26 CUKTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED RECION. 
for each class. We have already discussed the determination of C ai C b9 a, ft y, 8. To 
find C c9 a l9 ft, y l9 8 1? or rather those which depend upon the others after we have 
given particular values to those at our choice, we use the four equations furnished by 
S c 65 *S a = 1, 
which we shall give in the next section. At present it i- <nough for our purposes to 
find which of the coefficients in the complete connecting formula of a family vanish, 
and this can easily be done with the aid of the results of the preceding section ; either 
the symmetry of conditions at b and c, or else the relations characterizing the type, 
will tell us which of the coefficients a 1? ft, y l9 h t must vanish, and w oich may be made 
to vanish by a proper choice of bases in (20). We already know that it must be pos- 
sible to reduce all complete formulae (20) of families of the same class with the same 
multipliers to one and the same formula (see Theorem III of Part I, A, § G). In the 
following table we briefly characterize each class. 
Type I. Pa >p b 'p c > * 1, p a »p b ' Po ' * 1, Pa 'p b " Pc * 1, p a ' Pb ' Pc " * 1. 
There is but one class under this type. It is always possible to take a formula. (20) 
which no coefficients vanish. No 
&«*■-"• VV^I.- 
Using the term ev (wh 
fundamental branch to designate a member of a binary family which undergoes multi 
plicative substitutions only for all paths in T 8 , we deduce from the behavior of tin 
connecting formulae the fact that there can be no everywhere fundamental branch© 
belonging to families of this type. 
Type II. p a 'p b ' Pc > = l, pa » 
Pi 
1 
In every family under this and the two remaining types there is an everywhere 
fundamental branch. For this type all holes are ordinary 
Class 1. p _ ft = 0, while the other coefficients in (20) can never vanish 
Only a branch y a ', and its constant multiples, is everywhere fundamental, it: 
multipliers being Ra % pb ' y Rc \ 
^ Class 2. y = 7l = ; no other coefficients in (20) can vanish. Only a branch 
2/^, and its constant multiples, is everywhere fundamental, its multipliers being 
Pa ■> pb f p"- 
Class 3. p = ft == y = yi = . no other coefficient m (2Q) ^ v . mjs]u 0nly 
two linearly independent branches, y a > and y» 9 and their constant multiples, 
ywhere fundamental, their multipliers being p /, Pb% p/f and «.", p/', p.", 
respectively 
1 Equivalent to only three besides (13). 
