PART I. 
TTIE GENERAL CASE. 
While this memoir refers primarily to families belonging to a triply connected 
region, there are many ideas concerned which apply equally well to regions of higher 
connectivity. We will develop these more general ideas in a subdivision A, following 
this with a subdivision B, devoted to the more special properties true for the triply 
connected region alone. 
A. BINARY FAMILIES IN AN tt-TUPLY CONNECTED REGION. 
§ 1. THE REGIONS T n AND S. 
Let a, 6, c, . . ., I be n simply connected perfect 1 regions on the complex sphere, no 
two of which have a point in common. The remainder of the sphere constitutes the 
w-tuply connected region T n with holes a,b,c,...,L These holes may be continuous 
open line-segments, or even points, as well as two-dimensional regions. By n — 1 
cross-cuts joining a to b, b to c, etc., we can convert T n into a simply connected region 
which we shall refer to as the region S. 
§ 2. DEFINITION OF A BINARY FAMILY. 
In the region T n we are to consider a family of functions which we shall call a 
binary family, defined as follows : 
(1) Every member is single valued and analytic throughout S, and can be con- 
tinued over the cross-cuts, the branches thus obtained being again members of the 
o "~o 
family ; i. e. we have to do with a family of function-branches analytic in T n . 
(2) There are two linearly independent branches, y u y 2 , such that every member 
can be expressed in the form Cj y v + c 2 y 2 , where c\ and c^ are constants ; and, con- 
versely, if we give to c^ and c 2 any values whatever, the function c x y x + c 2 y 2 belongs 
to the family. 
We shall call the pair (y l9 y t ) a basis of the family. Any pair of linearly inde- 
pendent branches is readily seen to constitute a basis. 
1 By a perfect region is understood a region which includes all of its limiting points. 
