THE EOYAL ARTILLERY INSTITUTION. 
361 
Calling 
will 
it will be found more convenient to make the plane of xy pass through 
the point where the twist would be zero were the grooves sufficiently 
prolonged. Let the axis of x pass through one of the grooves ; and, 
for the sake of simplicity, we shall suppose the rifling to be given by 
one groove only. Let the axis of z be coincident with that of the gun; 
let AP (see Fig. 1) be the groove or curve des¬ 
cribed by the point P, and let P [x, y , z) be the 
point at which the resultant of all the pressures 
tending’ to produce rotation may be assumed to 
act at a given instant. Let the angle AON = <p. 
11. Now, the projectile in its passage through 
the bore is acted on by the following forces :— 
1st. The gaseous pressure G, the resultant of 
which acts along the axis of z. 
2nd. The pressure tending to produce rotation, 
this pressure P, and observing that it 
be exerted normally to the surface of the 
groove, we have for the resolved parts of this 
pressure along the co-ordinate axes, R cos A, R cos /*, 
and R cos v —A, [x, and v being the angles which 
the normal makes with the co-ordinate axes. 
3rd. The friction between the stud or rib of the projectile and the 
driving-surface of the groove. This force tends to retard the motion 
of the projectile ; its direction will be along the tangent to the curve 
which the point P describes. If /x x be the co-efficient of friction, and if 
a, P, y be the angles which the tangent makes with the co-ordinate axes, 
the resolved portions of this force are /x x P . cos a, /q R . cos p, /qP . cos y. 
12. Summing up these forces, the forces which act 
parallel to x are X — JR . {cos A — /q cos a}, 
„ V „ P= R . {cos /x — /q cos /?}, 
„ z „ Z— G + R . {cos v — /q cos y} ; 
and the equations of motion are 
M . — 2 = G + R {cos v — ^ cos y}, 
_ Tx — Xy 
’ dP p 3 ’ .. 
( 1 ) 
■( 2 ) 
,(3) 
p being the radius of gyration. Equations (1), (2), and (3) are identical 
with those I formerly gave. 
13. Now, in the case of a uniformly increasing twist, the equations 
to the curve which when developed on a plane surface is a parabola, 
may be put under the form 
x — r cos (j) ; y — r sin (j>; z 2 = h'ty .(4) 
Hence 
dx — — r sin 0 . d(j> ; dy — r cos 0 . d§\ 
