The theory of determinants in the historical order of development, by Sir Thomas Muir.

CONTINUANTS (SYLVESTER, 1853)41

417

Following on this comes the assertion that
"We may treat a proper continued fraction [i.e. with positive unit
numerators],in precisely the same manner, substituting. throughout,~- 1 in place of 1 in the generating matrix, and we shall thus, by the
same, process as has been applied to improper continued fractions,
obtain_____
Ni +A - Ni Di- ( - )
This would seem to imply that as yet Sylvester had not observed
that an alternative mode of representation was obtainable by
merely changing the sign of the units on one s ide of the diagonal.
The footnote contains two additional observations, the first
being to the effect that the new mode of representation
Ccgives an immediate and visible proof of the simple and elegant rule
for, forming any such numerators or denominators by means of the
principal terms [terml] in each; the rule, I mean, according to which
the ith denominator may be formed from
I qlq2q-q4...
(q q2,.., qj being the successive quotients) and the ith numerator.from.
q2q~q4...
by leaving out from the above, products respectively any pair or any
number of pairs of consecutive q-uotients as qpqp+i. For instance, from
qlq2q~q4q5 by, leaving out qjq2, M~3, q.,q4 and q4q5 we obtain
q3q4q5 + qlq4q5 + qlq2q5 + qlq2q3:
and by leaving out q~q2 q.,q4, qlq2* q4q5, q2q3.q4q5 we obtain
q5 + q3 ~ q1
so that the total denominator becomes
qlq2q~q4q5 + q3q4q5 + qlq4q5 + qlq2q5 + qlq2q3 + q5 + q3 +q1;
and in like maniner the numerator of the same convergent is
q2q3q4q5' {I + + + q~5+ qqq
q2q3q4q5 + -q4q5 + q2q5 + q2q3 ~41
The "rule " here spoken of is.- that enunciated for the more
general case of
+b b
a2 3+
M.D. II. 2Da.

M.D. IL: D