


188 MR T. MUIR ON EISENSTEIN’S CONTINUED FRACTIONS. 
length. Here he gives a second continued fraction for a series already dealt 
with, viz. :— 






l+a+a°+a° +a OR aes ae er Sg ee (XIX) 
Ih a 4 2 
ioe = ee ees 
S a a 7p a — x 
a 
and concludes as follows :—“ Adjicere liceat formulam generaliorem— 
(1—2)(1—po)(1—p*s).. =i nis Sai ee Ae audios eee: (XX.) 
(1—y)—py)—p*y)... 1+ PY~L Pn 
lp PY a 
ee te er py y as 
Er pees eel 7 
1—p® + 
This, it would appear, was the last communication written by EISENSTEIN on the 
subject of continued fractions ; and I am not aware that any one, with a single 
exception,* has attempted to solve the problems thus left, or made any reference 
to the subject, unless to record his inability to see how the results had been 
obtained. The object of the present paper is to supply this want. 
* In the year 1846 Hers, on the instigation of Jacont, attempted to establish E1sEnTErn’s 
results by means of Euzr’s transformation. In this however he failed, except in the case of the 
identities numbered VII. and XVII. above, but his letter in CruniE (xxxii. pp. 205-209) closes with 
an indication of a method of considerable complexity, by which he says (I.) might be obtained. Im- 
mediately following this he has another paper on the series 
(4 SGD a ee ea ee ese 
G=)@—=)) G—-—NeG=N@r=) Gr = 1) 
or say ¢(a, 8, y, g,”), of which Gauss’ hypergeometric series is a particular case, viz, when g =1. 
Treating this series step by step exactly after the manner of Gauss, he obtains the result 

g(a, 8+1, y+1, 9%) _1 
= a,x 
Sip aa Gs 
g(a, B, Y q> 2) 1 oe = age 
. gett i 
Bar Dy ae cai ei a+r] yr = il 
where dz, = C Yq yer and dor 41 = C Yq ) gon ; 


Cems Crees) Ne sae 
and this, he says, includes those identities of E1senstein which are given in CRELLE, xxvii. and xxviii. 
In regard to the first of these papers it must be remarked that the author’ s failure was greater than it 
Sigrit! have been, for there are at least siz of E1smnsTEin’s results obtainable by EvuLEer’s method, the 
additional ones being (V.) (VI.) (IX.) and (X.), which are derived from the transformation of the series 
1 ee 
Pp P Pp 
The result given in the second paper is very valuable in itself, but as explanatory of Ersensre1n’s work 
it is of little importance. Instead of saying that the latter's results are contained in it, it would be 
more correct to say that they are hidden in it, if, indeed some of them, as, ¢.g., those just referred to 
be there at all. 
