8 
half the angle turned through with plane grooves (by a pro- 
perty of the parabola) ; hence the work spent in friction 
with the parabolic grooves is double what it is with the 
plane grooves. This may be shown mathematically as 
follows : — 
I. To estimate the actual work spent in friction ivitli 
plane grooves. 
Let jjL — coefficient of friction. 
1 = the inclination of the grooves. 
K = work spent in friction. 
2 = the energy of discharge or the striking force with which 
the shot leaves the gun. 
Then, K = 
For if K - the mean pressure on the grooves, I = the length of 
the gun, then K = ^ R Z Jp + \ . . (i) 
A j ,, V ± R n 
And the energy oi rotation = -^ 2 = /- 
^ V + 1 
Jl^+ 1 
( 2 ) 
Hence (with a gun making one turn in 35 diameters) where 
. 1 . 
^ = pj- and fi = *3. 
K=|| = -0132. 
The equation shows, what is otherwise quite 
obvious, that with the plane grooves the work spent in 
friction is independent of the distribution of the pressure 
within the gun, and is proportional only to the energy of 
discharge ; and hence will be the same, whether the powder 
is quick or slow, provided the shot leave with the same 
velocities. 
This, however, is not the case with the parabolic grooves* 
It is obvious that the friction will involve the law of pres- 
sure in the gun. Consequently, we cannot calculate this 
work unless we make some assumption with regard to the 
law of pressure. 
