Let 



then 



or 



Mr. H. Hennessy on the Attraction of Spheroids. 27 



jx'—fx. + h, oo' = co + 7c, 

 l-l^±fih= V(l- ft 2 )(l- \i?±1^h—¥) cos k; 



h 2 



cos%= 1 + (l-ft«)Cl-^±W 

 A 2 is necessarily positive, /* must either be equal to or less than 

 unity. If |w,= l, the second term in the expression for cos' 2 k 

 would be infinite for any finite value of h. If j«, < I, p + h 

 must either be equal to or less than unity. If j«. + /i=l, the 

 second term would be again infinite for any finite value of h. 

 If both ju, and p + h are less than unity, the denominator of the 

 second term will be finite and positive, and therefore the above 

 equation could not be then satisfied for any finite value of h. 

 Hence in general /*=(), and jt*/ = fA, and therefore cos/rss + 1. 

 The double sign, with which the value of cos k is affected, 

 shows that, like cos w', it should be extended to both sides of 

 the origin of the co-ordinates; and when thus interpreted, we 

 may satisfy the above equation by making & = 0, or £=27r, 

 and consequently co' =00. Therefore, when 7=1, 



and 



U + 2a Tr=^ a - r) V + J Q J*-' 



3. If, which is permitted, the origin of the co-ordinates be 

 so taken as to make //,= !, 



/= i/a*-2arn> + rS ^ = ^ , 

 then 



J+Jo f 3 " I/+1 f 3 ~ ar\.±(a + r) ±{a-r)J' 

 re have the two values 



s^dy! _ — 2r /* _I V_ —2a 

 J +1 p ~ a*-r» J +1 f 3 ~ o^T 2 ' 



Whence we have the two values 



and therefore 



or 



~ du 



dr J 



, n du 4:ira 3 ay 



u+2a-r- = -. 



dr r 



