give Rotation to Rifled Projectiles. 207 



equations to the curve which when developed on a plane surface 

 is a parabola may be put under the form 



sc = r cos <p; y=rsin<jb; z 2 =kr<p.~ . . (4) 



Hence 



dx = — r sin <£ . d<f> ; dymr cos $.df>; 

 dz = — . d(f> ; <fc = ^-V4ar a + A v{ ; 



2; 



and we have, to determine the angles which the tangent to the 

 curve described by P makes with the coordinate axes, the equa- 

 tions 



dx —2.2'. sin d> \ 



cos a = -r- = — , — - ~ 



ds x/^ + k* 



~ dii 2z . cos 6 



COS 8= -f- = — 7 ====L_, /K\ 



__ <fe _ /f 



COay "ife - " VSSTf?' J 



14. In the Woolwich guns the driving-surface of the groove 

 may be taken, without sensible error, as the simpler form of sur- 

 face 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 in- 

 stance I shall examine the more general case, where the normal 

 makes any assigned angle with the radius. 



Assume then that on the plane of scy the normal makes an 

 angle o with the radius of the gun. The driving-surface of the 

 groove is then swept out by a straight line which, always re- 

 maining 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 o\ 



Now, the equations to the director being given by (4), and 

 that to the cylinder, which the generator always touches, being 



^ + ?/ 2 =(rcosS) 2 , (6) 



it is easily shown that the coordinates w l3 y l of the point of con- 

 tact of the tangent to the cylinder drawn from P parallel to the 

 plane xy } are 



oc x — r . cos h . cos (<£ — &)/ 



y l =zr . cos 8 . sin (0—5), 



and that the equation to the driving-surface is 

 c J* 



\ - 

 [_kr 



:;} P) 



?e is 

 cos -I -8 J. -f y. sin it — S>=sr.. cost. , (8) 



