Hi 



PROCEEDINGS OF SECTION A. 315 



If i ^ a, 



.» ^V = . ^ ^ ^"^ „^^ ^ [{a + bf^^{n-irZ)a-h\ 



b {n + 2j(n + 3)(n + 4) L i ^ ^ ^ » 



_ (a_5)'-3 { (n + 3)« + S }] (15) 



This integral becomes infinite for values of 7i less than — 2, and 



O 

 takes the form — when w = — 2. For this particular case we have — 

 O 



-^ z= 27ra + . {a- - b') log. --^ (16) 



r a — 



The result reduces to the value 47ra when b ■=. o. 

 For n ■=■ — 1 , we obtain the result-- 



w 



w 



— =?5r3«^ - ¥) (17) 



r 3 



§ 4. If the sphere be cut by the planes x ^=. c^ x ■=. d, where 

 h -\- a y c> d y b — a, and fff r" f/V be required for the volume 

 enclosed by the planes and the surface of the sphere cut off by the 

 planes, the part of the surface-integral arising from the spherical surface 

 will be — 



1 _^' - rt- 



b{n-\- 3j ]^, \ r- 



where c'^ = 0-4- -^c — b" 



d'^ = a^ + 2bd - b- 



Since the direction-cosines of the c-plane and c?-plane are respec- 

 tively (1, 0, 0) and (— 1, 0, 0), the integrals arising from the plane 

 surfaces are — 



— ^ f V" </S, and - — ^ f r" d^._. 

 « + 3j^ »+3j 



We may put f/Si and rfSj equal to "l-n-p dp where p is the radius of 

 a circle on the plane face having its centre on the a;-axis ; also, 

 since r'^ = c- + /a* for the c-plane, and d" + p"^ for the fZ-plane, we 

 have ill both cases r dr = p dp. The surface-integrals are therefore — 



25rc r'' „ ^ , , 1 27rd r .. „ H- 1 ^^ 



2^c r „ , , . , 27rd r 

 I f. " I clr and — l 



Hi 



n 

 Hence — 



{n + 2) {n -f 3) 



6 (w + 3j ( « + 4 « 4- 2 ) 



jc(c"'*2_ c»-.2-j_ d{d"^''-d"*') j ....(18) 



