150 Proceedings of Royal Society of Edinburgh. [sess. 
where in every case the coefficient of the product of two x’s 
vanishes if their suffixes differ by more than 1, and where 
c 0 2 =M Jl + 1 ). 
C i 2 = i( w ~l)(rc + 2), 
c r 2 = l(n - r)(n + r + 1), (r>0) 
C^n-l = 2 ^ » 
He was thus naturally led to the equation in z 
z-c<? 
KC 0 C 1 
KC 0 C 1 
z — C 2 — C 2 KC 2 C 3 
kG 2 C 3 Z ~ ~ G \ 
• kG 2<t _ 2 ^ 2(7 - 1 
= 0 . 
. Z - C 2 2o--1 ~ C 2 2o- 
where either c\ # is c|_ 1} or is and, if the latter, = 0. 
From a knowledge of Painvin’s paper he recognised the left-hand 
side of the equation as being the numerator of the continued 
fraction 
z - cs - 
*CqV 
z- cf - c 2 
z-c s 2 - c 4 2 - , 
but he ventured nothing in elucidation of it. Even the special 
case where 6 = 0 and where therefore k = 1 appears to have proved 
at the time too troublesome, although he knew otherwise that in 
this case the continued fraction 
and 
z(z - 2 2 )(z - 4 2 ) 
( z - l 2 )( z ~ 3 2 ) . 
(z - n 2 ) 
(z-n- l 2 ) 
if n be even 
(z — l 2 )(z - 3 2 )(z - 5 2 ) . ... (z-n 2 ) 
(z - 2 2 )(z - 4 2 ) .... (z-f^l 2 ) 
if n be odd ; 
for his words are — “ Einen directen Beweis fur diese Summirung 
des Kettenbruchs habe ich noch nicht aufgefunden.” 
