RATIONAL INTEGRAL CONTINUED FRACTION. 
539 
L( representing the leading coefficient in the (ith) simplified residue, and the sign of 
interrogation (?) denoting some function of (possibly a constant) remaining 
to be determined. And reverting to art. (l.), the quantity that would be called Ko,„. 
according to the notation employed in the formulae expressing in that article, will 
(abstraction being made of the algebraical sign and using for greater brevity (/), 
(/—I), See. to express \-\-oji-i, See.) come to be represented by 
Y 4(i.-3) Y — 5) 
(0 
&c. 
&c.’ 
a similar convention being supposed to be made respecting the numerator and deno- 
minator of each convergent as was made respecting them in the particular case 
treated of in art. (f), page 4/3. 
Art. (to,). I will merely add a very few words in generalization of the method of 
limiting the roots of fx given in the Supplement to the fourth Section. As an inferior 
limit to fx is identical with a superior limit to f{ — x), we may confine our attention 
to superior limits alone. Suppose then that 
J_ _l 1 l_ i_ 1 1 1 
fx <cii— Q2~ "”Qi — O')" Q'2 — (^)l~ (GI)2~ "*"(^)i’ 
where the partial quotients Q are each of any arbitrary degree in x, and have all one 
algebraical sign in the coefficients of the highest powers of x from Qi to Q^, and all 
the same sign (contrary to the former), in the coefficients of the highest powers of x 
from Q- to Q(,, and so on alternately, then 1°, a superior limit to the superior limits of 
the cumulants [QiQa.-.QJ, [Q'l, -..[(Q), (Q) 2 .-.(Q){i)] will be a superior 
limit to fx, so that it remains only to give a rule for finding a superior limit to a 
curaulant [Q^Qa, Qs.-.QJ, which, 2°, is to be found by making 
Q.-M,=0, Q2-M2=0, Q3-M3=0...Q;-M,=0, 
where 
r 1 ra ri - 1 
^ 1 , (X. 2 , being any quantities entirely independent and arbitrary except in regard 
to their being all of the same sign as the leading coefficient in the element Qi, Q 2 ,..Q;. 
W e may then find L,, ...L^ any superior limits to the roots of x in these i 
equations respectively ; L, the greatest of these, will be a superior limit to the proposed 
cumulant [Qi Q 2 ...QJ; and it may be observed that Mj M 2 . ..M,- are the general values 
which satisfy the equation 
,,11 1 ^ 
subject to the condition that for all values of e 
1111 
M,- M,- Mer2''-M, 
shall have a given invariable sign. The first part of the process, as just shown, con- 
sists in separating the type of the total cumulant which represents fx into partial 
