449 
EXPRESSED IN TERMS OF THE ROOTS OF fx. 
But since A’„+i, Ri’e infinite in value, 
h„ h„ 
9^1 H2 Ht 
L _ 
^n+ 1 ^n + 2 • • • 
Hence 
and 
But by what has been shown antecedently [taking- account of the fact of the leading- 
coefficient of being s in place of 1, which introduces the factor s'], we have 
wheie B) is the Bezoutian secondary of the (m— z— l)th degree in x to /and hut 
B- it has been proved =B; the Bezoutian secondary of the same degree to /and (p ; 
{—1 
hence — )' “-B;. 
Art. (29.). If now we return to the syzygetic equation, r./— ^©+^=0, ^ may he 
treated as known, having in fact been completely determined as a function of the 
roots, as well in its most general form, as also so as to represent the simplified residues 
to / and (p in the preceding articles; it remains to determine the values of r and t 
as functions of the roots corresponding to any allowable form of 3^, but I shall confine 
the investigation to the case where ^ is the lowest-weighted conjunctive, or which 
is the same thing, a simplified residue to ^and <p of any given degree in x-, each value 
of - will then represent one of the convergents to y when expanded under the form 
of a continued fraction. If he of the /th degree in x, r is of the degree (w — 1) 
and t of the degree {m — i—\). This being supposed, and calling n — i—\=v, 
m — i— \ I say that t will he represented by G and r by F, where 
G = ( - ) ''2 (^ - A, j {x -h^y..{x— A, J 
[K 
A,, . . . A„ 
yi-2 
“A 
?i 
ha 
K, 
1 
+ 
__ 
5 
and T is an analogous form F ; A, Ag.-.A^, as heretofore, being the roots of /and 
of p. To fix the ideas and make the demonstration more immediately seizahle, give 
m and n specific values; thus let w=5, w.=4, i=2, so that — 2— 1=2. Put ^ 
under the form ^, 0 , so that ^ in the case before us 
