520 MR. SYLVESTER ON THE FORMAL PROPERTIES OF THE BEZOUTIANT 
As an example of this property of the Bezoutiant, suppose 
bo^y -f- cxy^ + dy^ 
(p=ax^-\- i^x^y + yxyf + ^ 3 /^ • 
The Bezoutiant matrix becomes 
a^ — ha; ay—ca; al—da 
a^ — dot, 
ay — cot,-, + ; hy—c^ 
by — c(5 
al—dot,-, by—c^\ cl—dy. 
The Bezoutiant accordingly will be the quadratic function 
which on making 
becomes 
(a8 — bct)u\-\-{a^ — do(,-\-by — c^)ul-^c^ dyu^ 
-{■2{ay~cci)Ui.U2+2{d^—dci)u3.Ui + 2{by—c^)u2.'U3, 
u,=x^ u^—xy U^=y‘" , 
L.r^+M:r^3/+Nxy+Px/+Q/, 
where L, M, N, P, Q respectively will be the sum of the terms lying m the successive 
bands drawn parallel to the sinister diagonal of the Bezoutiant matrix, e. 
L =a(3 — 6a 
M = 2(07— ca) 
N=3(a^-r/a) + (67-c^) 
P =2(67-cj3) 
Q =c^ — dy. 
The biquadratic function in x and y ((3.) above written will be found on computa- 
tion to be identical in point of form with the Jacobian to/, p, viz. 
(3M‘-l-3Jirr/-|-c/)((3.v’+2ra3/+3S/)-(3“’+2f5*>/+r/)(*‘^'+2".y+''i^’)’ 
this latter being in fact 
3M.r®3/ + 3Na?y + 3 Pj2/® + 
The remark is not without some interest, that in fact the Bezoutiant, which is capable 
(as has been shown already) of being mechanically constructed, gives the best and 
readiest means of calculating the Jacobian ; for in summing the sinister bands trans- 
verse to the axis of symmetry the only numerical operation to be performed is tiat 
of addition of positive integers, whereas the direct method involves the necessity o 
numerical subtractions as well as additions, inasmuch as the same terms wi e 
repeated with different signs. Thus if 
y z= ax^ + bx'^y -|- cx^y^ -p dxhf -f exy^ -|- ly^ 
(p = ax® -f ^x*y -P yx^y‘^ + + ^3/^5 
using (r, s) in the ordinary sense that has been considered throughout, we obtain by 
