NOTE ON A DEFINITE MULTIPLE 

 INTEGRAL*. 



IN the XVIII th number of this Journal Mr Boole pointed 

 out the incorrectness of a theorem given by M. Catalan. The 

 following pages contain a brief demonstration of the result to 

 which he was led. Both he and M. Catalan made use of what 

 is generally known as Liouville's theorem, and thus perhaps 

 rendered their analysis less simple than it would otherwise 

 have been. 



Let us transform the integral 



/<*BI ...... !&*>*/(<*& + ...... +a n x n ) 



by the assumption 



a^x v + ...... a n x n OM^ ...... (a 2 = Sa 2 ), 



and by (n 1) other linear relations connecting^ ...... x n and 



u^ ...... u n , and such that 



Then, as is well known, dx t ...... dx n is to be replaced by 



du v ...... du n , and thus 



fdx l ...fdx n f(a l x l + ... +a n x n ) = fdujaujdu z ...fdu n ... (1). 



Let the integrations on the first side of this equation include 

 all values of the variables which do not transgress the limits 



were B is supposed to be greater than A. Then, as So; 2 = S 

 the corresponding limits on the second side of the equation are 



* Cambridge Mathematical Journal, No. XX. Vol. iv. p. 64, February, 1844. 



