INTEGRALS OF AN ELLIPSOID. 59 



Hence — 



r fpr '^''ds= (2k + 2« + 3) r r C(x^ + ^^ + z-^y" 



(^^^1 + — V' dV (23) 



Putting k zero in the above equation, 



( (pr "" dS=C2n+ 8) (" f ("(or^ + ?/' + z'^ydV (24) 



Putting n zero in (23), 



where V is the volume ot tlie ellipsoid. 



Let g = /^ + 'L' -f- ^V' while F i r) = r'^". 



\a* b* c*/ 



Then g = ~2/,^^^ '■^^^ points on the surface, and hence — 



(26) 



Putting n zero in (2G), 



JX4^- (^'^■+^)i'.rj"0^+:-)'^ ^ '''' 



Prom integral (10) vve have — 

 hence making ?« = 2 in (24) we have — 



SSS ^^'^ + ^' + ^^'^'' '^ ^' ^ [ H ^""^^ I "' + 2 s («o] .^ - (2«) 



Jf in the fundamental equation of Part I. — viz., 



//(.„+„„ + ,„.,. .=j;Cj-(;;« + ^; + ^f).. (^) 



, a'x b'l/ c'z 



we make a = , y =:r^, w = - , 



j9* /)* />•* 



we obtain — 



.y2 V 2 



Using integrals (7) and (11), it follows that— 



/J-^^-^.(.)[[.(L)-+,.(L)]| 



V 

 35 



+*C-^)V' (*"' 



