CURTISS. 
BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
39 
The differential equation of every Q family whose symbol is (35) is, therefore, 
(38) 
dx* "*" 
V 
x 
£+i 
v' 
V 
ft 
X 
1 
dx 
+ 
i 
? 4 
a: 

* t J^(i? 
i) 
where the accessory parameters x A t must satisfy a system of k equations which we 
obtain from the conditions that (38) be satisfied by the k branches 
1 -f c™ (x - s t ) + c 
2 
(•) 
(* 
*,) a +..., (1 = 1,2,...,*). 
but 
These equations we will not here develop for the general case, 
selves with deducing them for families all of whose apparently singular po 
content our- 
simple 
where cr i 
i( 
1.2 
K 
The differe 
quati 
of such 
family, its regular singular points being at 0, oo, 1, may be 
(39) 
dx 2 "*" 
X 
i 
x 
£+! 
V 
V 
tt * = n 
X 
1 
1 
t = l 
X 
s 
dQ 
d x 
+ 
x 
+ /*' /*" + 
p'p" 
x 
1 
+ 
S*-«.J 
* 
Q 
x(x 
1) 
0. 
This is to be satisfied by a branch whose development about s^ is 
(40) 
1 + a, (x — *,) + a 2 (x 
s t ) + . . . ; 
substituting this in (39) we obtain a system of equations to be satisfied 
the 
coefficients of (40). Of these, however, only the following 
impose any condition 
the accessory parameters 
A 
*<(«« 
a 
l 
0, 
2a 
2 
2 a, + 
a x A t 
A< 
*i(*i 
+ 
8,(8, 
1 
1) 
\8 t 8 t 
1 
1 
+ 
1 
X'-V 1 
+ — 
1/ 
». 
. 
8 
1 
i= i *♦ ~~ 8 j J 
*<(*< 
— — + /*V + - — =■ + 2, rr IT 
o, 
*=» 
where the symbol 2' indicates a sum in which j is to assume all integral values from 
1 to n inclusi\> 
j=i 
pt the 
These equati 
pose one condition on the 
accessory parameters A. Giving 
have the n equations 
ly, the values 1, 2 
we 
(41) 
2 
At 8, ( 
\' + \ 
II 
+ 
i/W 
*t 
1 
« 1 1 
^•-1 *," */J 
+ A 
<« - 1) [ 
V j 
l 
+ 
2—1 
0. 
1 
A term used by Klein in his lectures since 1890, 
