PROCEEDINGS OF SECTION A. 



317 



The direction cosines of the normal to the surface at P will be 

 COS. cos. 0, COS. 9 sin. 0, sin. 0. 



Hence Ix + my -\- nz =: b cos. 6 -\- a. 



The element of surface r/S may be regarded as formed by the revo- 

 lution of a circular arc of length adO about the axis of :: ; hence rf S ^ 

 'Itt {b -\- a cos. 6) a dO. 



The limits of 9 will be 2/7- and O, hence we have — 

 \{^ 2 F (r) uY = 2^a j| (^ + & cos, y + » cos. ^) 



[ r-'^^ F (r) f/;-"l d9 



(23) 



[ 



- (rt + 6 co^. 9^ {b -\- a COS. 9) 



d9 = O (24) 



Let A- = OandF (r) = r'". 



Then since r^ = a^ + ^^ + 2a& cos. 9, we have — 



27r« 



lif 



7-'" d\ = 



2n + 3 



C 

 X (« + 6 cos. 9) [b + rt COS. 9) [a- + b^ -f -lah cos. ^)" f/0 (25) 



f 



(« + 6 cos. ^) {b + rt COS. ^) 

 a'' + 6'^ + 2 a6 cos. 0) I 



d9 — O. 



,(26) 



The single integral in (25) can (if n be a positive integer) be ex- 

 panded in powers of cosine 9 and integrated. 

 It will be noticed that — 



cos.''" ^^ 9 d9 = Q 



.lit 



cos.'"' 9 d9 = 



_ (2m — 1) {2m - 3) 3. 1 



2'" - ' [m 



Thus— 





r- d\ = 2ira^b («' -f ¥) (27) 



r' dY = 27r'a'b { (a' + b^ + aW j . . 



.(28) 



r 



L I • R * * Y aDJ 



"M- A '<v 



i^/* " • 



