
140 MR T. MUIR ON EISENSTEIN’S CONTINUED FRACTIONS. 
Assuming now the truth of (V.) we may establish (IX.) and (X.). For, 
taking the reciprocals of both sides of (V.) and increasing them by 1 there 
results 
1 ihe; 
7 mre | if a 2 ae 
Bir oe 4050 Ere ie 


and writing —p for z and taking reciprocals we have 
i a de 
p pp pet on pee P 3 
whence, by means of (a), (IX.) is obtained; and (X.) is derived from it by 
changing the sign of p. 
Again, assuming (XX.) to be established, we may deduce the results from 
(XI.) to (XVIII) inclusive. Taking first the case of (XI) we put pe” for y, 
p~ for p, and pe for x in (XX.), and on simplifying the resulting right- 
hand member, we find 
(1 — pet) (1 — pte") (L—p et)... 1 oe 
—. —_—_. nowt 


: TL : ‘ ! 
and if we take w to stand for + and .°. e” for 2? in EISENSTEIN’s notation we 
K 
shall have 
(1—p—e#*) (1 — per?) (L—p—*et) . 
(L—p—tevt) (1 —p*ewt) (1 — pet) . 

=A+Bi 
.. also 
(1—p—e-*) (1—p-e- #2) (1 —p—Se- #8). 
(1—p—te-“?) (1—p—e- #2) (1—p—e-44) ... 

=A-—Bi 
and hence, by multiplication (the first factor of the first numerator with the 
first of the second, and so on) 


(1—2p— cos o + p—) (1—2p cos o+ p_*) (1—2p* cosw@+p—?) .. 
(1—2p— cos w +p”) (1— 2p cos w+ p) (1—2p~ cos a+p-").. =A +B 
Now the first number of this is known to be equal to 

