412 
Now 
MINUTES OF PROCEEDINGS OF 
fdF\ . z fdF\ z 
(%J“ C0S ^’ 
(dF\ \ (x z , y . 2 \ 
UJ =-^U C ° S /7r + ? Sm /7,j ; 
but since in the case we are now considering (#,y, 2 ) is a point both in the 
surface given by equation (7) and in the directing helix, we have from (4), 
x , z y . , . z 
- = cos (b = cos — , - = sm d> = sin ; 
hr r hr 
therefore (§) = -p 
- {©’♦ O’* (f 
VP= I 
Hence 
) 
* $ sin <f> 
COS A =- v-^ r y , 
Vl + ^ 
h cos d> 
cos tx = y , 
Vl4-£ 2 
Vl +^ 2 . 
Vl + / 
.(9) 
Now substituting the values of the direction cosines given in equations 
(5) and (9), in (1), (2), and (8), we have as the equations of motion. 
d^(f> Fr h — fx 1 ' 
W ~ Vl + $' Mp 2 * 
( 10 ) 
( 11 ) 
and hence the normal pressure on the rib of the projectile, 
r ’ h —• [x x dft' 
But if ®r be the angular velocity of the projectile, and h be the pitch of the 
rifling, we have the following relation between the velocities of translation 
and rotation, 
_ dcf) 27r 27r dz 
dt ~~ h h * dt' 
d 2 (f> 27r d^z 
Hence 
