CURTISS. 
BINARY FAMILIES IN A TIUPLY CONNECT] REGION. 
15 
(8) 
«/. Va") = B fa', fr") = (,y/, #,") 
• • 
& w, #"), 
which 
the same as (6), includin 
the 
Our proof is thus completed. 
A generalization of this result which we shall need 
proof is readily supplied from the proof of Theorem 1 1 1 
i ted constants C a , 
C 
••' 
^s*% • • • • Oi 
(1 
1V - Jf(y',y")and(y,y 
any two corresponding loses of tiro kindred binary 
families, and if(f, y") is related to fundamental bases of its family by thef 
V'?) = A'W,9.")'**W 9 9?)m. . . 
v te'. v!'\ 
then (f, y") is also related to properly chosen fundamental bases of its family by the 
formula 
<f. V") = A' UJ, y») = B> (&', &") = . . . = L < {§ f t ^ 
the associated constants C a , C b , C 
in both formulae. 
cy 
C b as well as the coefficients, being the same 
§ 7. RELATIONS BETWEEN KINDRED BINARY FUNCTIONS. 
Before proceed 
d (y\ V') are . 
be convenient to introduce a few 
pond 
ba 
f two kindred binary f 
if V, y 
ft 
we will 
peak of y' and y', and also of y" and y 
ponding branches 
We will call functions whose branches are members of a binary family, binary func- 
tions ; and by two kindred binary functions, we mean functions two of whose branches 
are corresponding branches of two kindred binary families. 
We proceed now to develop certain relations between corresponding branches be- 
longing to kindred binary families; but by the principle of the permanence of func- 
tional 
the equations developed will be in f 
between the kindred 
binary functions to which those branches belon 
Let (/, y") and (If, y") be corresponding bases of two kindred families 
ition of Theorem IV, § 6, we have 
gth 
f y 
y" y" 
Det A' . 
y a 
y« 
a 
ya 
y a " 
Det B'. 
y b " y»" 
DetZ'.l^' # 
W' yi" 
From these equations it follows that the determinant 
y' y' 
y" y" 
defines a function undergoing only multiplicat 
for, using the notation of (4), we have 
substitutions for every path 
T. 
