PEOCEEDINGS 
OF THE 
Ciimtribgc ^^rlosopljkal jfadetg. 
On a Definite Integral. By Prof. T. J. I’a. Bkomwich, 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 ( Oamb . Phil. Trans. Yol. 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 
I Ve~ u dx^dx 2 ... dx ny 
J (n) 
the limits of integration being — oo to 4- oo for all the n variables 
x x , x 2 , ..., x n \ while U, V are quadratic forms 1 containing these 
n variables and a constant^- 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 0 , V 0 be the parts of U, V respectively which do not 
contain x 0 ; then, clearly, U 0 is also definite and positive. Hence 
we can find real linear functions of xf, x 2 , ..., x n (say y 1} y 2 , ..., y n ) 
such that 
U 0 = % r 2 , V 0 = 'Zcryf, (r = 1, 2, . . . , n) 
1 So far as the form of U, V is concerned, x 0 is on the same footing as the 
variables x x , x 2 , . . . , x n ; it is only in the integrations that x 0 is distinguished from 
the rest as a constant. 
VOL, XI, PT, YI. 
30 
