20 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION . 
Pa'po'pc'^h Pa"Pt>'Pc' = h />«W = 1 > Po'Pb'pe"*U 
m 
but these four equations are inconsistent, since from the first three follows 
Pa Pb p c " = 1. 
By a proper choice of notation every binary family in T 3 can l>e brouyfd under one, and 
but one, of the above four types. 
§ 3. CLASSIFICATION OF FAMILIES UNDER EACH TYPE. 
The types have been determined by means of characteristic relations among the 
multipliers. In classifying under the types we make use also of the second condition 
that two families be kindred, namely, the requirement that they have a common con- 
necting formula. As we have already remarked, we need use only the part (12) in 
this discussion. We proceed as follows : 
Suppose we are given a formula (12) for a given family; and suppose als 
given a second family with the same multipliers, and belonging to it a formula 
(17) 
YJ = aY b ' + /3 IV', 
Jr«" = 7l7 + SF 6 "; 
then, according to our criterion, the two families are kindred token and only token, by 
chowe of fundamental bases in the second family, (17) may be changed to 
(18) 
U = ay h ' + /3 y b '\ 
y a " = 7 y*' + « h n * 
ivliere the associated constants C, as well as the coefficients, have the same value as in (12). 
If this can be done, the two families, which are of the same type since they kave the same 
multipliers, will belong to the same class ; if not, to different classes. Here, then, is given 
the basis of our further classification. Let us observe that this is a classification of 
families by means of their groups, but it affords a classification of the groups as well. 
To see how a formula (17) may be changed, let us recollect the degree of arbitrari- 
ness which enters into our choice of fundamental bases; we may multiply each member 
of a fundamental basis by any constant * 0, and still have a fundamental basis ; more- 
if the hole is ordinary, this is the only way in which we can pass from one such 
basis 
We 
may 
illustrate our course of procedure by considering the case where both 
and b are ordinary, while none of the coefficients in (12) and (17) vanish. We a 
here write, instead of (17), the new formula 
