Motion of a Spinning Projectile, 337 



Again replacing V by u, we get 



da, _g 



dt it 



(6) 



Dividing corresponding sides of equations (2) and (6) we 

 get 



du u 3 {n 



Consequently, 



«=IG *} (*> 



that is, 



— *&-h) (9) 



But since X is ^always small, it is approximately equal to 



dY 

 tan X, which is -^ . Therefore, 



dX ■ ■ , 7 /l 1\ 



_=tanX -^- 2 -_j 



2X 



= tanA °~2v( eT - 1 ) b ^ (5) ' 



Integrating this, and adjusting the constant so that Y = 

 when X = 0, we get 



2X 



Y = x(ta»X +^)-g 9 (/-l).. . (10) 



The horizontal range of the shot fired at elevation X is the 

 value of X that satisfies the equation 



2X 



0= X (tanVH | $)-gp-l).. . (11; 



But the value of X that satisfies (11) is the value of X at 

 the intersection of the line 



Y = — Xtan Ao 



and the curve 



which is the ourve described by a shot projected horizontally. 

 The ordinate in (12) is negative for all positive values of X. 



