670 Proceedings of Royal Society of Edinburgh. [sess. 
Mum, Th. (1874, Feb.). 
[Further note on continuants. Proceedings Roy. Soc. Edinburgh , 
viii. pp. 380-382.] 
Proceeding from the fact, obtained in the previous paper, that 
the order of a continuant may be depressed if the first element of 
the main diagonal be unity, viz., that 
and that 
K 
- (b l + a^)(b 2 + a 2 ) 
b 2 b 3 . . . 
V 1 K ( b 2 b i ■ 
L a 2 a S . • - 
. J \ a i + 0 1 a 2 a 3 
rnb-L b 2 . . . 
) = ra-K ( bl ' ' 
i ct 2 a 3 . . 
• / \a^ a 2 a 3 . 
ma 
the author shows that 
f b 1 ip 1 + a i ) b% (b 2 + a 2 ) b 3 
\1 rtj a 2 a 3 
and this is at once used to obtain as an equivalent for 
d Y d 2 d 3 .... 
• • • • 
the continued fraction given in Stern’s monograph of 1833 ( Grelle’s 
Journal , x. p. 267). 
The fact communicated by Cayley, but now known to be found 
in Nachreiner’s paper of 1872, viz., that any continuant is expres- 
sible by means of a simple continuant, is noted; likewise the 
identities 
K (aqar 1 , a 2 x , a 3 x ~ l , a^x ,...)„ = K(a 1 , a 2 , . . . , uf) if n be even 
and = ar 1 K (a 1 ,a 2 , ... , a n )if n be odd. 
Wolstenholme, J. (1874, May). 
[(Evaluation of a simple continuant whose diagonal is univarial) 
(Problem 4391). Educ. Times , xxvii. pp. 45-67 ; Math, 
from Educ. Times , xxi. pp. 83-85.] 
The problem is to prove that 
K(a, -1 a~\ . . . , a) n = sin m+ 1)0/ sin 0, if a = 2 cos 0. 
Two proofs are given, the more interesting depending on the 
fact that 
K n = - K n _ 2 , 
