246 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
But the first of these equations may be written 
'0 _ 
7 , , c.x, 
o b + x+ J-J: 
x 0 
and substituting for xjx 0 from the second equation, and so on, we obtain 
ultimately 
Xn 1 
x o b + x- b - f 
a 2 
b 0 + x- 
b n + x 
so the coefficient /3 00 is equal to the continued-fraction (2). 
Now if E denotes the unit matrix 
I 1 0 0 ... 0 \ 
0 1 0 ... 0 
0 0 1 
\ 0 0 1 
0 
the substitution (5) is evidently represented by the matrix M + ofE, where 
M denotes the matrix (4) : and as the substitution (6) is the reciprocal of 
this, it is represented in the symbolic matrix notation by This 
may be expanded as if M and E were algebraical quantities, giving 
E _ M M 2 _ AB 
™ r/i ™4 + • ' * 5 
tAj tAJ tAJ vU 
the numerators of all the fractions being matrices: and therefore /3 00 , 
which is the leading coefficient in this substitution, is equal to 
1 _ W* _ b M 
ryt rJl ry > 3 ™4 * * 
tAS tAJ %Aj tAJ 
where b {n) denotes the leading coefficient in the matrix M” . Since we have 
already proved that (3 00 is equal to the continued-fraction, the theorem is 
now established. 
Example 1. 
As an example of the theorem, consider the continued-fraction 
Here the matrix is 
1 
3 +x + 
M = 
3 
(■5O 
1 0 
