9 
II. To estimate the actual work spent in friction with 
parabolic grooves when the pressure on the studs is 
constant. 
Let x = -^hc the equation to the developed grooves, and let s be the 
d 'll 
length of the grooves. Then, if we assume that / = 1, and that 
as 
(the work spent in friction with the parobolic grooves) 1 - 
we have the work of rotation 
dx 
I 
And the work of rotation 
2hfi 
K; 
Since i = 
Kj, — ^ '2/ S, 
And for plane grooves K == — 
K& 
i 2 
K 
- 2 . 
An expression for this work might have been obtained 
without assuming ^ = I, but so long as i is less than ^ 
the difference is very small. 
Hence we see that on this assumption the work spent in 
friction with the parabolic grooves is twice as great as with 
the plane grooves. This assumption is not an unreasonable 
one, for the declared object of the increasing twist is that it 
may equalise the pressure of the studs on the grooves 
throughout the gun. However, it is not to be supposed 
that this object is always attained, for one kind of powder 
has a different law of force from another. It is necessary 
therefore to consider other laws of force. We cannot obtain 
a general expression which y^ill include all, but we may ex- 
amine several laws of force which will enable us to see how 
far the law of force affects the results. 
In all cases the force diminishes from the breech to the 
muzzle, and the law inay be roughly expressed by P — — ^ 
y 
