8 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
§ 3. THE GROUP OF A BINARY FAMILY. 
Starting from any given point of S, let us continue simultaneously a chosen basis 
(y x , y 2 ) along a closed path which may, if we please, intersect one or more of the 
cross-cuts, but must lie within T n . After making this circuit, y x and y 2 have become 
branches y x and y 2 , linearly independent, and still members of the family. Defini- 
tion (2) of the preceding section gives us the equations 
where a, ft y, 8 are constants, and aS — fiy £ 0. The basis Q/ lf t/ 2 ) has undergone 
the linear substitution 
A 
symbolically 

($v W = (" sjbtv y a ) ■ A <*n 3/ 2 ) 
If we had continued our basis along some other closed path in T n9 (y { , y 2 ) would 
have undergone, in general, another linear substitution 
*-J ? 
If we continue (y l9 y 2 ) along the first path, and then along the second, it underg 
the linear substitution 
AB =(**' + &*/ a? + 08' 
\va'+ ay 7 £' + 88' 
For a closer treatment of this subject, one may consult Klein's Hyper geometrische 
Functioned The points of chief importance are these : Corresponding to the fact 
that all closed paths in T n may be regarded as obtained by combination and repe- 
of n closed fundamental paths, namely single circuits about each hole alone 
have the theorem that all the substitutions which a basis (y b y 2 ) can undergo are 
combinations and repetitions of n generating substitutions, each of which corresponds 
a single circuit about one, and only one, of the holes a, b, e, . . ., *, respectively. 
At this point, however, we should note a matter to which Riemann alone of the 
iters on this subject seems to have given especial attention. This is the fact that 
the above definition of a fundamental path, and hence of a generating substitution, 
1 Lithographed lectures 1893-94, pp. 98-101. 
We 
I 
to it. 
use 
