VIEWED WITH RELATION TO THE METHOD OF INVARIANTS. 515 
and ip (x, y) becomes 
til ““ 1 
say T{x, y), 
andif B'(w;, be the Bezoutiant to G and T; B(u„ u^...u^) being that to fandp, 
then, on making u^, the same linear functions of m', 
as ifx-^-gyY', {fxYgyT~\hx-\-'ky)^ 
are respectively of 
j :*", x'^~\y ...x.y"'~'^ ; y"^, 
B will become identical with B'. I was led to suspect the high probability of the 
truth of this proposition concerning the invariance of the Bezoutiant from the follow- 
ing considerations : 1st. That for the particular case where f and ^are the differential 
derivatives in respect to x and y respectively of the same function F(x, y), the 
Bezoutiant of / and p, which then becomes the Bezoutoid of F, determines the 
number of real factors in F, which obviously remains the same for all linear trans- 
formations of F. 2ndly. That taking/ and (pin their most general form, the invariant 
to their Bezoutiant, /. e. the determinant of their Bezoutiant is an invariant of/ and p, 
being in fact the resultant of these two functions ; now as every concomitant (an in- 
variantive form of the most general kind) to a concomitant is itself a concomitant to 
the primitive, so it appeared to me, and is I believe true (although awaiting strict 
proof), that any form satisfying certain necessary and tolerably obvious conditions of 
homogeneity and isobarism, a concomitant to which is also a concomitant to a given 
form, will be itself a concomitant to such form ; this principle, if admitted, would 
be of course at once conclusive as to the Bezoutiant being an invariantive concomi- 
tant to the functions from which it is derived. 
Art. (61*.). Since the publication of the two papers above referred to on the Calculus 
of Forms, I have made the important observation that every species of concomitant, 
however complex, to a given system of functions, may be treated as a simple invariant 
of a system including the given system together with an appropriate superadded 
system of absolute functions ; thus an ordinary covariant involving only one system 
of variables, as m , v,w... cogredient with x, y, z... the variables of a system S, is in fact 
an invariant of the system S combined with the system nx — vy, vz — wy, wx—uz, &c., 
u, V, w ... being treated as constants; so again a simple contravariant of S is an 
invariant of S combined with the equation ux-{-vy+wz-\-&ec. ; so again, to meet the 
case before us, a covariant to the binary system / and p expressed as a function of 
?/„ where u,...a,„ are cogredient with x^-'\y ...y^-\ may be regarded 
as an invariant of the ternary system/ p, G, where 
n = +m .7/— V . . . -p ( - )“- ‘ . , .a:—', 
(Mi, 7/2, ...//^_j being here to be treated as constants), and accordingly the differential 
equations which serve to define in the most general and absolute manner such cova- 
