GENERAL THEOREMS, &c. 213 



not transgressing the two inequalities mx + ny < h, m'x + riy ^ h\ 

 we shall in effect integrate over the surface of the quadrilateral 

 OMAN'. But if the two lines did not intersect within the 

 positive quadrant, then one or other bounding inequality would 

 be inoperative, and we should in effect integrate over the sur- 

 face, not of a quadrilateral, but of a triangle, as in the case 

 contemplated by Liouville's theorem. It is manifest that we 

 may have, instead of two limiting inequalities, any larger num- 

 ber we please, and that our integrations may thus be made to 

 extend over an irregular polygon of a greater or less number 

 of sides. I do not believe that any writer on multiple inte- 

 grals has considered the case in which the limits arc given by 

 more than one inequality, but the restriction to that of one is 

 clearly unnecessary. 



Let us suppose there are r variables x, y, ... z, and that we 

 have to evaluate the integral 



f dx...f dze- ax -- cz <j)(mx+...pz) <, (m t x + ...p t z) ...... (1), 



subject to the two inequalities 

 mx+ ...z^h, 



m ...p, h ; ra, ... p t , h, being all positive ; and <f> and (f> t any func- 

 tions whose values may be represented within the limits of inte- 

 gration by Fourier's theorem. 



Let the value of the integral in question be /; then, by con- 

 siderations analogous to those of which I made use in a paper 

 which appeared at the commencement of the last volume of the 

 Journal* ', we shall have 



where 



1 rft rh, r 00 f 



I fa du I $ t u l du t I da. I da, . G, 



6r = I dx . . . I dze- ax '"~ cz cos a (mx+. . .pzu) cosa, (m t x+ . . . 



J Jo 



and the lower limits of integration with respect to u and u may 

 be any negative quantities. 



I remark in the first place, that 



/< r< r 00 r 



da I d^ff-fcij da da,H, where 



./O J J -oo ^ -oo 



* Page 150 of this volume. 



