7 
ference of velocity was such as to make three per cent 
difference in the energy of discharge — a result somewhat 
greater than what would have been due to the legitimate 
friction, unless the coefficient of friction between the studs 
and the grooves was excessively high from some cause, such 
as the cutting of the studs into the grooves. However, it 
would seem that the conclusions at which I have arrived 
are in accordance with actual experience, and help to explain 
what was otherwise to a certain extent anomalous. 
Although these conclusions cannot be definitely proved 
without the aid of mathematics, they may be shown to be 
true (or reasonable) under certain circumstances, as follows ; 
The work spent in friction will, both with the parabolic 
and plane grooves, be equal to the coefficient of friction mul- 
tiplied by the mean pressure on the studs and again by 
the length of the grooves (or by the length of the 
gun — nearly). Now, the coefficient of friction and the 
length of the gun are the same in both cases ; hence this 
work will be proportional to the mean pressure on the 
grooves throughout the gun. Again, if the pressure on the 
parabolic grooves is constant (which it is the object of these 
grooves to make it), then the mean pressure in both cases 
will be inversely proportional to the angle which the shot 
turns through while in the gun. This follows directly from 
the fact that the speed and consequently the energy of 
rotation with which the shot leaves the gun is the same in 
both cases; for this energy is nearly equal to the mean 
pressure multiplied by the arc through which the studs 
turn,* and hence the mean pressure is equal to the energy 
divided by the arc. 
We have then the work spent in friction proportional to 
the mean pressure ; and the mean pressure inversely pro- 
portional to the angle turned through by the shot in the gun ; 
therefore the work spent in friction is inversely propor- 
tional to the angle turned through by the shot in the gun. 
Now, the angle turned tlirough with parabolic grooves is 
* This is always true for plane grooves, but it will only be true for para- 
bolic grooves when the pressure on the studs is constant all along the grooves. 
