Mr Bromwich, Theorems on Matrices and Bilinear Forms. 79 



orthogonal substitution in order to complete the reduction. The 

 further substitution is always easily calculated by forming RAS 

 and comparing it with B. 



Sylvester's numerical example (Comptes Rendus, t. 108, 

 p. 653) is 



A = 8^3/j -x^y 2 - 4ty 1 x 2 + 7x 2 y 2 . 



Then A A' = ^oc 1 y l — 39 {x 1 y 2 + x 2 y 1 ) -f Q5x 2 y 2 , 



A' A = 80x 1 y 1 — 36 {x x y 2 + x 2 y^) + 50x 2 y 2 , 



the roots of our determinantal equation are 



\ = 26 or 104. 



Then on calculation 



X l = (x, + x,)l V2 , Bi = (2a* + 3# 2 )/Vl3 , 



X 2 = (-x 1 + x 2 )/ V2 , B 2 = (- 3# a + 2x 2 )/ Vl3 . 



Thus, as the roots of the determinantal equation are different, we 

 have 



A = VIoX^ + Vl04X 2 Ho. 



2. Expressions for functions of a bilinear form. 



Let A be the given form and let <f> (r) = \ rE — A | denote the 

 fundamental determinant of the form ; further, let yjr (r) be the 

 quotient of (f> (r) by the H.C.F. of all the first minors of <f>(f). 

 Then we know 1 that yfr ( A) = ; and that yfr (r) is the expression of 

 lowest dimensions in r which vanishes when r is replaced by A. 



We shall use the notation 



t/t (r) = (r — a) a (r — by (r -c)y ..., 



and it should be remembered that every factor of <£ (r) appears in 

 -\|r (r) ; but possibly to a lower power in case the factor is repeated 

 in <f> (r). 



We start with considering a rational function of the form ; 

 there is of course no theoretical difficulty in calculating such 

 functions directly. It is, however, worthy of notice that the 

 following method is practically easier, whenever the roots of <f> (r) 



1 Frobenius, Crelle, Bd. 84 (1878), p. 12 ; Berliner Sitzungsber., 1896, p. 601. 

 Ed. Weyr, Monatshefte fur Math, unci Phijs. Bd. 1 (1889), p. 187. Muth, Elemental-, 

 theiler (Leipzig, 1899), p. 34. H. F. Baker, Proc. Lond. Math. Soc., vol. 31, 1899, 

 p. 195. 



