CURTISS. BINARY FAMILIES IN A TRIPLY CONNKCTED REGION. 9 
we still need to specify how such a path is to be taken relatively to the cross-cuts 
which change T n into S; this is a matter of convention, but unless we add sucli a- 
specification, inconsistencies will arise in the computations which follow. The con- 
vention we here adopt is practically Riemann's. 1 — Given a point P from which a 
fundamental path about hole i is to start and to which it is to return. We divide S 
into two regions by joining I to a by a cross-cut such that that part of S in which P 
is lies to the left of a point which traces out its boundary so that it meets the holes 
in the order I, . . ., c,b, a. A fundamental path about i is to have the hole i, but no 
other hole, interior to it, and is to be deformable into a path which crosses the cross- 
cuts at only two points. A generating substitution corresponds to such a path 
traversed positively, its inverse to that path traversed negatively. 
With this convention we see that a path formed of n fundamental paths about 
l f . . ., c, b, a, successively, each taken positively, is equivalent to a path enclosing 
no hole. If the n generating substitutions for a given basis are S a , S bi S c , . . . , S„ this 
gives the relation 
(1) 
Si . . . S c St, S a = 1, 
where 1 stands for the identical substitution. 
A different convention would have changed the order of the product in (1). 
The totality of substitutions which a basis undergoes when continued along all possible 
closed paths in T„ constitutes a group whose n generating substitutions correspond to 
single circuits about each of the holes. 
This group, which we call the group of the binary family, is discontinuous, and, 
in general, infinite. If we had used instead of (y x , y 2 ) any other basis (?//, ?//), we 
should have obtained a new group simply isomorphic with the old, and therefore, as 
an abstract group, indistinguishable from it. In so far we are justified in speaking of 
the group of the family as we have done; if there is need of a more specific expression, 
we shall speak of the group corresponding to a basis. 
§4. THE DOUBLY CONNECTED REGION. 
There are certain well-known facts for the case where the number of holes of T n is 
two which we will deduce as briefly as possible in this section. If, in this case, we 
continue a basis (y„ y 2 ) of a binary family along a fundamental path about the hole 
a in the positive direction, we shall have the generating substitution 
<* • »•> - (»' «') <*> ' ** 
1 " Beitrage," etc. VVerke, pp. 70-71. 
