200 Captain Noble on the Ratio between the Forces tending to 

 and in the case of the rifled gun, 



M - d i= G '-7m^ k+li - 



(15) 



Now if the velocity-increments in the two cases be taken as equal, 

 we shall have from equations (14) and (15), 



G'=G + 



E 



\/l+k* 



0*i*+ 1). 



(16) 



And the second term of the right-hand member of equation (16) 

 represents the increment of pressure due to the rifling. 



Let us now examine the pressures which subsist when a poly- 

 gonal form of rifling is adopted ; and we shall suppose the polygon 

 to have n sides. The equations of motion given in equations (2) 

 and (3) hold here as in the last case, and the values of a, /3, y 

 given in (5) remain the same. The driving- surface is, however, 

 different, being traced out by a straight line which always remains 

 parallel to the plane of ocy, meets the helix described by P, and 



77- 



touches the cylinder whose radius is =rcos— (see fig, 3, where 



P A represents the generating line 

 drawn from a point P of the helix 

 to touch the cylinder B C). Now 

 the equations to the helix being 



ic=rcos<j), y = rsm<j) } z = krcf> > (17) 



while that to the cylinder is 



# 2 -f-z/ 2 =f r.cos- ) =^r^ suppose, (18) 



if we draw from the point P(#, y } z) 

 of the helix a tangent in the plane 

 z = kr(f) to (18), the coordinates of 

 the point of contact (see fig. 3) will 

 be 



Fig. 3. 



#, = r, .cos 



*/] 



= r '- sin (*-^)-- 



(19) 



Now the equation to the tangent drawn through the point a?, y x 

 of the circle w 2 -ry <2 =r l < * is 



(20) 



xx \ +yy^ = r i 2 - 



And substituting in this equation the values of x x and y } derived 



