
136 MR T. MUIR ON EISENSTEIN’S CONTINUED FRACTIONS. 
No demonstrations are given of these identities ; but they are said by the 
author to be only particular cases of a very general equation, the complete 
theory connected with which he hoped to give on another occasion. 
At p. 193 of the same volume of CreLie he returns to the subject, not 
however for the purpose of giving the promised theory, but to add several 
other results similar to those before given. These are 
gate t te ES el ea es bak (V.) 
L—27!4+z2-4—2-94 eta 5 
and one derived from this by changing the sign of z, and the signs of both sides ; 






z a a a es A 
fe eee pe t(t—1)? Ske hae (Vig 
ee Ne Bx 1 ae 2(2—1)2 
EL pa ee 
o—-1 —... 
where ¢= zx; and 
ee eee ay eae es ae ee oo. (VIL) 
ee il —Sat_ (L-o%™ x 
=) =e ec = 24) 
(oe 
o-— 
where o is a primitive root of the equation 2”=1. 
As has been said these additional results are also given without proof, and 
reference is again made to their author being in possession of a general method, 
for he adds: “ Nous supprimons ici un grand nombre d’autres formules et de 
conséquences analogues.” 
Tn the next volume of Cretzez he takes up the subject again, giving another 
series of results connected with the theory of elliptic functions, viz :— 







aK 
Ts ety rae (IX.) 
DE ae Pp 
pi+l— 
Q2hK 2 
<i = sles D Pa). aaa (X.) 
am aaa ea p 
I 
where K, &c., have their usual siguification as given above. 
Putting, further, 
at 
J/-l=%t and z= eX, 
