16 
CURTISS. 
BINARY FAMILIES IX A TRIPLY CONNECTED REGION. 
yd 
Va 
!l 
yd 
yd' 
"iY 4>d (*) 
c 
a x y' $*'(*) 
af"M'{x)+^(x-a{)*<l> a f (x)log(z-aJ (x-a.rfd'ixj+^x-a^d^logix-a,) 
(x 
(X 
*lY 4>d (X) 
atf" <f>d' (x) 
<hf f d ('• ) 
a x Y" <£a" (x) 
But since the multipliers of kindred families are the same, a 
both be integers, so that we may write 
a and a 
a" must 
yd yd 
V " V " 
a x y+«"$ a {x\ 
where <f>Jx) is single valued and anal) 
holes. Hence we have the equation 
about a ; similarly for the other 
(9) 
y' y' 
f f 
«i) a ' +i " (x 
btf'+fi" 
.{x-LY+*'4>(z)i 
where <j> (x) is single valued and analytic everywhere in T n , except possibly at the point 
infinity, which, in (9), is supposed not to be an isolated singular point. If it is, it is 
easy to see how this discussion is to be modified. Equation (5) gives the relation 
a' + a" + P + &' + . . . + X' + % 
a 
an integer, 
that (f> (x) has at most a pole in the point infinity, and must be single valued there, 
In particular, since the 
ponents concerned may be 
ed or diminish' ?d by any 
we may so choose them that <j> (x) will be analytic in the point infinity 
Between any three kindred functions there 
ich 
may 
V 
easily deduce. Let (/, y"), (y, y"), y 
binary families, and let y, y, y t be any three 
families. 
have 
be corresponding bases- of three kindred 
iponding branches of 
The bases chosen may always be so taken 
if 
y 
y' + /% 
rr 
pecti 
fa 
y 
ay' + /3y», § = a$' + /3$". 
Having so chosen the bases that these relations hold, we obtain the equation 

y 
y 
y 
y' 
y' 
¥ 
= y 
y" 
y" 
f 
y' y 
f y" 
y 
y' 
J 
v'l, =y y 
y" y"\ + y y" y" 
The coefficients of y, y, % here have a form analogous to (9), so that, since 
pliers of kindred families are the same, we have only to divide through by 
the multi- 
suitable 
powers of 
«i), (x 
*i) 
), to obtain the 
1 The notation here is readily understood by comparison with that used for the hole a. 
»• 
