MR T. MUIR ON EISENSTEIN’S CONTINUED FRACTIONS. 143 
sides, there results another special form (C), say, which gives for a series of the 
same kind an expression of the form 
pace? ; 
aa 82 Be 
i 
These results serve to establish (I.) ([V.) and (VIL). Writing 1, R“, R~, 
fim. 100 C,, G, C, 6... in (B) we obtain (I). Im the case of (IV.) we 
replace = by its known equivalent 
4p 4p? 4p? 
el pet ee 

i + 
and writing the terms of this in order for },, 6,, 6,, 6, ... in (C) and putting 
x = 1, the required result is obtained after some simplification. (VII.) is got 
from (B) in an exactly similar manner. The remaining identities (V.), (VI.), (IX.), 
(X.) have been already referred to in the footnote as being obtainable by EuLER’s 
method. 
It is hard to derive from a study of E1sensTEIn’s work any positive infor- 
mation regarding the method or methods which he employed. This is due not 
only to the fact that he confines himself in general to the statement merely of 
results, but also that occasionally he “ darkens by elucidation,” a certain air of 
mystery being sometimes induced by the few inevitable words which connect one 
result with another, and by the seeming capriciousness with which he selects 
his special cases, often preferring an intricate process of substitution for one 
which is more evident and equally effective. It seems, however, highly probable 
that the method was more akin to Gauss’ than EULER’s, the majority of results 
being closely allied to those obtained by the former method. One thing is 
certain, EISENSTEIN knew nothing of EuLEr’s transformation, for, otherwise, he 
would not have attached importance to those of his results which are obtainable 
by it, knowing, as he must then have done, that an infinitude of such results 
lay ready to the hands of any one, viz., one result, at least, in connection with 
every series in existence ; and, besides, in one place he changes a series into a 
product, then the product into another series, and from this second series derives 
his continued fraction, whereas the result could have been got from the jirst 
series with the greatest ease by EvuLEer’s method. Although we could have 
wished more information, and an improved style in conveying it, it deserves to 
be said, however, that the method has given a number of interesting and useful 
results in analysis, and two, viz., (I.) and (XX.) which are decidedly notable. 

VOL. XXVIII. PART I. 20 
