10 
where y is the distance of the shot from the breech, and 
a and X are constants for each class of powder. 
Although with this value of P the equations of motion 
cannot be solved rigorously, an approximate solution may 
be found as follows : — 
III. To find the ratio of luork spent in friction with 
parabolic and plane grooves luhen the law of 
force is 
p^_V. 
a-vy 
The equations of motion are 
^ dx 
P = P^+ . . 
as p 
( 1 ) 
( 2 ) 
Neglecting the ju R as small in (1), and taking p (the radius of 
curvature) = h, we have if ^ = 1 
/Pdy = \\og. 
a + y 
2 \ 
a + y 
or Kj, = fjiy^B>dy = ^^ {21 + a) log. - ^ | 
d X 
And for plane grooves p is infinite and ^ = 
K jit J^Bjdy = y\i log. 
Kfe . a 1 
log. (l + 0 
If « = 0 SO that P = - 
y 
K. „ 
K 
If a is very great, so that P = 
^_3 
K “2 
And the former law more nearly expresses the condition 
of most guns. 
