1904.] The Longitudinal Stability of Aerial Gliders. 105 



where Si is the area, 2«i the breadth of the lamina, £1,^ the co- 

 ordinates of its centre, Vi the resultant velocity of its centre, 

 oil = tan -1 ui/vi the angle which the direction of motion makes with 

 the axis of y, ft the angle which the plane of the lamina makes with 

 the axis of y, so that (aj - ft) is the angle between the plane and 

 the direction of motion of the centre of mass, and finally, 

 lh = Vi cos ft + & sin ft. 



The summation is to be extended over all the planes of the 

 system. 



If, now, u, v be the component velocities of the centre of gravity 

 the velocities ui, V\ of the centre of the lamina will be given by 



Ul = u-7 h o, n = v+£id, 



and for steady motion, since = 0, U\ = u, V\ = v. 



We may therefore write the expressions for the forces in the form 



mX = 2KSi («*i 2 + ^^/(tan-^i/^i - ft) cos ft 



= 2ESi (u 2 + v*- 2u ni 6 + 2*&0)/( tan" 1 ( u ~^ \ -ftjcosft, 



I + J 



neglecting terms in & 2 . 

 Similarly, 



mY = 2KSi (# + - 2itift0 + 2^0) / ( tan" 1 _ ft \ sin ft , 



mG = - SKSj (u 9 ~ + 1- 2 - 2^ + 2«£i0) / ( tan- 1 fc^if) - ft ) 



Differentiating these expressions with respect to u, v, 6 we have : — 

 mX u = 2 [KSi (2u - f( ai - ft) cos ft + KS^/' (a x - ft) cos ft] ; 

 or, since 9 = in the steady motion, 



mX tt = 2 [2KSii*/(«i - ft) cos ft + KS^/' (*i - ft) cos ft]. . . . (8), 



Similarly, 



mX, = 2 [2KSx«/(«i - ft) cos ft - KS^/'^i - ft) cos ft]. 

 mY, = 2 [2KSi%/( ai - ft) sin ft + KS^/' ( ai - ft) sin ft], 

 mY v = 2 [2KSi*/(«i - ft) sin ft - KS^/'fo - ft) sin ft], 

 mX« = 2 [2KSi ( - w n + tft) /( ai - ft) cos ft 



- KSi (vr n + ?^"i)/'( a l - ft) C0S ft]' 



