422 Prof. Bromwich, On a Definite Integral. 



When this is expanded as the sum of a number of partial 

 products, each term appears as the product of a number of known 

 integrals, such as 



f+°° r+ co f+°° „ 



e-* 2 dz = Tr 1 l\ ze~ zi dz = 0, z*e~ e dz = W' 2 > 



J —<x> J —00 J — 00 



and so the value of the whole integral is 



u ~ * 7r n ' 2 e ~ lx « 2 (%%c r + pec,?). (r = 1, 2. . . ., n) 



All the constants in this are known with the exception of %c r ; 

 now c 1} c 2 , ..., c n are the roots of an equation in X which, when 

 expanded, takes the form 



u \ n - (Sb rs % rs ) \ n ~ l + . . . = 0, (r, s = 1, 2, . . ., n) 



where f> s is the minor (with proper sign) of a rs in u , so that % rs 

 is a second minor of u. 



Hence u Xc r = %'b rs t; rs , 



and substituting we have, finally, the value 



where the accented % indicates the omission of the zero suffixes. 

 This result agrees, save as to notation, with Mr Black's. 



