251 
1915-16.] On the Theory of Continued-Fractions. 
then 
b 1 0 0 0 . . . \ 
a-L b l 1 0 0 . . . 
0 a 2 b 2 1 0 . . . 
0 0 a 8 6 8 1 . . . 
1 b {1) 6 (2) 
ftd) 6 (2) 6® 
6 < 2 > 6 ® 6 (4) 
= cq 2 a 2 
and similarly, in general, the persymmetric determinant of the n th order 
formed of 1 , b (]) , b (2) , b (3) , b (4) , . . . has the value a^ -1 a 2 n ~ 2 . . . a n _ r 
It is easily shown in the same way that the persymmetric determinants 
formed of b (1) , b {2) , b {3) , . . . can be evaluated in terms of b, b v b v b s , . . . 
Hence we see that if the leading elements in all the powers of a continuant 
matrix are given, the elements of the matrix itself can be obtained in terms 
of persymmetric determinants formed of these quantities. 
Y. The Differentiation of Continued-Fractions. 
We shall now proceed to find the differential coefficient of a continued- 
fraction 
1 
b + x - 
b-y + X 
b 2 + x 
On 
b n + X 
with respect to x. So far as I am aware, no investigation on the subject 
has been published hitherto. 
Denoting the continued-fraction by S, we know from § 2 that S is 
equal to the leading coefficient in the matrix when E denotes the 
dS 
unit matrix and M denotes the matrix (4). So — ^ must be equal to the 
1 
leading coefficient in the matrix 
; that is to say, it must be 
(M + ccE) 5 
equal to the leading coefficient in the matrix reciprocal to (M + ccE) 2 . W e 
thus have immediately the theorem : 
if 
i 
■ a, 
0l+X b 2 + X ~ • • • 
b n + x 
S = 
