MR T. MUIR ON EISENSTEIN’S CONTINUED FRACTIONS. 139 
Of the twenty identities there are at most only five which are independent. 
These are (I.), (IV.), (V.), (VIL), and (XX.). From (I.) are deducible (IL), 
(III.), (VIIT.), (XIX.). Thus, substituting p for 2 in (1.), where p is one of 
the m» roots of unity and m is odd, the left hand member becomes 
1 + px sect sees pe gee 
ore a? epee hel ob oe hep tae 
+ perme” 4 pemtDgamtr gp + pom —Diyon—t 
ae 
or 
1+pet+.... pee 
Sa clear i Wee ee ee tp epee ee 
+ orld + pe ti... ape ak Toe ee 
ie 
or 
int + pu a pate fy ey Je pe et 
Cwm ia ae er 
Making the same substitution in the right hand member, which thereby 
becomes terminate, and multiplying both members by 1 — 2”, we obtain (II.). 
(VIIL.) is derived from (I.) in a perfectly similar manner. Again, in (I.) 
writing p—’ for x and p for R, and dividing both members by p, the continued 
fraction becomes after some reduction 


and the series becomes 
Now, it is well known that 
2K 2 2 
JS Sl+statet mt... : = ia) 
hence, the truth of (III.) is apparent. Lastly, putting x for = in (1.), and 
making some simplifications, we obtain a result which is at once transformed 
into (XIX.) by writing 2’ for z. 
VOL. XXVIII. PART I. 2N 
