420 Prof. Bromwich, On a Definite Integral. 



where Ci, c 2 , ..., c n are the roots of the determinantal equation in 

 \, namely, 



|\^„-F„| = 0. 



It is to be observed that any number of the quantities 

 d, c 2 , ..., c n may be equal without modifying the reduced forms 

 of JJ , V ; for the quadratic JJ 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 



When the new variables y are substituted in U, it will take 

 the form 



1y r 2 + 2x Xd r y r + kx 2 , (r = 1, 2, . . ., n) 



where the constants d r , k depend on the original coefficients of 

 U and on the substitution giving y x , ...,y n in terms of x 1} ...,x n ; 

 but at present we do not need to determine d r and k explicitly. 

 Now put 



z r = y r + dr% , = 1, 2, ..., n) 



and then U = 2^ r 2 + l^o z > 



where I is a new constant whose value is required subsequently. 

 To find I, suppose that the original expressions for U, V were 



and write for brevity 



u = | U | = J a rs | , (r, s = 0, 1, 2, . . ., n) 



u = | U 1 = | a rs | , (r, s = 1, 2, . . ., w) 



so that w is the minor of a 00 in the determinant u. 

 Now we have determined y 1} y 2 , ..., y n , so that 



Uo = yf + yi + ... + y n \ 



and the determinant of these coefficients is unity. Hence, by 

 a well-known theorem, 



u = M\ > 

 where M = d Jf u —'^ . 



[SDi , .. ., X n ) 



Similarly, by considering U, we find 



u = IN 2 , 

 where N = H*», ^, --, *n) d(y u ..., y n ) ^ 



\X , X x , ..., X n ) V \0C-i, ..., x n ) 



Hence I = u/u , 



which gives I in terms of known quantities. 



