EVALUATION OF DEFINITE MULTIPLE INTEGRALS. 151 

 Sdxjdy... * to...)* {/to0} 



= / ^rudu I da I dx I dy . .. $ (xy . . .) cos a (/. u) ... (1) ; 



"If J h j J / / 



the limits on the first side being given in the manner already 

 mentioned. Those of the integrations with respect to x, y, &c. 

 are arbitrary, provided they include all values of the variables 

 which satisfy the given inequality. 



Again, if in the given multiple integral the limits were 



determined by the single relation f> ^ h, joined to the con- 



ditions that x } y, &c. were to have no values less than certain 

 assigned limits; e.g. if we were to consider only positive 

 values of the variables, the formula (I) would still apply with 

 a slight modification. The inferior limit of integration with 

 respect to u would be arbitrary, provided it included all the 

 values which could be given to /. by admissible values of 

 the variables, while the inferior limits of integration for 

 x, y, &c. would be determined by the particular conditions of 

 the case. 



Let us take, as an example of the method, the integral 



fdxfdy...x a - 1 y b - 1 z c -\..f(mx + ny + ...) ......... (A), 



m, n, &c. being all positive, and the limits being given by 



mx + ny + . . . h } 



no negative values of the variables being admitted. In this 

 case (1) becomes 



fdxjdy ... ory 1 ...f(mx + ny + ...) 



= -( fuduf dal dxl 



7T J hi JQ JQ JQ 



and the integrations with respect to #, y, &c. may be con- 

 veniently extended to infinity ; (\ it should be observed is an 

 arbitrary quantity < 0). 



I first seek the value of 



/co / 



dxl dy ...x^y*' 1 .,. COBOL (mx + ny+ ...-u) = (B). 



^0 ^0 



Integrating first for x, we get 



