PROCEEDINGS 



OF THE 



(&umhxxtyz ||{ril0j30pJHal Bmtty. 



On a Definite Integral. By Prof. T. J. I'a. Bromwich, M.A., 

 St John's College. 



[Read 5 May 1902.] 



The following integral presented itself to me in the first place 

 as a good illustration of the theory of reducing two quadratic 

 forms to canonical types. But I find that it has been treated 

 by the late Mr Black and that his solution was published from 

 his papers by Prof. Hill in 1897 {Camb. Phil. Trans. Vol. xvi. 

 p. 219); according to Prof. Hill the integral in question is of some 

 importance in the theory of statistics. 1 hope that the alternative 

 investigation given below may not prove uninteresting. 



The integral to be evaluated is 



J (n) 



the limits of integration being — oo to 4- oo for all the n variables 

 x lf x 2 , ..., x n ; while U, V are quadratic forms 1 containing these 

 n variables and a constant x . In order that the integral may be 

 finite it is necessary and sufficient that U should be positive for all 

 real values of the variables ; that is, U is a positive, definite form. 

 Let U , V be the parts of U, V respectively which do not 

 contain x ; then, clearly, U is also definite and positive. Hence 

 we can find real linear functions of x x , x 2 , ..., x n (say y 1} y 2 , ..., y n ) 

 such that 



U = % r 2 , V .= Xc r y r 2 , (r=l,2,...,n) 



1 So far as the form of U, V is concerned, x is on the same footing as the 

 variables x lf x z , ..., x n ; it is only in the integrations that x is distinguished from 

 the rest as a constant. 



VOL, XI, PT. VI. 30 



