156 EVALUATION OF DEFINITE MULTIPLE INTEGRALS. 



As a verification of our analysis we may remark, that in this 

 case 



2 __^__ =e -,^ ); 



for we have, what is probably a known result, and which at any 

 rate may be easily proved, 



-. a 



where F (p) (A) denotes the p ih derived function of F(A) : and 

 this formula applied to the case where F (a) = e~ au gives the 

 above written result. 



By differentiating (11) for a. b, c, &c. X, //,, i>, &c. times 

 respectively, X, //,, v, &c. being integral or fractional, and dividing 

 by ra x , n 11 , p", &c. we should obtain the value of 



mx + ny + . . . x . . . 



which would include every case of (3). But the investigation 

 would be complex, and I shall therefore only indicate it. 



In a future number of the Journal I may perhaps apply the 

 method to some other cases, and particularly with regard to 

 such multiple integrals as 



the limits being given by the series of inequalities, 



&c. > &c. < &c. 



In theory, such an integral is reducible to a multiple integral of 

 as many variables as there are limiting inequalities. But it is 

 not easy to find cases in which this reduction can be actually 

 effected. 



