132 Proceedings of Royal Society of Edinburgh. [s 
and if, by way of example, we take the fourth and fifth con- 
vergents, these will be in the umbral notation represented by 
h h 
and 
a Y a 2 a 3 a A 
b \ b 2 b 3 b 4 
respectively. Hence 
b 2 b 3 b 4 b 5 
a 1 a 2 a 3 a 4 a 5 
&i b 3 b± b 5 
n 5 d 4 -n 4 d 5 
.. a 2 
a z 
«4 
% 
V 
a 2 
a 3 
«4 
°i _ 
<* 2 
a 3 
h 
b 4 
h 
A 
h 
K 
b 4 
b 3 
_ a 2 
«; 3 
«5 
v 
« 2 
a 3 
a 4 
“i _ 
a 2 
a 3 
~ \ 
A 
\ 
h 
b 2 
h 
a 2 
a z 
a 5 
X 
a 2 
a s 
a 4 
a 1 
~ b 2 
h 
K 
\ 
h 
b 3 
h 
_ a 2 
a 3 
a 5 
v 
a l 
a 2 
h 
a 3 
a 4 
~ \ 
\ 
b s 
\ 
A 
b 2 
h 
h, 
1 
B 
1 
0 
1 
0 
0 
0 
X 
0 
1 
C 
1 
B 
i 
0 
0 
0 
0 
1 
D 
1 
c 
1 
0 
0 
0 
0 
1 
0 
1 
D 
1, 
= 1 
X 
1 
= 
1, 
b 2 b 3 b 4 b b b 1 , 
4 5 
b A K 
a~ a. a. 
b 2 b 3 b, b. 
as was to be proved. And the demonstration is evidently 
general in its nature.” 
In regard to this there has to be noted, first, the use of 
^2 $3 ^ 
b 2 b 3 b 4 
when it would have been equally effective to use 
2 3 4 
2 3 4 ; 
and, second, the use of a theorem for expressing the product of a 
five-line determinant and one of its secondary minors as an 
aggregate of products of pairs of four-line determinants. 
Following on this comes the assertion that 
“We may treat a proper continued fraction \i.e. with positive 
unit numerators] in precisely the same manner, substituting 
throughout J - 1 in place of 1 in the generating matrix, 
