244 Proceedings of the Royal Society of Edinburgh. [Sess. 
I think it would be a mistake to propose the problems in the form : 
Given two continued-fractions , to find a third continued-fraction 
which is equal to their sum or 'product; or, Given a continued-frac- 
tion whose elements are functions of a variable, to find another 
continued-fraction which represents its differential coefficient : for I 
doubt if the problems so formulated possess any simple solutions. But 
Sylvester showed long ago that any continued-fraction may be re- 
garded as the quotient of two determinants : and if we regard con- 
tinued-fractions in this light, advancing boldly from their theory to 
the theory of determinants, and aiming to express the products or 
sums of derivates of continued-fractions in the form of determinants, 
the situation becomes much more promising: we are, in fact, infusing 
into function-theory the marvellous flexibility and comprehensiveness of 
the determinant calculus. 
In the present paper we are chiefly concerned with the expansion and 
differentiation of continued-fractions from this point of view. 
The ultimate purpose of the work lies in its application to function- 
theory and the solution of differential equations: the continued-fractions 
are then non -terminating, and the determinants associated with them are 
of infinite order. But as the present paper is occupied only with the 
formal algorithms, the order may for convenience be supposed to be finite, 
and questions relating to the convergence of infinite processes need not 
here be considered. 
II. An Expansion-Theorem. 
Our first object will be to show that the elementary algebraic identity 
JL-L_A i 2 _ ^ 
b + x x x 2 + x 3 x 4 + ’ ’ ' 
admits of a generalisation, which furnishes the transformation of a 
continued-fraction into a power-series, and which may be thus stated : 
Any continued-fraction 
1 
b + x- 
\ + x 
a 2 
— a n 
b n + x " 
may be expanded as a power-series in 1/x , in the form 
1 b (1) b {2) b® 
X X 2 + X 3 X* + 
(2) 
. ( 3 ) 
