
TRANSFORMATION OF INFINITE SERIES, ETC. 471 
developes readily a special property, from which he succeeds in showing that 
Fa, 8+1,y7+1, a) 1 
F(a, 8 112) a(y—B)a 
y(y+1) 
(8 +1) (y+1—a)x 
(y+1)(y+2) 
(a+1)(y+1—B)x 
(y+ 2) (y+3) 
(B +2) (y+2—a)x 
(y+38) (y+4); 
ie 



This is but a particular case of (A) when the reciprocal of both sides is 
taken, and so also are one or two other previously discovered theorems of less 
generality and importance. 
The only perfectly general method hitherto known for the transformation of 
series into continued fractions is that discovered by EuLER,* and it is important 
to notice the radical distinction between the continued fractions given by the 
two methods for any particular series. Taking, for example, the series for 



1 : 3 
tan —1z, viz., «— 5a +z a’ —...., the two continued fractions are found 
to be 
x we 
ee and 
ge? roo 2 yo 
a ores ne 3730? B22 erie ale 372? 
os ee B+ 
700 + nee a 
the former being that given by EuLer’s method. Now, if in the case of the 
left-hand fraction we stop at any particular partial denominator, the portion 
taken represents exactly a corresponding number of terms of the series for 
tan —'2: whereas, if we stop at a particular partial denominator of the right- 
hand fraction, we find that the portion taken represents an injinite series which 
Sor a certain number of terms agrees with the given series for tan —'a. 
The results of the present paper are thus seen to refer only to znjinite series, 
the continued fractions coinciding, it may be, in the limit with these series ; 
and it is a problem of some interest to determine the correction required on 
stopping at any particular partial denominator, or, as is said in the case of 
series, to determine “‘the remainder after 2 terms.” STERN’s method is unsuited 
for this; but now that the theorems have been found, other methods of proof 
will in all likelihood appear, to which there may not be the same objection.t 
* Opuscula Analytica, t. ii. p. 138. 
+ One such method has since been discovered by the author, and communicated to the Mathe- 
matical Society of London, in a paper read February 10th, 1876. 
VOL. XXVII. PART IV. 61 
