248 Proceedings of the Royal Society of Edinburgh. [Sess. 
Calculate first a series of quantities a 00 , /3 00 , a 01 , a n , j3 01 , /3 n , a 02 , a 12 , 
a 13 , /3 02 , . . . by the formulce 
a oo ~ 1 
:> * = 0 when i > /t 
ft,fc = 0 when z>&J 
Aft = a 0ft + c 2 a 3* 
Aft = a ift + C 4 a 2fc 
Aft = a 2ft + C 6 a 3ft 
( 8 ) 
(9) 
a o, ft+i — c i A* 
ai J ft+i = c 3 A& + Aft 
«2,ft+i = %Aft + A* 
( 10 ) 
Then if the continued-fraction (7) 'is equivalent to the power-series 
1 6 < 2) 
JC a; 2 £ 3 
. + 
( - lp&w 
• (H) 
~ ~ x n+1 + ' ' ' ’ 
the quadratic form 
.... 
o o 
is equal to 
( a 00 X 0 + a 01 X l + a 02 X 2 + • • • ) 2 + c i c 2( a ll^l "h a 12-^2 a 13^3 "t - * * * )“ 
+ + a 23 X 3 + . . . ) 2 + . . . (12) 
and therefore the b’s can be determined from any of the equations of 
the type 
b [i+k) = d Qi a 01t + CyC^ a lk + c 1 c 2 c 3 c 4 a 2i a 2& -f (13) 
This solution of Stieltjes’ does not appear at first sight to have any 
direct connection with the solution given in § 1. The two can, however, 
be brought into relation in the following way 
First, by the process known as “ contraction ” of continued-fractions * 
we can (as Stieltjes was well aware) reduce the continued-fraction (7) to 
the form 
1 
CnC n 
CqCa 
X + Cj- 
X + C'o + Co ; ! : 
2 3 X + c 4 + c 5 
which is of the type (2). To complete the identification with the continued- 
fraction (2) we write 
\ — b , C-jC 2 — , c 2 "h C Z — ^1 » C 3 C 4 — a 2 ’ C 4 "h — ^2 ’ • * • 
(14) 
* This process, which is due to Euler, Nova Acta Petrop ., 2 (1784), p. 36, consists in the 
repeated use of the identity 
c-f a n =(c + a) a ^ 
1 + 
& 
(/3+y) + 8‘ 
