NOTE ON A DEFINITE MULTIPLE INTEGRAL. 161 



Transform the integral in x by assuming 



x v = r cos 0j , # 2 = r sin t cos 2 . . . # n = r sin # x . . . sin O n _ l ; 

 and that in u, by similar assumptions, 



u^r cos 0'j, w 2 = r sin #' t cos 0' 2 ... u n r sin 0' t ... sin 0'^. 



I may be allowed to mention that this transformation, which 

 appears to have been given for the first time by Mr Boole, in 

 the last number of the Journal, had occurred to me before I had 

 seen his paper. His analysis leads at once to the conclusion 

 that dx l . . . dx n is to be replaced by 



r n ~ l sin"' 2 t . . . sin M _ 2 drd6, . . . d0 u _ l . 



This may be also proved by successive substitutions in the man- 

 ner pointed out in the case of three variables by Mr A. Smith, 

 in the first volume of the Journal. 



Thus (1) becomes, since 2a? 2 = 2w 2 = r z , 



n - : dr /sin"" 2 l d0 l ... Jd0 n _ l f{r (^ cos ^ + ... a n sin l ... sin 0^) } 



r 

 J 



= 



cos 



The limits for 6 and & are the same. 



That this equation may subsist for all values of A and 5, it 

 is necessary and sufficient that 



/sin"^^ ... JW<9 n _ 1( /{r (a x cos 0, + ... a n sin l ... sin n _J} 

 = /sin w - 2 6>' 1 /(ar cos^) dffjam* r *0' t d0\ ...fdO'^ ....... (2). 



With respect to the limits of 6 and 0', it is not difficult to 

 perceive that if O l . . . O n _^ are taken between the limits and TT, 

 Xj^ ... #_! will receive all the values of which they are capable, 

 namely, all that included between r and + r ; and that the 

 same set of values cannot occur more than once. But in 

 order that x n may vary from to r, it is necessary to extend 

 the superior limit of d n _^ from TT to 2?r. Thus the limits of 

 n-1 are and 2?r, while those of the other variables t . . . 6 n -^ 

 are and TT. And similarly for & '. 



On the second side of (2) we have the factor 



J n 



(A). 

 11 



