Motion of a Spinning Projectile. 361 



Hence 



X ~\m*P-rf) 



1 + 



8(/-l)mT 1 



{ E 1 M« / - 1) / F ' +S) + BUff-W- 1 )^.-*]. , (72) 



(73) 



E x , E 2 , ¥ 1} and F 2 being four independent constants. 



57. On retracing the transformations we find that 



£=g^ xe «TT-i</+i>, . . . 



X having the value given by (72). On integrating this last 

 equation we should get two more constants (one real and 

 one imaginary), thus giving six independent constants in 

 the value of -^r. This is the full number of constants, for 

 equation (51) is of the third order in i/r, which contains two 

 variables e and 77, and we need three constants for each. 



58. If cf> is always sinall, then i/r is sure to remain small, 

 consequently z will remain small also, as equation (19) 

 shows. That is, we may be sure that the shot will be stable 

 with its nose foremost if we know that cj> is always small. 

 Stability will be assured, then, if we know that E x and E 2 

 are small and that the terms in the expression for 6 do not 

 greatly increase as T increases, the functions such as e imT 

 which have imaginary indices,, being treated as unity. Now 

 the largest term in </>, neglecting the factors with imaginary 

 indices, is 



( T VB,M*/- 1 >r-*tw) 



KmFP-r/J * 



Now, 



MT 



1 / rf 



= ™ + V m 2 -7j^' 



and the greatest value this can have is 2m, which occurs 

 when T = co, and its least value, on the assumption that 

 m 2 T*>r/, is m. Thus MT -1 does not greatly increase as T 

 increases, and the other factors involving T in (71) decrease 

 as T increases. If , then, m 2 T 2 >r/we may safely assert that 

 <f> remains small if it is small at the beginning, which is all 

 that is necessary for stability of the axis. 



