CURTISS. 
BINARY FAMILIES IN A TIMW.Y CONNECTED REGION 
•7 
If X' 
X 
and 
that k 
kof (a 
A 
fx" S. 
X y i.e., X" + fi + v = 0, the two i >ta of ( 
X 
// 
For a given family here, A 
/ 
X 
" 
n 
\) v ' be a member, while A 
Vf" be a member. Of these 
in ary and sufli 
X 
n 
ar 
necessary Mid suflic 
be only possible c\> 
3 
r< 
fundamental branches, families of Class 1 have the former, those of Clan X the latter 
and those of Class 3 the f 
We may, then, characterize the cla *es as f 
Class 1. A 
" 2. A 
<t 
3. 4 
X" 
V 
X" 
/*'«, 
/*"«, 
/t'«, 
X' 
8 * 
X". 
X' 
/* 
/ 
X' * X". 
X" 
/* 
/f 
Type IV. The doubtful cases come under four beads, namely: (a) X' + /a' + y' 
(6)- X' + fi' + i/ 
X' 
X" * 0, 
X' + fx' + v' 
/*' 
M " 0, (,/) *' + /+• 
z^ 
V 
// 
* 0. 
These we take up in order. 
(a) X' + ll' + v' 
0. Here X" + /*" + *" 
0, and since A' 
X , ^ 
M 
// 
,• 
j' 
// 
are positive integers or zero, they must ill this case all be zero. Tin' three regular sin- 
m 
gular points are each logarithmic, so that we have to do here with only the cla other 
than 1. 2. 3. 4 
d only 
X 
/* 
is such 
siblc. We h 
fundamental branches 
x*' (x 
x K ' (x 
1) v '[k" lOg X + K'\0g(x 
1ft 
(see page 46), where k # 0, k" ± 0, and k' + k" * 0. 
Each class here is charao 
k" 
terized by the ratio — . 
fC 
This ratio we can find in terms of 8 by substituting branches 
1 and #e"log#+#c / log(a:-l) in the functional determinant of the family a: *'{x 
thus obtaining the identity 
<?, 
i 
k u log x + k! log (x 

II 
i) =: + _• 
a; a: 
r 
1 
£ X A'+A»-2A^_1 (^ _ 1 )r'+,"-*'-l (s 
* 
X 
#(.£ 
f 
') 
Hence we have 
# 
n 
S 
K 
t 
s 
1 
Each value of s ( other than 0, 1, oo ) gwes a class 
For all these classes the a€ces§ory 
parameter A has the value 
V 
V* 
s. 
We see here that ctatf* actually exist corre- 
sponding to every value of 
K 
n 
K 
r 
