UNDER THE FORM OF A CONTINUED FRACTION. 
421 
The right-hand members of these (w) equations I shall henceforth term the Bezon- 
tians to U and V. 
[The determinant formed by arranging- in a square the n sets of coefficients of the 
n Bezontians, and which I shall term the Bezoutian matrix, gives, as is well known, 
the Resultant (meaning thereby the Result in its simplest form of eliminating the 
variables out) of U and V.] 
Eliminating dialytically, first .r”’* between the first and second, then a”"' and 
between the first, second and third, and so on, and finally, all the powers of .v between 
the 1st, 2nd, 3rd, wth of these Bezoutians, and repeating the first of them, we obtain a 
derived set of {n) equations, the right-hand members of which 1 shall term the secondary 
Bezoutians to U and V, this secondary system of equations being 
Qo . U - Po . V= -f K3^"-^+ . . . 
(jKiQo — KiQi)U — (,Ki.P 0 — Ki.Pi)V=Li..z’" L„_, 
((iKi . 2 K 2 — aKj . iK2)Qo+ ( aKj . Ka — K, .2K2)Qi-l- (Kj , ,K — ,Ki . 1 K 2 ) . Qa)!! 
. > ( 11 .) 
-21^2 2Ki-iK2)Po+( 2K.1-K2 Kj . 2 K 2 ) Pj -j- ( Kl • i ^2 — , Kj . j Ka) Pg) V 
&c. = &c. 
And we can now already without difficulty establish the important proposition, that 
the successive simplified residues to y, expanded under the form of an improper con- 
tinued fraction, abstracting from the algebraical sign (the correctness of which also 
will be established subsequently), will be represented by the n successive Secondary 
Bezoutians to the system U, V. 
For if we write the system of equations (11.) under the general form 
— I-b.V=A;.j?”~'-|-B;a?”“'“'-j- See., 
the degree of and H, in x will be that of Q;_i and R_i, i. e. /— 1 ; and the dimen- 
sions of A„ B;, &c., in respect of each set of coefficients is evidently (/) ; consequently, 
by virtue of art. (2.), A,a;"“^-l-B^'“^-t- &c., which is the /th Bezoutian, will (saving at 
least a numerical factor of a magnitude and algebraical sign to be determined, but 
which (when proper conventions are made) will be subsequently proved to be -j-l) 
represent the /th simplified residue to as was to be shown. 
Art. (6.). More generally, suppose U and V to be respectively of n-i-e andw dimen- 
sions in a:. 
* V is supposed to be taken as the first divisor, and the term residue is used, as hitherto in this paper, 
throughout in the sense appertaining to the expansion conducted, so as to lead to an improper continued 
fraction, in that sense, in fact, in which it would, more strictly speaking, be entitled to the appellation of excess 
rather than that of residue. 
