1903-4.] Dr Muir on the Theory of Continuants. 
135 
is the differential-quotient of the numerator, Spottiswoode did 
nothing but report the fundamental result reached by Sylvester. 
The full passage (p. 37 4) is as follows : — 
“ The improper continued fraction 
A 
1 
0 . 
. 0 
0 
1 
B 
1 . 
. . 0 
0 
0 
1 
c . 
. . 0 
0 
0 
0 
0 . 
. . M 
1 
0 
0 
0 . 
. . 1 
N 
in which any number of rows may he taken at pleasure, and 
the formula will give the corresponding convergent fraction. 
The same holds good for the continued fraction 
if we write 
1 B 1 . . . 
0 1 C . . . 
Sylvester, J. J. (1853, Sept.). 
[On a fundamental rule in the algorithm of continued fractions. 
Philos. Mag. (4), vi. pp. 297-299.] 
Without any reference to his previous paper on the subject 
Sylvester here announces that if 
he the denominator of the £ th convergent to 
C - 
where 
V = A 0 . . 
(^1 > $ 2 » • * • > ^ i ) 
