CURTISS. 
BINARY FAMILIES 
TRIPLY CONNECTED REGION. 
1^5 
not at our disposal. To show this, let us introduce in place of the old bases (y a % yd'), 
W> V"), new bases (Y a % Y a "), (Y b ', 27'), satisfying the equations 
Y ' 
■*■ a 
Y" 
- 1 a 
Y b \ 
Y b ", 
associated constants C having the values C 
a, ^b' 
We must have 
YJ 
Ya" 
Y b ' 
Y b " 
\ yd, 
h yd + v x jr.", 
\yb\ 
H yb + v 2 y b "- 
By comparing with (19) we easily deduce the equations 
*i — \, /*i 
/*a» v \ = V 2 
; \ jt 0, r^ 0; 
Ca 
V 
1 
a 
i> 
\ 
i 
c b 
^2 
giving the equation, 
C a 
a 
b 
C a 
a 
K 
b 
Hence no matter what basis we use in ( 1 9 ) 
the same n 
same group 
t of a family belonging to this c? 
pliers, and for which k is the same, and only such families 
the same for a given family ; i. e. 
Accordingly all families here with 
have the 
But 
pable of taking on an infinite number of 
for famil 
with the same multipl 
We can at least see no restrictions on its value he 
pt that we are to avoid cases 1, 2, 3, 4), and in the last subdivision of Part II 
shall give an example where k can assume an infinite number of 
of k, then, g 
We get cases 1, 2, 3, by putting 
0,oo, 
5. Each value 
1, respectively. 
In Case 4, k has no meaning. Cases 1, 2, 3, 4 give each a class, which we number 
correspondingly, and we have besides these an infinite number of classes corresponding 
each to a different value of k. 
§4. TABULATION OF CLASSES. 
It is for many purposes useful to know the characteristics of the complete 
con- 
necting formula 
(20) 
yd 
ya 
1/ 
ay b ' + fi y b " = a, !fc > + £ y c », 
Vyb'+S y b " = 7l yj + B, y", 
