676 Proceedings of Royal Society of Edinburgh. [sess. 
The paper concludes with two other curious theorems of similar 
character. * 
* The rather noteworthy theorem in continuants which is the basis of this 
striking result in continued fractions can be still more widely generalised as 
follows : — 
The continuant 
! \ 
+ A 
h 
^ + i'A + 5 , Cl 
s 2 7\ 
- 3 + ^ A + S 0 C 2 
s 3 r 2 
is unaltered in value by adding 
?i + f!A + v> 
*4 *3 
h + s AA+s 4 c 4 
T l C l y r 2 C 2 " S 1 C 1 5 r 3 C 3 _ S 2 C 2 J r 4 C 4 ~ S 3 C 3 > ~ S 4 C 4 
to the elements of the main diagonal, and changing the elements of the upper 
minor diagonal into 
Putting in this r 2 , r 2 , . . . = s 1 , s 2 , . . .we have the identity 
- + A 
Cj — — + A + 
S 2 
— + A + SoC 9 
c 3 —f + A + S3C3 
c 4 — - + A + s 4 - 4 
Spl 
— — + A + SjCj -^-^2 
% S 2 
c n — + A + s 9 c 9 
- 3 - + A + S3C3 ~b 4 
3 , s 4 
c _i + A + s 4 c 4 
i + A 
and specialising further by putting each of the s’s equal to s we come back to 
the theorem of the paper above reviewed. 
