524 MR. SYLVESTER ON THE FORMAL PROPERTIES OF THE BEZOUTIANT 
T=0, 
■£?«1 
do^'jn-^r 1 
d 
dcc.2n^^ 
and 
.T=0. 
From the hemihedral symmetry of T, which only changes its sign \\hen the oider of 
the coefficients in / and <p is simultaneously reversed, it is obvious that one of these 
equations cannot be satisfied without the other being so too. Looking then exclu 
sively at the first of them, we see that this is satisfied by virtue of the equations 
( d d } 
T=0 
Hence then the differential equations to T being satisfied proves that it is an inva- 
riant, and, as above observed, its form shows upon its tace that it is a combinant. 
Precisely in the same way it may be demonstrated, that to two functions each of 
the same even degree {2m) as 
+ 2mai . ‘3/ + 2 ' — ^^2 • + • • • + O-zm • 
and a,, -f 2 mai . ' . 3 / + 2 m . + . . . -f a.,„ 
there will be a quantity 
G = Uo • “ 2 m —‘2:ma^ . a^ra- 1 + ^2 • « 2 »i- 2 ± “ 2maiff 2 m- 1 + «o • «2m5 
which, although not a combinant, will satisfy the differential equations necessary to 
prove it to be an ordinary invariant to the two given functions. 
Art. ( 68 .). Now let us consider the three forms/, <p and the subsidiary form 
/= a^x^-\-ma^ . a;'"' L?/ + • • • + «m -i/”* 
<p—h.x^-\-mb^.x'"'~'^.y -\- ... 
n=M,. 3 /”‘-'-(m-l)z^ 2 .//'‘"L^r±&c. +{ — )aT'x”'-\ 
where Mj, U 2 , ... u„, are to be treated as constants. 
f l-2...(2f+l) A/. /V’-"- f 
x-Zi+i'J m[m—\)...{m—2i)\jdx' dyJ *' 
1.2...(2i +l) /'^d , ^ 
E 2 <+ 1 • m{m — \)... [m —2i) ' ^dij) ' 
Make 
