EVALUATION OF DEFINITE MULTIPLE INTEGRALS. 155 



the upper sign is to be taken when u is > 0. 



By differentiating this for a, we have the value of 

 a. sin a. uda. TT ^ 1 



= att 



(a 2 +a 2 ) (6 2 +a%. L 2 ^ (6 2 -a 2 ) (c 2 -a 2 ) ... 

 and so by repeated differentiations we find the values of all the 

 integrals which enter into I G cos a uda and J H sin a. uda. 

 Thus 



when w is > 0, and 



when w is < 0. 



But ^ a^_ x + &c. + a* = 0. 



Hence ^=0, when w is < 0, and therefore in (E) we may 

 make \ 0, so that the limits of integration for u are and h. 



Again a* +^> 1 a*~ 1 + &c. = (a + a) (a + b) . . . 



and 2a(J 2 -a 2 )(c 2 -a 2 )... = {2a. 



Therefore 



2a (& 2 - 2 ) (c 2 -a 2 ) ... (& - a) (c -a) ... 



cvu 



and jF= STT r-, r (10), 



mn ... (b a) (c a) . . . 



and therefore finally 



{ Q dxf Q dy...e- (mMby+ - ) f(m 



l I fue~ au du 



V 



(^> a) (c a) ... 



If a = Z> = c = &c. = ^4, the first side of this equation 



