1904.] The Longitudinal Stability of Aerial Gliders. 101 



The angular velocity of rotation of the body will be ddjdt or 0. 



We shall use m to denote the mass of the body, mk 2 its moment of 

 inertia about the centre of gravity. 



Whatever be the law of resistance, the resistances of the air on the 

 several parts of the system will in general be functions of u, v and 0. 

 These resistances are always equivalent to two forces, which we shall 

 call mX and mY, along the axes, and a couple mG about the origin, so 

 that X, Y, G, denote the forces and couple, divided by the mass of 

 the body. 



The equations of motion of the body are — 



fdu dd\ a v 



m Hft _ v -tt ) = mc J cos v - mX, 

 \dt dtj J 



m [di +w dt) = -^^e-mY, 



(!)• 



In steady motion u, v and 9 are constant, and equations (1) give 



= g cos 6 - X, = - g sin - Y, 0= - G (2). 



Knowing the forms of the aeroplanes and other parts of the system 

 and the law of resistance, X, Y, G are known functions of v and 6. 

 Moreover, in steady motion, 6 = 0. * 



Equations (2) thus determine the values of u, v, 0, for steady 

 motion. 



Fluctuations about Steady Motion. — We must now examine what 

 happens when the system is slightly disturbed from its state of steady 

 motion. Let the disturbance be represented at time / by small 

 increases 8u, Sv, 80 in the values of u, v, 6, and let u , v Q , 9 be their 

 values in steady motion, so that in the disturbed motion, 



u = u + 8u, v = v + 8v, = Q + 86. 



Also let X , Y , G be the forces and couple in the steady state ; 

 then in the disturbed state we have, neglecting small quantities of the 

 second order, 



X = Xo + Su — + 8v d * + 86 d ^ (3), 



du dv dO 



and two similar equations for Y and G. 



We shall denote differential coefficients such as dX/du . . . dQ/d() 

 by X n ...G e . 



Substituting in the equations of motion, we obtain, to the first 

 order, 



