MULTIPLE INTEGRALS. 217 



^nd replacing a, a, by values which make both these factors 

 vanish. It hence appears that 



(mn l 



~~ 



[c (mn t m t n) + a (np t n t p) + b (pm t p?n}} . . . ' 



the denominator being the continued product of r 2 factors, 

 each of the same form as the one written down. Of course the 

 other quantities F are obtained in the same manner. 



Let us now take the more general case in which there are 

 s limiting inequalities, s being less than r, and in which the 

 function to be integrated is 



...^(m^+.^p^e-^"- 

 the inequalities in question being 



<. 7, 

 ... pgZ ^ fig . 



"We shall arrive at a perfectly analogous result in this more 

 general case. In the first place the integral sought may be 

 thus written, 



I 1 <f> l u l du l ...I <j) a u 8 du 8 l da^... I da g Gr, where 



CO * 00 



= I dx... I dz e~ aaj -- c2 cosa 1 (w 1 a?+. . -p^-u^ . . .cosa 8 (m,cc+. . .p 8 z-u s 



J ^ 



o o 



Now a little consideration will convince us that 



.00 .00 ^ - + 00 .+ 00 



d^ ... da 8 G- = - I da : ... da 8 H, where 



^0 Jo " J - J-oo 



-oo /.oo 



H = I dx ... I dz e--- cos [x2<a>m + . . . + z^cup Saw] : 



^0 - 



for if we take the expression 



cos [x Sam -f . .. + z Sap Saw], 



make a x negative, add the resulting expression to the original 

 one : then in the two terms thus got make 2 negative, and as 

 before add the results, we shall, continuing this process, get in 

 all 2* terms, which will be found to be equal to 2' times the 

 continued product of the cosines involved in Gf. 



