UNDER THE FORM OF A CONTINUED FRACTION. 
431 
used in the same conventional sense as employed throughout) be 
A^+By-'+C^'-^+ &c. 
BV-' +CV-^4-DV-^+ &c. 
The residue next following, obtained by actually performing the division and duly 
changing the sign of the remainder will be 
which is of the form 
^|B'M-AC'^j^‘-^+ &c. 
Thus the leading coefficients in the complete unreduced residues will be 
A; B'; ^,j^B'M-AC«|, 
and when reduced by the expulsion of the allotrious factor will become A; B'; 
B'.M— A C% and consequently, when B' the leading coefficient of one of the simpli- 
fied residues vanishes, the leading coefficients of the residues immediately preceding 
and following that one will have contrary signs. 
First, let 
yj? = -f- ~ ‘ + &c . , <px= ax" + * + &c . 
As regards the numerical ratio of each Bezoutian secondary to the corresponding 
simplified residue, it has been already observed that there are always unit coefficients 
in the latter of these, and the same is obviously true of the former ; hence if we call 
the progression of the leading coefficients of the simplified residues 
B-i , B2 5 B3 ^ B^, &c., 
and that of the leading coefficients of the Bezoutian secondaries 
Bi ; B 2 ; Bj ; B 4 , &C., 
we have 
Bi = + Ri B2=4:B2 B3=+R3 B4=+R4, &c. 
It may be proved by actual trial that Bi = Ri and B2=R2. Moreover, since the 
signs are invariable, and do not depend upon the values of the coefficients, we may 
suppose B2=0 (which may always be satisfied by real values of the quantities, of 
which B2 is a function) ; we shall also, therefore, have R2=0, and consequently B3 
has the opposite sign to that of Bj, and R3 the opposite sign to that of Rj, which is 
equal to B, ; hence when B2=0, B3 and R3 are equal, and consequently are always 
equal ; in like manner we can prove that R4 and B4 have the same sign when 
R3 and B3 vanish, and consequently are always equal, and so on ad libitum^ which 
proves that the series B,, Bj, . . . B„ is identical with the series Rj, R2, . . . R„, and 
consequently that the Bezoutian secondaries are identical in form, magnitude and 
algebraical sign with the simplified residues. Secondly, when fx and (px are not of 
the same degree, it has been shown that the secondaries formed from the non- 
MDCCCLIII. 3 L 
