514 MR. SYLVESTER ON THE FORMAL PROPERTIES OF THE BEZOUTIANT 
by the ordinary method of M. Sturm, had been long felt by me to be a desideratum, 
and as an object the accomplishment of which was indispensable to the ulterior deve- 
lopment of the theory, and it is certain that I did not in anticipation exaggerate the 
importance of the result to be attained. 
Art. (60.). It may happen that the Bezoutiant tof and <p (each of the with degiee) 
may become a quadratic function of less than m independent variables, or the Bezou- 
toid to/ (a function in .t of the mX.h degree) of less than (m— 1) independent variables. 
This will take place whenever / and <p have roots in common, or whenever F has 
equal roots. The number of independent relations of equality between the roots of 
/and ?), and the amount of multiplicity, however distributed, among the roots of F, 
will be indicated by the number of orders thus disappearing out of the general form 
of the Bezoutiant and Bezootoid in the respective cases*. In what particular mode 
the form of each would be affected according to the manner of the distribution of the 
equalities and the multiplicity requires a specific discussion, which I must reserve for 
some future occasion. 
Art. (61.). I shall devote the remainder of this memoir to a consideration of the 
properties and affinities of Bezoutiants or Bezoutoids, regarded from the point 
of view of the Calculus of Invariants. For this purpose it will be more convenient 
hereafter to convert all the functions which we are concerned with into homogeneous 
forms, and I shall accordingly for the future use /and (p to denote functions each of 
.r and y, which I shall write under the form 
/= tfo • * -3/ + ^ • •/+•••+ • X”' 
Tft 1 
-y + .3/' -h 
In what follows a knowledge of the general principles of the Method of Invariants is 
presupposed, but a perusal of my two papers on the Calculus of Forms in the Cam- 
bridge and Dublin Mathematical Journal, February and May 1852, will furnish 
nearly all the information that is strictly necessary for the present purpose. The firsi 
point to be established is, that B, the Bezoutian of /c and px, is a Covariant to the 
system/, p ; the variables in B being in compound relation of cogredience with the 
combinations of powers of x and y. 
That is to say, I propose to show that if /, g, h, k be any four quantities, taken for 
greater simplicity subject to the relation fk—gh=\, and if on substituting/r-fg^/ for 
X and hx-\-ky iov y,f{x,y) becomes 
Ao.x”‘+OTA,.a^''‘-k^+m.^^A*.j;"‘-k/-fA„../‘, say G{x,y), 
* I have elsewhere defined how this word order, as here employed, is to be understood. If F, a homoge- 
neous function of x^, x^,....Xn, can be expressed as a function of Mj, Mj- ■ ••«»-> (all linear functions of .r,, 
F is said to be a function of k— i orders, or to have lost i of the orders belonging to the complete form. 
