of the Trajectories of Shot. 273 



and its mean value must therefore be taken over a small arc. The tables 

 being such as they are, there can be no doubt that, if it were possible, the 

 most convenient independent variable would be the velocity itself. On 

 the other hand, all attempts at humouring the equations of motion so as 

 to introduce the velocity as independent variable are of no avail. "When 

 the trajectory is very flat, it may be possible to get results which are not 

 very objectionable; but no general solution is hereby attainable. The 

 forms of the equations, however, suggest as a possibly good substitute 

 the horizontal component of the velocity. Accordingly, taking this quan- 

 tity as independent variable, I now proceed to find the distance-integrals 

 by a method of approximation. 

 Since 



doc 



and 



du- m 

 — = — av n cos 

 dt r 



dx 1 m _, 1 

 — = — - cos n 1 ch : 



'p 



X = - 1 cos 



Similarly, 



l c» . 



- I si 



n-l 



„_2 , du 

 sin # cos* d> — 

 u n ~ L 



i 



§ 6. Our business now is to substitute for <f> its value in terms of u. 

 To enable us to do this, put <p = a — \p and expand in powers of \p : we 

 have 



cla 



L= ^^sec w+1 a^-^^ ) sec w + 1 atana^+ etc.Y (6) 



JL-JL 



u n p* 



Also 



cos n_1 ^=cos M_1 a-|-(n— 1) cos n-2 a sina^/ 



{(n-2) cos n - 3 a sin 2 a-cos' 1 " 1 a}^ 2 

 +etc 



VOL. XXVI. 



U 



