produce Translation and Rotation in the Bores of Rifled Guns. 201 

 from (19), we obtain as the equations of the generator, 



xr l .coal(j) J + yr x . sin f </> ) = ^i 2 , z=krcj), (21) 



and as the equation to the driving-surface, 



( Z 77"\ . {% 7T\ 7T , " 



x - cos \k;-n) + y- sm {i;--J =rcos n- • ■ < 23 ) 



Now 



(^) l{?.. cos (* ?)_£. sin (* ZE)) ; 



\dz J k \jr \kr nJ r \kr nJ j 



or, since P(#, y, z) is a point at once in the helix and the skew 

 surface, 



(—)=- sin-- 

 \dz J k' n 



J7<*F\ 8 , /*F\* ^F\ 2 \ i 1 / " / . 7TN 2 



i(arJ + (%J + (iJ / = * V *"+( m ») • 



And substituting these values of (-7-), &c. in (8), we have for 

 the direction cosines at the point P, 



k . cos (4 I 



\kr nJ 



Also 



cosA,= — 



cos a= — 



\kr nJ 



k . sin 



v^HD 1 



cos v= 



. 7T 



sin — 

 n 



(23) 



And putting the values of oc } /3, 7, X, ^ v in the equations of 

 motion (2) and (3), we have 



M.J=G-R< 



H x k 



sin- 



Vl + k? 



+ 



v^+(^ 



(24) 



