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 work spent in friction tuith 
parabolic and grooves luhen the laiv of 
force is 
X 
P- 
a+ij 
The equations of motion are 
(1) 
( 2 ) 
'ds^'' ^ ^ ds 
_ dx ir 
P = P^ + 
d8 p 
Neglecting the ju R as small in (1), and taking p (the radius of 
curvature) = 6, we have if = 1 
/R<^y-=/p|rf^ + ^log. “ + ^ 
orKj = iy^R(^ 2 / = ^| (2Z + a) log. 
a 
(X 1 - ( 
a / 
d X 
And for plane grooves p is infinite and prp-i 
'll 
/I ... I Cl 
.*. / ^dy = fiXi log. 
1 
a 
^‘ = 2 + -- 
K I 
log. (l + 0 
If a = 0 so that P = - 
. y 
^-^=2. 
K 
If a is very great, so that P = - 
ct 
h 
K 
3 
2 
And the former law more nearly expresses the condition 
of most guns. 
