1904 - 5 .] Dr Muir on the Theory of Continuants. 
657 
the other gives 
(b 2 - d)Y + bfi 2 Q' bf 
P + ftjQ d + b 2 — + 
+ 
b\ 
if P _ ^ 1^2 
Q' d + 
d+br ~ b ^ + d+t 
+i 
b r b r j r i # 
d 3 
and, the first continued fraction in the one case being the same as 
the second continued fraction in the other case, save that the last 
denominator of the former is d + b r+1 and of the latter d , it thus at 
once follows that If the continued fraction 
or S, say, 
d + 
b A i b 
~d + 23 
be convergent , then 
b 2 
6o 2 
= b ® + ^ 
d + b i~ h + d + b~- b 3 + 1 ^ + d + b 1 
and. 
V 
= - b-L S &1 
<i + t> 2 -b 1 + d + b , S + d-b i 
consequently 
d 
b 1 + L 
b . 2 
d S + d-b 1 
d + b i h 2 + d + i 2 _ h ^ 2'S + d + \ 
2 + b i + d + b -b + ^ *= - . — 
Z a + ° 2 °i + d + b z -b 2 + 2 S + d-\ 
and therefore finally 
i t ft+ _v 
| 2 ° l + 
h 2 
U 2 
d + b \ b i+ d + b" i -b s + > 
1 - + 1 + *L b 3 1 
PEOC. EOY. SOC. EDIN. — VOL. XXV. 
'd\* 
9 / . 
42 
