526 MR. SYLVESTER ON THE FORMAL PROPERTIES OF THE BEZOUTIANT 
to me desirable that a commencement of a table containing them should be made 
and placed on record. Tlie remaining pages of this memoir will aocordinglj. be 
devoted to the ascertainment of them. 
The theory of the Bezoutoid being included within that of the Bezoutiant, need 
not hereafter call for any special attention ; I may merely notice that^^e^Bezoutoid 
to a function of the degree (m) will be a numerico-linear function of ^ G’s 
if m be odd, and of the G’s if m be even. 
It will be more convenient hereafter to denote the G’s as Gi, G 3 , G 5 respectively, 
in lieu of G„, G^, G^, &c., and to continue at the same time to give to the Ts and 
Q’s the same subscripts as the corresponding G s. 
Art. (69.). 1 st. Suppose m~2, 
f=iax^ -\-2hxy-\-cy^ 
(p=zax^ -\-2^xy-{-yf 
CL~u^.y — u^‘X. 
Then 
Ej ./= {ax-\-hy)l-\-{hx-\- cy)n 
El . 9 = + ,%) 1 + + 73/) 
Ti = (ax+ hy) (f3^ + yy) — {hx + cy) {ax + ^y) 
= (a|3 — ba)x^+ {ay— ca)xy~]r {by ~ c^)f 
Q^ = a^~uy—2u,.u,xy+ul.x^ 
G^={a(i — ba)u^i-{-{ay—ca)uy2-\-{by — c^)u2- 
Let us now form in the usual manner the Bezoutiant to /, <p; this is the 
quadratic function which corresponds to the matrix 
{2a^ — 2ba)\ {ay—ccc) | 
{ay—ca)-, {2by — c(B) J 
le. lB={a(3-ba)u!-i-{ay-ca)ti,.u,+ {by-c^)t4=G, or B = 2G,. 
2nd. Suppose w=3. 
y= no?® + 3 bxy + 3 cxy^ dy^ 
9 = aa;® + 3(3xy-i-3yxy^ + hy^ 
n = ■— 2^2 .i/x + W3 . 
We have then 
E,.{/)=:{ax^+2bxy+cy^)^+{bx^ + 2cxy-i-dy^f^ 
E, . {<p)=={ax^+2f3xy+yi/)^-i-{(3x^-i-2yxy-E^y^h 
T,= {ax^+2bxy-{-cy^){(3x^-h2yxy-i-^y^)-{bx^+2cxy-j-dy-){ax--h2f3xy+yyn 
= (a(3— ba)x ^ + 2 (ay — ca)^V + + («^ “ da))x‘'y- + 2 (3>^ — dft)xy‘‘ + ( c^—dy)y 
Q, = n® = uly^ — 4Wi . W 2 + ( 4^2 + 2 Mi . M3)3/ V — AiL^.Usyx^ + M3 . x\ 
