362 
MINUTES OF PROCEEDINGS OF 
and we have, to determine the angles which the tangent to the curve 
described by P makes with the co-ordinate axes, the equations 
cos a = — = 
dx —2 z. 
sin^i 
*/\z 2 + k 2 * 
0 dy 2z . cos 6 
cos fi = -f = ..■■■■. " , 
ds Vkf+M 
dz k 
cos y = — = _ . 
ds 4z 2 + k 2 
(5) 
14. In the Woolwich guns the driving-surface of the groove may 
be taken, without sensible error, as the simpler form of surface where 
the normal to the driving-surface is perpendicular to the radius, the 
surface itself being generated by that radius of the bore which, passing 
perpendicularly through the axis of z, meets the curve described by the 
point P ; but in the first instance I shall examine the more general 
• case, where the normal makes any assigned angle with the radius. 
Assume, then, that on the plane of xy the normal makes an angle 8 
with the radius of the gun. The driving-surface of the groove is then 
swept out by a straight line which, always remaining parallel to the 
plane of xy , intersects the curve described by P, and touches the right 
cylinder whose axis is coincident with that of z, and whose radius 
— r . cos 8. 
Now, the equations to the director being given by (4), and that to 
the cylinder, which the generator always touches, being 
x* + y 2 = (r cos 8) 2 , . ...(6) 
it is easily shown that the co-ordinates x v y x of the point of contact of 
the tangent to the cylinder drawn from P parallel to the plane xy, are 
x 1 — r . cos 8 . cos (0 — S), 
y x = r . cos S . sin (0 — S) 
m 
and that the equation to the driving-surface is 
x .cos -< — 
{lr~ 8 } +2, ' Sin {fc- 8 } “ r - 
cos S. 
( 8 ) 
15. The angles which the normal to this surface makes with the 
co-ordinate axes are given by 
© 
cos X = 
VOMfMSf 
with similar expressions for cos ja and cos v. But 
//dF\* /dF\\ /dF\» 1 / - 
Vi*) + + = 
z 2 (sin fi) 2 + k 2 . 
