VIEWED WITH RELATION TO THE METHOD OF INVARIANTS. 
517 
it will readily be perceived that the matrix formed by the twenty-five coefficients, 
viz. — 
i 3 2 A^^ j A, 
^4, 4 
■^3, 4 A3^ 3 Ag^ 2 
4, 2 •‘■‘4, 1 ■‘*4, 0 
Ag^ 1 A 
3, 0 
4 A2^ 3 Ag^ 2 A2^ 1 A2^ 0 
■^1,4 A,^3 Aj 2 A,^ 1 A,_o 
Afl, 4 Aq^ 3 Aqj 2 Aq^ 1 Aq^ g, 
will be symmetrical about its dexter diagonal (that one, namely, which passes through 
A 4_4 and Ag^o)? and will be identical with the Bezoutian square corresponding to the 
system/, p ; in fact, using the notation previously employed in the first section, it 
becomes 
(0,1) (0,2) (0,3) (0,4) (0,5) 
(0,5) (1,5) (2,5) (3,5) (4,5), 
(r, 5) being used in general to denote the difference between the cross products of the 
coefficients of and in f and <p. Restoring now to m its general value, and 
taking/ and <p homogeneous functions of x and y, and making 
y)<^W,y')-.f{^,'y')<p{oo, y) 
xy' — x^y ’ 
we see without difficulty that 
^ = 2 A,., , { .y”- ‘ -’•.r'* .y*"- * -* } , 
where A^ , is the term in the rth line and 5th column of the Bezoutiant matrix to / 
and Ip. This is the identification, the idea of which, as before observed, is due to 
Mr. Cayley. 
Art. (63.). If, now, we consider the system of functions 
f{x,y) = a^.x^-\-ma,.x^-\y-\- + • 3 /’" 
^(^5 y)=6o-2^’"+w&..j?’""‘.y+ 
n(a7,y) y'«->_(m-l)w^_ 2 .y'"-^± + (-)'”-’. 
