198 Captain Noble on the Ratio between the Forces tending to 



Now X, fju, v being the angles wbich the normal to (7) makes 

 with the axes, 



— \ 



dx / 



cos\ = 



cos /a: 



cos v= 



(: 



{(f)'+(f)*+©y 



W/ 



dx) \dy) + \dz) J 



{( 



(8) 



\dz) 



Now 



/d¥\ . z /d¥\ z 



(d¥\_ Iff 

 \dz)~ k\r 



z , y . 

 cos!— + ^-. sin 

 Ar r 



kr}> 



but since in the case we are now considering (x } y, z) is a point 

 both in the surface given by equation (7) and in the directing 

 helix, we have from (4), 



rp >)> 



-=cos<f>=cos 



r T 



= sin 6 = sin -=— ; 

 kr r ~ kr 



and 



Hence 



• ( d -)=-- 



•' \dz) k 



#sin<j& 



cos\= — 



k . cos <f> 

 cos Lb = — ■ = , 



cosv= — 



V'l + F 



(9) 



Now substituting the values of the direction cosines given in 

 equations (5) and (9), in (1), (2), and (3), we have as the equa- 



