UNDER THE FORM OF A CONTINUED FRACTION. 
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syzygetic function, or more briefly, a conjunctive of / and (p, and A, and L, may be 
termed the syzygetic factors to so considered. If we divide each term of the 
equation (15.) by the allotrious factor (M), we have 
where R, is the /th simplified residue to (f, p ) ; and if we call and so 
as to obtain the equation 
T,./-P,.p=R„ (16.) 
we see that the fraction formed by the component factors to any simplified residue 
(/) P)> will be identical in value (although no longer in its separate terms) with 
one of the corresponding convergents to j, exhibited under the form of an improper 
continued fraction. I shall in the next section show how, not only the successive 
simplified residues, but also the component syzygetic factors of each of them, and 
consequently the successive convergents, may be expressed in terms of the roots of 
the two given functions. 
Since the preceding section was composed the valuable memoir of the lamented 
Jacobi, entitled De Eliminatione Variabilis e duabus Equationibus Algebraicis,” 
Crelle, vol. xvi., has fallen under my notice. That memoir is restricted to the con- 
sideration of two equations of the same degree, and the principal results in this 
section as regards the Bezoutic square and the allotrious factors applicable to that 
case will be found contained therein. The mode of treatment however is sufficiently 
dissimilar to justify this section being preserved unaltered under its original form. 
Section II. 
On the general solution in terms of the roots of any two given algebraical functions 
of y. of the syzygetic equation, which connects them with a third function, whose 
degree in (x) is given, hut whose form is to he determined. 
Art. (14.). Lety and p be two given functions in x of the degrees m and n respect- 
ively in X, and for the sake of greater simplicity let the coefficients of the highest 
power oi X in y and p be each taken unity, and let it be proposed to solve the 
syzygetic equation 
7,.f—t..p-\-’^,=0, (17.) 
where is given only in the number of its dimensions in x, which I suppose to be (/) ; 
but the forms of r,, t„ are all to be determined in terms of h^, Aa . . . the roots of 
/ and ^ij ^ 2 j • • • the roots of p. 
I shall begin with finding ; and before giving a more general representation of 
^ 4 , I propose now to demonstrate that we may make 
^ i X Qv — hj^ {x AjJ . . . (j7 — hg ) } , 
3 L 2 
( 18 .) 
