1904 - 5 .] Dr Muir on the Theory of Continuants. 
655 
P r + Q r = (w + ctj) 
l -n 
Tb (Xo 
n — do OL q 
and 
P r + nQr = v 
1 - n 
n - a. 
1 
n — a.c 
a r +l 
a r + 1 
a r -f 1 
1 a r + 1 
a r _ 2 a r + 1 
- n — a r _ 1 a r 
a r +l 
a r + 1 
a r + 1 
1 a r + 1 
• • • «r-2 a r + 1 
—n — a r _ x a r 
where the two determinants, being of like formation, may he suitably 
denoted by V 2 and Y 1 , and whence by division there is thus obtained 
n(P r + Q r ) 
P r -t- nQ r 
(n + af) 
v, 
With the Y notation the recursion-formula 
Y^ = fXjY^x + (n + fti-f i) Y 
is next established,* from which it readily follows that 
+ 
n + a 2 
a 2 
+ 
and therefore 
n + a T 
+ 
n + a r _! 
a r-l + 
n + a r 
a r -F 1 , 
* It would have seemed more direct and natural to have shown that V 1 is 
transformable into the continued-fraction determinant 
“1 
7l + a 2 
- 1 
a 2 71 + a ,3 
a r n + d f 
-1 a r + 1 
and therefore Y 2 into the determinant got from this by deleting the first row 
and first column. 
