the pressure of the air on the tail plane depends on the normal velocity of the wind relative to it. Hence the normal velocity of the plane itself is limited in magnitude, and the angular velocity with which the machine turns about its centre of gravity must therefore be less when l is large than when lis small; that is, the long–tailed machine turns more slowly than the one with a short tail, and this is a disadvantage.

Some time ago when leaving my laboratory I happened to notice a large number of cheels gliding overhead that seemed to me to show something unusual in their mode of flight. I guessed that they were after locusts, a flight of these insects having passed over Agra three days previously. On reaching my house I found, with the help of my binocular, that my conjecture was correct. The following are the notes that I took at the time:—

IIth August, 1915, 3.0 p.m. A few small thin clouds. Air rather hazy at a height. Wind S, occasionally moving small branches.

A large number of cheels were gliding in a way that seemed to me unusual. They were at heights between 200 and 400 metres. They were generally slow–flex–gliding and did so in all directions and with frequently changing course. Two were seen rapidly losing height, feet foremost. Occasionally they dived or at least glided steeply downwards at speed.

The first direct statement of the problem of controlling an aeroplane, without the permanent exercise of physical effort by the pilot, appears in this paper. During the state of transition from straight flying to flying in a circle, a pilot exerts a force to pull over the rudder and in the usual case it is necessary to maintain some force in order to make a continuous turn. Professor Bryan proposes the design of an aeroplane such that turning having been started, maintenance of turning is obtained without any effort being necessary to keep the rudder in position. So far as the pilot is concerned the same result would be obtained by balancing the control surfaces, but the effect on the aeroplane is quite different. In the case of balanced controls a lateral force is still exerted by the rudder and so stresses are introduced in the tail structure of the aeroplane ; on the other hand, in an inherently controllable aeroplane, there is no lateral force on the rudder when turning and, therefore, no stresses in the tail.

Engineers have, until recent years, fought shy of anything relating to aerial navigation. Those who ventured, in spite of the odium attached to that study, to look into it at all, became very soon satisfied that the great obstacle in the way was the lack of a motor sufficiently light to sustain its weight and that of an aeroplane, upon the air. Fifteen years ago the lightest steam motor was the marine engine weighing 60 pounds to the H.P., while the gas engine weighed very much more; the locomotive weighed 200 pounds per H.P. During the past fifteen years a great change has taken place. Steam motors have been produced weighing only 10 pounds per horse–power, and gas engines have been lightened down to 12½ to 15 pounds per horse–power, so that the status, so far as engineers are concerned, is very greatly changed, and there is some hope that, for some limited purposes at least, man will eventually be able to fly through the air.

The investigation described below was undertaken in order to obtain a more accurate knowledge than can be obtained from any published results as to how the resisting torque due to the blades of a fan dynamometer varies with their shape and size, with the diameter of the fan, and with the density of the fluid in which the blades rotate.