1915-16.] On the Theory of Continued-Fractions. 
249 
These equations (14) can all be comprehended in the statement that the 
matrix 
M 
is the product of the two matrices 
1 0 0 0 ... 1 
and 
o 
o 
O 
O 
<N 
0 c 3 1 0 0... 
0 c 4 10... 
0 0 c 5 1 0 . . . 
0 0 Cg 1 • - . 
.... 
0 0 0 c 7 1 . . . 
and as Stieltjes’ equations (9) and (10) are evidently formed of the sub- 
stitutions which correspond to these latter matrices, the matrix M and its 
powers now enter the problem quite naturally, and the harmonisation of 
Stieltjes’ formula with those of § 2 presents no further difficulty. Stieltjes’ 
constants a ok , a lk , a 2k , . . . are found to he simply the elements in the first 
row of M*. 
IY. Relations involving Persymmetric Determinants. 
The problem converse to that of § 2 is the conversion of a power-series 
into a continued-fraction 
1 _ 
x 
1 
b + x 
&d) b ® b < 3) 
O ”1 o 
+ . 
bl + x b„ 
+ X~ 
in such a way that the w, th convergent to the continued-fraction, when 
expanded as a power-series in ljx, shall agree with the above series as far 
as the 2 n th term inclusive.* This converse problem has been much more 
* This latter condition excludes the comparatively worthless method of converting 
infinite series into continued-fractions which is commonly given in elementary text-books 
on algebra, and which may be expressed by the equation 
i-i+i-i. 
abed 
1 + Y(&- oQ + lP/(b-g)(c-b) 
for in this equation the n th convergent to the continued-fraction coincides with the series 
only so far as its n th term, instead of as far as its 2 n th term. 
