12 



NATURE 



[March 3, 1910 



ever, to a property which I call the principle of in- 

 dependence of height, this distinction does not affect 

 stability to any appreciable extent. The important 

 point is whether the weight is sustained partly by the 

 front and partly by the rear planes, as in certain 

 Voisin machines, or is wholly supported by the front 

 planesj the rear ones acting merely as a tail in the 

 neutral position. For a monoplane with neutral tail 

 the condition of stability takes the form given by 

 Lanchester, when the necessary substitutions have been 

 made by making use of the condition of equilibrium. 

 The reason why Lanchester's method leads to a cor- 

 rect result is to be sought in considerations of the 

 peculiar nature of the oscillations, and in especial in 

 the relative smallness of their modulus of decay. For 

 a machine of the Voisin type, with sustaining surfaces 

 arranged tandem, the condition of stability is nearly 

 as simple, and certain modifications are sufficient to 

 cover the case when the propeller thrust does not 

 pass through the centre of gravity provided that this 

 thrust is constant. 



A very convenient plan in such cases is to suppose 

 the actual machine replaced by an equivalent mono- 

 plane, with neutral tail, although if the inclinations 

 of the planes be varied for vertical steering the 

 equivalent monoplane will be changed. 



The most remarkable result, however — and Mr. 

 Harper was the first to point this out to me — is the 

 important effect on stability of the direction of motion 

 in the vertical plane. Longitudinal stability falls off 

 rapidly when the aeroplane begins to rise, even if 

 other things are constant. A monoplane would, under 

 theoretical conditions, become unstable when ascend- 

 ing at an angle to the horizon of less than twice the ' 

 angle of attack (or inclination of the main plane to the 

 line of flight). 



The effect of head resistance is to increase the 

 stability, and a further increase occurs if the thrust 

 of the propeller, instead of being constant, decreases 

 when the velocity increases. By the use of three 

 planes instead of two, an additional increase of 

 stability can be obtained. On the other hand, if the 

 aeroplane be gliding downwards the longitudinal 

 stability is greater than in horizontal flight. 



I think the above conclusions indicate a source of 

 danger which may possibly have led to mishaps when 

 aeroplanes have risen too rapidly in the air. 



Captain Ferber's investigations, on the other hand, 

 refer mainly to the stability of a single aeroplane as 

 dependent on fluctuations in the position of the centre 

 of pressure consequent on variations of the angle of 

 attack. He assumes Joessels's formula, introducing 

 two arbitrary constants in place of the numerical 

 coefficients. The difficulty I have several times 

 pointed out is that, if a plane is turning over, its 

 rotational motion may affect the position of the centre 

 of pressure, as well as possibly the resultant thrust, 

 and no experimental information is apparently avail- 

 able on this point. For this reason the use of narrow 

 aeroplanes is to be recommended, stability being 

 secured by a tail or by two planes placed one behind 

 the other. Moreover, the theory of narrow 

 aeroplanes gliding at small angles affords the 

 simplest introduction to a general study of 

 aeroplane stability, just as geometrical optics 

 in which aberration is neglected affords an 

 introduction to a general study of lens construction. 

 It is to be remembered that both the symmetrical and 

 asymmetrical oscillations are determined by equations 

 of the fourth degree, each in the form of a determinant 

 of the third order containing the dynamical constants 

 and resistance coefficients, and when this determinant 

 has been expanded, four conditions of stability have 

 to be satisfied, one being that Routh's discriminant 

 NO. 2105, VOL. 83] 



BCD — .\D^ — EB^ shall be positive. Fortunately, for 

 purposes of approximation, CD — EB may be substi- 

 tuted for the last in many of the systems occurring in 

 aviation. It will thus be seen that stability is a very 

 complicated problem, and that approximate methods 

 are essential. 



Asymmetric stability is far more difficult of in- 

 vestigation than symmetric. It is necessary to take 

 account of the separate effects of straight or horizontal 

 aeroplanes, vertical fins, and bent-up or V-shaped 

 planes. The late Captain Ferber's solution is based 

 on the substitution for the actual planes of their pro- 

 jections on three coordinate planes (p. 46 of his paper). 

 Unfortunately, even assuming the sine law of resist- 

 ance, this substitution does not seem to give even the 

 correct first approximation which is all the author 

 claims. In particular, if the aerodrome is rotated 

 about any axis in its plane of symmetry, couples are 

 set up on the main aeroplane which have an im- 

 portant effect on the stability, but are appar- 

 ently not included in his scheme. The final result is 

 a biquadratic with one root equal to zero, and Captain 

 Ferber regards an aeroplane as stable when it describes 

 a helix ; whereas such an arrangement should reall\ 

 be regarded as lacking in stability. The couples in 

 question are taken account of by Lanchester, who 

 uses what he calls "aerodynamic and aerodromic 

 radii " to represent their effects. For a narrow aero- 

 plane gliding at a small angle, the effect depends on 

 the moment of inertia of the area of the plane about 

 the vertical plane of symmetry. A horizontal tail of 

 negligible lateral dimensions does not affect the asym- 

 metric stability. 



To secure stability, recourse must be had to vertical 

 fins, or to bent-up aeroplanes or aerofoils. 

 The effect of vertical fins (neglecting " wash ") depends 

 on their areas, and the first and second moments of 

 these about the axes, and in studying them it is 

 necessary to have recourse to the "principle of parallel 

 axes." The sections in "Lanchester" on "fin reso- 

 lution " practically embody this principle, but are a 

 little difficult to follow ; they suggest the path taken 

 by an explorer who had not a compass to guide him 

 to the mathematically direct road in the form of the 

 principle in question. His conditions of stability seem 

 reasonable deductions from the hypotheses he makes, 

 but the conclusions must not be regarded as final. 

 Both the necessary and the sufficient conditions of 

 stability are really far more complicated, and it 

 is highly improbable that the problem could have 

 been carried much further v^-ithout the elaborate 

 use of analysis which I have found necessary, and. 

 the assistance of an independent calculator, which Mr. 

 Harper has kindly provided. The only way of pro- 

 ceeding was to calculate the coefficients in the bi- 

 quadratic for particular arrangements of fins and 

 planes, starting with the simpler ones, and passing 

 to more complicated ones when one has become 

 thoroughly familiar with the different terms and their 

 meanings. 



For an aeroplane with one vertical fin only, the 

 conditions of asymmetric stability require that the 

 centre of pressure of the fin should be slightly in 

 front of the centre of gravity of the machine, while at 

 the same time it should be at a height above the 

 centre of gravity large compared with its distance 

 in front. Two of the conditions of stability are 

 difficult to reconcile wnth the conditions of equilibrium, 

 the difficulty increasing as the velocity increases and 

 the angle of attack diminishes ; moreover, they are 

 inconsistent unless a certain relation holds between 

 the radii of gyration of the machine and of the main 

 supporting surface. It is doubtful whether this con- 

 dition would be consistent with practical requirements. 



