410 
MINUTES OE PROCEEDINGS OF 
Take (fig. 2) as the plane of xy> the plane passing through the commence¬ 
ment of the rifling at right angles to the axis of the gun. Let the axis of x 
pass through the groove under consideration, and let the axis of z be that of 
the gun. Let AP be the helix, and let (see figs. 1 and 2) P(x,y, z) be the 
point at which the resultant of all the pressures on the groove may be 
assumed to act, the projectile being in a given position. Let the angle 
AON=<t>. 
Let us now consider the forces which act upon 
the projectile. We have, firstly, the gaseous 
pressure acting on the base of the shot. Let us 
call this force, the resultant of which acts along 
the axis of z, G. Secondly, if R be the pressure 
between the projectile and the groove at the point 
P, this pressure will be exerted normally to the 
surface of the groove, and if we denote by X } //,, v 
the angles which the normal makes with the 
co-ordinate axes, the resolved parts of this force 
will be 
R cos X, R cos R cos v. 
Thirdly, if be the coefficient of friction between 
the rib of the projectile and the driving-surface, 
the force ^R will tend to retard the motion of 
the projectile. This force will act along the 
tangent to the helix which the point P describes; x 
and if a, ft y be the angles which the tangent makes with the co-ordinate 
axes, we have as the resolved portions of this force pc^R cos a, p x R cos ft 
fx x R cos y; and summing up these forces, we have the forces which act 
parallel to x — X — R (cos X — /x x cos a), 
„ y — Y — R (cos /z — /x x cos ft, > .(1) 
„ z = Z — G + R (cos v — cos y); ) 
and the equations of motion will be 
(fiz 
M— — G + R (cos v — /z x cos y), . (2) 
Fig. 2. 
__ rg-Xy 
dP “ M P * 5 ... w 
p being the radius of gyration. 
We proceed to determine the value of the angles a, ft y, X } /z, v. Let the 
equations to the helix described by the point P be put under the form 
x = r cos <p, y — r sin <£, z = .(4) 
h being the tangent of the angle at which the helix is inclined to the plane 
of xy. Then 
dx = — r sin dy = r cos (f)d(f), dz = krd<f> } 
ds = r V1 + k 2 . d(j) } 
