VIEWED WITH RELATION TO THE METHOD OF INVARIANTS. 519 
functions in their corporate combined capacity qud system. That the Bezoutiant 
possesses this property is evident ; for if instead of/and (p we write and 
any such quantity as ar.h,~a,.b^ {a„ being- coefficients in f, and a„ b, the corre- 
sponding ones in p) becomes 
so that B, the Bezoutiant, becomes increased in the ratio of {ki'—k'i)”^, i. e. remains 
always unaltered in point of form and absolutely immutable, provided that ki’-k'i be 
taken, as we may always suppose to be the case, equal to 1. 
We derive immediately from this observation, the somewhat remarkable geometrical 
proposition, that the intersections with the axis of made by any two curves of the 
family of curves u=-kf{x)-\-q.<p{x), (/and p being functions of of the sa.me degree) 
give rise to a constant number of effective intercalations, whatever values be given 
to X or for the two curves so selected. 
Art. (65.). B(7 /i, Mg, ... M„j) being a covariant of the system/and p, and Wg, ... u 
cogredient with .r’"'-', ... y^~\ it follows from a general principle in the theory 
of invariants, that on making m„ Wg, ... respectively equal to the quantities with 
which they are cogredient, B will become an ordinary covariant to/ and p. By this 
transformation B becomes a function of x and y of the degree 2 (mi— 1) in x and y 
conjointly, and linear in respect to the coefficients of / and also in respect to those 
of p. The only covariant capable of answering this description is what I am in the 
habit of calling the Jacobian (after the name of the late but ever-illustrious Jacobi), 
a term capable of application to any number of homogeneous functions of as many 
variables. In the case before us, where we have two functions of two variables, the 
Jacobian 
\df c 
J(/^)= 
dx’dy 
d£ 4 
dy’ dx’ 
dx ’ dx 
dj_^ 4 
dy ’ dy 
We have then the interesting proposition that the Bezoutiant to two functions, when 
the variables in the former are replaced by the combinations of the variables in the 
latter, with which they are cogredient, becomes the JacobianT. So in the case of a 
single function F of the degree m, the Bezoutiant, i. e. the Bezoutoid to 
on 
dx ’ dy 
making the (m-1) variables which it contains identical with x^'-\y ■, ...y^~" 
respectively, becomes identical with the Jacobian to i. e. the Hessian of F, viz. 
'd^F 
d?-F 
dx^ ’ 
dxdy 
d^F 
dxdy ’ 
* I have subsequently found that this proposition is contained under another mode of statement, at the 
end of Section 2 of the Memoir of Jacobi, “De Eliminatione,” above referred to. 
t For a strict proof of this proposition see Supplement to Third Section of this memoir. 
MDCCCLIII. 3 Y 
