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 x 1 y 1 - x^y 2 - + 1x 2 y 2 . 
Then A A' = 65^^! — 39 (x-^y 2 + oc 2 y x ) -1- 65x 2 y 2 , 
A' A = 80 x x y^ — 36 (x x y 2 + x 2 y^) + o0x 2 y 2 , 
the roots of our determinantal equation are 
X = 26 or 104. 
Then on calculation 
Xi = (a?j + x 2 )/V 2 , Hi = (2a?, + 3a? 2 )/Vl3 , 
X 2 = (—x 1 + a? 2 )/V2 , H 2 = (— 3a?j 4- 2a? 2 )/Vl3. 
Thus, as the roots of the determinantal equation are different, we 
have 
i = \/26l 1 H 1 +Vl04Z 2 H 2 . 
2. Expressions for functions of a bilinear form. 
Let A be the given form and let <£ (r) = | rE — A \ denote the 
fundamental determinant of the form ; further, let yjr (r) be the 
quotient of <£ (r) by the H.C.F. of all the first minors of </> (f). 
Then we know 1 that yjr (J.) = 0 ; and that yjr(r ) is the expression of 
lowest dimensions in r which vanishes when r is replaced by A. 
We shall use the notation 
y/r ( r ) = (r — a) a ( r — by (r — cf . . 
and it should be remembered that every factor of </> (r) appears in 
yjr (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 </> (r) 
1 Frobenius, Crelle, Bd. 84 (1878), p. 12 ; Berliner Sitzungsber., 1896, p. 601. 
Ed. Weyr, Monatshefte fur Math, und Phys. Bd. 1 (1889), p. 187. Muth, Elementar- 
theiler (Leipzig, 1899), p. 34. H. F. Baker, Proc. Lond. Math. Soc., vol. 31, 1899, 
p. 195. 
