508 Proceedings of Royal Society of Edinburgh. [sess. 
That 0 is a factor in the latter case is evident from a consideration 
of the fundamental identity 
(6 h ^6 }, ^e . . . . ) = 40 ....) +,b l (6 l ^ b 
which shows that if, as is easily seen to be the case, the continuant 
of the 3rd order has 0 for a factor, so also must the continuant of 
the 5th order, and therefore also the continuant of the 7th order, 
and so on. 
(3) The fact that the change from a univarial to a bivarial 
diagonal necessitates no change in the coefficients on the right- 
hand side of the identity prepares one for an analogous widening 
of other theorems in which continuants with a univarial diagonal 
are involved. Thus, denoting the continuant in (I) by <£„ we have 
the important condensation theorem — 
$2 m 
0<f) + b 1 
b 2 6<b + b 2 + b 3 b 3 
b± 6$ + b± + b b 
(II) 
6 $ + 5 2 m- 2 + ^2m-l j 
3> 
= 6 
6(f> + b^ + b 2 b 2 
b 3 0cj) + b 3 + 5 4 & 4 
b b 6<f) + /> 4 + b 5 
(ir 
0cf> + b 2m _ 3 + 5 2m _ 2 . 
Dividing & n as it appears in (II) by the cofactor of its first element 
we obtain a continued fraction, and dividing the equivalent con- 
tinuant in (I) by the cofactor of its first element we obtain another 
continued fraction : and as, when n is even, the two divisors 
differ only by the factor <£, the two continued fractions differ to 
the same extent. We thus have 
