247 
1915-16.] On the Theory of Continued-Fractions. 
and raising it to the w th power we obtain 
M n = I i(w + l)(w + 2) \n(n + 2) £n(n- 1) \ 
-n(n + 2) l-|w(% + l) -w(w-2) 
\ \n{n- 1) £(w-2)(?i-3) / 
so the expansion of the continued-fraction as a power-series is 
1_3 + _ | .( -l) n (7t + l)(» + 2) 
x X 2 x 3 ' c lx n+l 
Example 2. 
Consider next the continued-fraction 
_ — a(l — a)(/3 - 5) 2 
a; + (a0-a5 + 5) ^ + ( aS _ a/3 + ^ ' 
Here we may write the matrix M in the form 
/ a/3 — aS + 8 a(j8-8) \ 
\ (l-a)(jB-5) aS-a/3 + /3/ 
and its n ih power is 
M w = / a/3 w + (l- a )5 M a()8 w -8 n ) \ 
\ (1 - a)(i8 M - S w ) (1 - a)/3 w + adn ) 
so the continued-fraction is equivalent to the power-series 
^ (-ir( a r + 5^-08") 
n 
I did not originally arrive at the theorem by the way indicated in the above proof : it 
was suggested by combining Sylvester’s theorem * * * § that any continued-fraction can be ex- 
pressed as the quotient of two continuants with Cayley’s theorem f that a matrix always 
satisfies a matrix-equation of its own order, and Kronecker’s corollary J to Cayley’s theorem, 
in which the elements of the reciprocal of a determinant are expanded in descending powers 
of a parameter, the coefficients being elements in the powers of a matrix. The combination 
of these theorems gives the result under discussion readily. 
III. Stieltjes’ Method of converting Continued-Fractions 
into Power-Series. 
So far as I am aware, the only previous discussion of the problem of 
converting a continued-fraction into a power-series is due to Stieltjes. § 
He took the continued-fraction in the form 
x + -+ 
1 + 
x+ 
1 + . 
+ 
X + 
and arrived finally at the following solution 
(7) 
* Phil. Mag. (4), 5, p. 446, 6, p. 297 (1853) ; Math. Papers , 1 , pp. 609, 641. 
f Phil. Trans., 148 (1858), p. 17 ; Coll. Papers , 2, p. 475. 
J Berlin Monatsb., 1873, p. 117 ; Berlin Sitzungsb ., 1890, p. 1081. 
§ Annales de la Fac. des Sc. de Toulouse , 3 (1889), H. 
