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 
r+ oo r+oo r+oo 
I e~ z *dz — 7 r 1/2 , I ze~ zl dz = 0, I z 2 e~ z dz — r 1/2 , 
J —00 J —oo ^ —oo 
and so the value of the whole integral is 
~ i 7 r n/2 c ~ 1x2 + p^o 2 )- (r = 1, 2, . . n) 
All the constants in this are known with the exception of 2c r ; 
now Ci, c 2 , c n are the roots of an equation in A, which, when 
expanded, takes the form 
u 0 \ n - (tb rs i; rs ) W 1 - 1 + . . • = 0 , (r, s = 1 , 2 , . . ., n) 
where is the minor (with proper sign) of a rs in u 0 , so that £ rs 
is a second minor of u. 
Hence u 0 Xc r = 2'& rs f rs , 
and substituting we have, finally, the value 
u 0 ~ 6,2 7r nli e ~ tW5 ° 2 / Uo [%u 0 2<'b rs t; rs + a) 0 2 ^b rs u r u s ], 
where the accented % indicates the omission of the zero suffixes. * 
This result agrees, save as to notation, with Mr Black’s. 
