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LX. On certain Definite Multiple Integrals. 

 By the Rev. Brice Bronwin*. 



T N the integrals treated of in this paper, let the limits of in- 

 * tegration be given by the equation 



Or 1m CD 



? + p + + J? = h ^ 



including both negative and positive values of the («) variables. 

 Let P (r) = 1 . 2 . 3 ... r ; and for convenience let D, D„ &c. 



stand for -r-, -jj, &c. respectively. The general term of the 



series expressing the value of 



^ = // ... <p(g — x, h—y, ......) dxdy 



the odd powers of x, y, &c. obviously vanishing. But by a 

 well-known theorem, integrating for[positive values of x 9 y, &c, 

 and doubling the result; making s=p + q + ..., this term be- 

 comes 



p(p-i)p(?-|)- .«... 



By another known theorem 



and the above term is changed into 



on "^ f P V + "2~)(*D)^(gD^... , , , 

 2 ' 1 * * g " P(2s + n) P(y)P(g)... - * W< V f 0; 

 And the sum of all the terms of the order s is 



2 * x8 " s -""pVT^r {(, * D),+(SDi)2+ " -} ' <ffei- " ) 



= a§ ...f(s) to abridge. 



Hence \J/ = a § ... 2/(s), (6.) 



s having all integer values from zero to infinity. 

 Suppose <f> (g, h ...) such that 



(D' + D^ + ...)♦& *.») = <>. 

 * Communicated by the Author. 



