420 Prof. Bromwich, On a Definite Integral. 
where c 1} c 2 , . .., c n are the roots of the determinantal equation in 
X, namely, 
I Xtfo- Vo 1 = 0. 
It is to be observed that any number of the quantities 
Ci, c 2 , ..., c n may be equal without modifying the reduced forms 
of U 0} V 0 ; for the quadratic U 0 is definite, and so the invariant 
factors of the determinant are linear (Weierstrass, Berliner 
Monatsberichte, 1858, p. 207, or Werke, Bd. I. p. 233 ; particularly 
§ 4). 
When the new variables y are substituted in U, it will take 
the form 
% r 2 + 2x 0 Xd r y r + kx 0 2 , (r = 1, 2, ..., n) 
where the constants d r , k depend on the original coefficients of 
U and on the substitution giving y lt ...,y n in terms of x x , ...,x n \ 
but at present we do not need to determine d r and k explicitly. 
Now put 
Zr = y r + drX o, 0 = 1, 2, n) 
and then U = + lx 0 2 , 
where l is a new constant whose value is required subsequently. 
To find l, suppose that the original expressions for U, V were 
U = Xa rs x r x s , V = Xb rs x r x s , ( r , s = 0, 1, 2, . . ., n) 
and write for brevity 
u = | U | = | a rs | , (r,5= 0, 1, 2, ...,n) 
u 0 =\ U 0 \ = \a rs \, {r,s = \,2, ...,n) 
so that u 0 is the minor of a 00 in the determinant u. 
Now we have determined y x ] y 2 , ..., y n , so that 
bio — y-I + yl + • • • + Vn> 
and the determinant of these coefficients is unity. Hence, by 
a well-known theorem, 
u 0 = if 2 , 
where M = . 
d(x 1} ..., x n ) 
Similarly, by considering U, we find 
u = IN\ 
where N = ffr’ ~ — H = = M. 
0 \Xq , Xi , . . . , X n ) O yXi , . « • , X n j 
Hence l = u/u 0 , 
which gives l in terms of known quantities. 
