110 Proceedings of Royal Society of Edinburgh. [sess. 
where each integer except the first in the upper minor diagonal 
appears twice. 
Another having a like interest is 
x p 
i % P~i 
- 1 X p - 1 
- 1 J X p - 1 J 
-p + 1 x 1 
-p + J a? 
-p 
which is the case of § 4 where b = d= 1 , /? = c = J . 
(6) The theorem which is the basis of the two identities of § 3, 4 
is the following : — 
If in the continuant of the n th order 
x \ 
~ Pn-l X b 2 
- Pn-2 X 
= x(x 2 + l 2 ) (x 2 + 2 2 ) . . . (x 2 +p 2 ), 
-@2 X b n-l 
-Si X 
the difference betioeen the element following x in any odd row and 
the element preceding x be constant , equal to h x say ; and the corre- 
sponding difference in the case of any even row be also constant , 
equal to /5 : say ; then 
x2 +¥i 
is a factor of the continuant , the co-factor being the similar con- 
tinuant of the (n — 2) ih order whose minor diagonals are got from 
those of the original by striking out bj , b 2 from the one and 
Si 3 S 2 f rom lh e other. 
(7) If the last row of the continuant he an exception in this 
matter of a constant difference, the continuant instead of resolving 
into two becomes transformed into another continuant. 
Thus, for the case of the 6th order, wdiere in the given con- 
