1903-4.] Dr Muir on the Theory of Continuants. 
133 
and we shall thus, by the same process as has been applied 
to improper continued fractions, obtain 
N j+1 D S - N*D j+ i = ( J - l) 4 x ( J - 1)‘ 
=(- iy.” 
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 side of the diagonal. 
The footnote contains two additional observations, the first 
being to the effect that the new mode of representation 
“ gives an immediate and visible proof of the simple and elegant 
rule for forming any such numerators or denominators by 
means of the principal terms [term ?] in each ; the rule, I mean, 
according to which the i th denominator may be formed from 
• • • T 
(q ly q 2 , . . . , qi being the successive quotients) and the ^ th 
numerator from 
?2?8?4 • • - ft 
by leaving out from the above products respectively any 
pair or any number of pairs of consecutive quotients as q ? q ? + v 
For instance, from b Y leaving out q x q 2 , q 2 q 3 , q 3 ? 4 
and q^q b we obtain 
+ Ms + q } q 2 q 5 + : 
and by leaving out Msi-ftft* q^q A q h , we obtain 
ft + ft + ft ; 
so that the total denominator becomes 
ftftftftft + Ms + Ms + ftft? 5 + ftftft + ft + ft + ft l 
and in like manner the numerator of the same convergent is 
ftftftft { 1 + 4- + — + i 
( Ms ftft Ms ftftftft ' 
i.e. 
ftftftft + ftft + ftft + ftft + 1 ” 
The “rule” here spoken of is that enunciated for the more 
general case of 
Cl, + -i- On 
