336 PROCEEDINGS OP THE AMERICAN ACADEMY 



where a =. T«, b = Tp, c = Ty, s = To'. Hence, for the general 

 ellipsoid, 



iS = I s I sni a:V {a^ 7, cos x) 



y * 



s. I W [a- CDS'' a;), 



y cosx 



, aV — 62c2 

 and, II nv = -ni — , 



5'= I — 2m cos x\J{\ — m" cos t) -f- 2 sin — i(m cos x) , 

 y 



V. „r^=/£[si„2. + 2'^];, 



where v = sin — ^ (m cos x). 



If the surface be a prolate ellipsoid of revolution, then b z= c ^ a. 



a2 — Ifl , 

 w = — 5 — , and 



a result which agrees with that obtained in 20°. 



Repeating this process of integration for cases 11. and III., we 

 obtain in succession 



II'. S= fcPsirir — 2?^", 



J 4y« L J „o 



y 



where v = Ch — ^ (mCha:) and rn^ = — — —-, — ; 



^ ' a-S' 



III'. ^= r^rSh2r + 2r1', 



J 4niL Avo 



y 



n h c^ — o''S 



where v = Sh — ^ (wShx) and 7n- = r— . 



For cases IV. and V., we shall have q\ = ax -\- g, q'^ = g'x, and 



Yq\o'.2 = aa'x- -\- ^yx, 

 whence 



