CURTISS. — BINARY FAMILIES IN A TIJIPLY CONNECTED REGION. 13 
§ 6. KINDRED FAMILIES. 
There are two views which we may take of the group of a family; we may look 
upon it as an abstract group in which the members are any collection of elements 
obeying certain formal laws, or as a concrete group whose members arc linear substi- 
tutions which a given basis undergoes, these substitutions having coefficients which 
are uniquely determined when the path of continuation is given. It is from this 
latter point of view that we proceed to study the ^roup. We shall suppose the holes 
a, b, c, . . ., I always given, and limit our discussion to families having the tme holes. 
It is entirely conceivable that we may have two binary families with 1 lie same 
group; hypergeometric families, which we shall study in Part II, readily furnish u 
with examples of this. But first, — exactly what is meant by families with the same 
group? To answer this briefly, — two binary families belonging to T„ are said to have 
the same group if it is possible to choose from, the one family a basis (y', y"), and from 
the other a basis (y', y"), 1 sucn ^ iai whni continual together over the same path 
what path in T n that may be, the latter basis undergoes always the same swbsti 
u the former. Such families we shall call kindred families ; (//', y") and (Tf, if 
responding bases. No matter what values we give to the constants m„ n x , m z , % 
provided m x n^ — m? n x ^ 0, the bases 
m 2 « 2 
) (/, y") = S (y\ y"), 
and 8 (y', y") correspond. Hence : 
I. If two binary families are kindred, to every basis of the one corresponds a basis 
of the other, i. e., a necessary as well as sufficient condition thai two binary families be 
kindred is that for every basis of the one there exist a corresponding basis of the other. 
We should note the vagueness of the word u corresponding M as used here. Ob- 
viously the same group may belong to an infinite number of bases of the same family 
(for example, the constant multiples of a given basis), so that this correspondence is 
not of a finite number of bases to a finite number. 
A second theorem of great importance, whose truth is immediately obvious, is the 
following : 
II. A necessary and sufficient condition that a basis of one binary family corre- 
spond to a basis of another such family is that the two bases have the same generating 
substitutions. 
1 This is not to be confused with a notation used in §§ 3 and 4 
