516 MR. SYLVESTER ON THE FORMAL PROPERTIES OF THE BEZOUTIANT 
riant of/, <p, or invariant to/ <p, O, say I, will take the form 
d . d\ / d . , d\ / d . , d 
■{( 
• • • + (®- 1 )“— -at) } 
’^1 
dUm-i 
"^-‘^’du 
du-^ 
These equations may be proved to be satisfied when I is taken =B, the Bezoutiant to 
/ <p, and thus B may be proved to be a covariant to / (p, but the demonstration is 
long and tedious. An admirable suggestion, well worthy of its keen-witted author, 
for which I am indebted to Mr. Cayley, will enable us to prove the mvariantive 
character of B by a much more expeditious method. 
Art. ( 62 .). For greater simplicity begin with considering functions of a single 
variable ; and in order to fix the ideas, suppose (m) to be taken 5 , and write 
fx=^ax^-\-hx*-^cx^-\-dx^+ex+l 
(px =■ ax^ - 4 - jSx* -{- -\-sx-{-X, 
and let this is of course an integral function of .r and x, since the 
numerator vanishes when x=x ' ; and we have by performing the actual operations, 
! a^-boc)x'.x‘+{ay-irCa)^.x'ix+xl)-\-{al-da)xV(X-\-xx!^-x^) + (a^-e<‘)\ 
\(hy-cf)x^x‘+{U-d^)x‘x\x^:^) + (ht-e?:)x!^(x'^+xx‘+x')\ 
-!f(h'k—lf}(x‘+x‘‘^+xx'+x’) J 
j^Ucl—dy)iex'‘-\-(a—ey)xx‘(x+x) + (A-ly).(x!‘+xx!+x:‘-)) 
— el)xx' ■^{dX—ll){x-\-x')'^ 
-J- (eX — ^2) ? 
and if we arrange ^ under the form 
A 4,4 j^\T'-fA 4,3 a^CF®-l-A 4,2 .rV"-f A 4 , , j:kF-l-A 4 ,o 
4 - A 3, 4 x'^x ^ -h A 3, 3 + A 3 , 2 x'^x^ 4 - A 3 , , x^x' - 4 - A 3. 0 x^ 
4 -A 2,4 a;^F^ 4 -A 2,3 ^'^F® 4 -A 2,2 -i-’V^+Aa, , x'x' -fAa.o 
- 4 - A , ,4 xx* -l-A,, 3 -rT' +A 1, 2 4 -Aj,o^F +A,,oX 
+Ao, 4 a;'" d-Ao.sF* 4 -Ao. 2 ^'" +Ao,,^ +Ao.,; 
+ 
