142 MR T. MUIR ON EISENSTEIN’S CONTINUED FRACTIONS. 
into 
i! Pp eo 
{qe sia. eth Beene Peers 
+ 5-1" * @-1(P-1)” * @—De—De_D * 
- (B) 

3 
L Pp YP P : 
P+ 5a! + GD)" t+ Genet 
Now it has been shown elsewhere* that 

CAC t+C 2 +60 +... ay ; A 
by +b,2+b 27 +b07+... 1 — SE age oe: ) 
i 
A A A Ap aan 
where a,= =, a=; , @=7 >. »--- G=Z a 
ee came TS ONES An—2An-1 
by by by bs b,-1 
OP Obie a wees 
00 bb, a hose 
and A, = Ste eee 
O, 0 Gein nS Gore 
Oy BCC esa. tae ees 
Cp xGi egrets wn ey 

the number of rews containing b in the determinant being the same as the 
number of rows containing ¢ or one more according as m is even or odd. 
Putting n=1 in this, and substituting the terms of the numerator of () for 

Co, Cy Cp... and the terms of the denominator for bo, b,, 0. . . . in (A) we obtain 
1—a)1—pa)(l—prn). . - coal, a—y 
(1—y)(1—py)— py) .- - |p! a pa—py 
eS ee) 
ae 
which, when the right-hand member is simplified, is identical with (XX.). On 
putting 6,=1, }.=6,=b, =...=0in (A) we have a special form (B), say; 
giving for a series in ascending powers of z an expression of the form 
a 
1-2 ag 
et 
si fs 
and on putting ¢, = 1, q¢ =Gq=¢,=...=0, and taking reciprocals of both 
* Trans. R.S.E. 1875-6. 
