THE ROYAL ARTILLERY INSTITUTION. 
413 
and 
£ _ Mp 2 V1 + j$ 2tt d?Z- 
r Jc — p^ 1 h dt 2 
( 12 ) 
Now substituting in this equation the value of -^-derived from (10), we have 
or 
P = 
__2t rp 2 Vl + ff* 
rh 
{G- 
R 
Vl +k* 
-f 1)}, 
R 2^ Vl + k 2 
G 7ir(k — fxj + 2 Trp 2 (jjl + 1) 
(13) 
And this equation gives the ratio between the pressures producing translation 
and rotation. 
"We now proceed to determine the increment of the gaseous pressure 
due to the resistance, &c. offered by the rifling to the forward motion of the 
shot. We shall imagine a smooth-bored gun to fire a shot of the same 
weight as that of the rifled gun. We shall further suppose that the two 
projectiles are delivered with the same velocity; and we wish to know, the 
same ballistic effect being produced by the two guns, what is the increased 
pressure which the rifled gun has had to sustain. Now the equation of 
motion in the case of the smooth-bored gun is 
.(14) 
and in the case of the rifled gun, 
M m — G '— = 
dt 3 + 
0*1*+1). (15) 
Now if the velocity-increments in the two cases be taken as equal, we 
shall have from equations (14) and (15), 
G'—G+ —i=== (jije + 1).(16) 
VI + h 2 
And the second term of the right-hand member of equation (16) represents 
the increment of pressure due to the rifling. 
Let us now examine the pressures which subsist when a polygonal form of 
rifling is adopted; and we shall suppose the polygon to have n sides. The 
equations of motion given in equations (2) and (3) hold here as in the last 
case, and the values of a, /?, y given in (5) remain the same. The driving- 
surface is, however, different, being traced out by a straight line which 
always remains parallel to the plane of xy, meets the helix described by P, 
and touches the cylinder whose radius is —r cos ^ (see fig, 3, where PA 
