Motion of a Spinning/ Projectile. 353 



these angular deflexions are Ve and Ytj, which are approxi- 

 mately the same as ue and uij. Hence 



5«i»B(/-l)(«-«) (40) 



if 



w 



-J« J ? = E{(/-l)(y-,)+//3}. . . . (41) 



if 



41.^ Equations (38), (39), (40), and (41), together with 

 equation (2), namely 



du _ u 2 u 2 



_____ or -j-, 



and the expression for the resistance 



_, Wu 2 Wu 2 

 ±1 = r or — — - 



will give the values of a:, y, e, and r]. Since the mathe- 

 matics is just the same whether we use I or Z 1? we need only 

 consider one of them. We shall therefore use only I in the 

 following investigation. 



42. Writing T for -we get 



a 2 



x- T =l, 



so that 



dT=dt, 



the only difference between T and t being the instant from 

 which they are measured. Differentiations with respect to 

 t are therefore the same as differentiations with respect to 

 T. We shall use T as variable instead of t. 



43. From equation (6) we have 



dot _ du __ </,„ 

 dT ~ dt ~~l ' 

 Consequently, 



d 2 a _ g 



44. Now, rewriting equations (38) to (41) in terms of T 

 instead of u, we get 





u 





dT _ 



dt 



dT 

 du 



du I 

 di ~~ u 2 



