CUKTISS. 
BINARY FAMILIES IN A TKIl'l.Y CONNECTED REGION. 
1!) 
(i;>) 
Pe' + Pc" 
W'l + " 2 , 
and this equation will be found most essential for the classification we wish to make 
To obtain the values of m x and n» let us use t lie relation 
& 
i 
in 
in 
i 
2 
S b Sm 
From formula (12) we can easily compute S b S a for the basis (yd, ?/,/'), and in this way 
we find 
m \ = Pa Pb [8 
CaP] 
a 
A 
r.- pi ^ . 
w 2 = [>«' P*" ( 
7 + C a a) + p a 'p b ' C b (8 
C m /5)]| + &>.' V a 
*V<W]|. 
where A stands for the con- vanishing determinant aS 
If 
\v 
pb 
sides of (15) by A, and perform a few simple transformations, we obtain the equation 
(16) 
a S -TT- 1—7—1) (Pa P ' Pc 
Pa Pb Pc Pc 
1 ) (Pa Pb' Pc " 
1) 
C a a/3p n '( Pl ; 
pi. ) 
+ 7 f , <W P<! pJ 
Pc 
1 ) (p a ' p,!' pJ - 1) + C b /3 8 pi! ( Pa ' - Pa ") + c a c b & Pa ' Pl : 
o 
The presence of products of multipliers p a 'p h 'p c ', p a 'p,'p r ", etc., in this equation givi 
us a hint that an important principle of classification is introduced when we state, for 
a given family, whether such a product i or is not, equal to unity; such knowled e 
is, in fact, essential for the discussion of (16). That this has a deeper signili nice will 
be manifest when we have observed subsequently that whenever such a product is 
equal to unity, there exists a member of the family which undergoes only multiplica- 
tive substitutions when continued along any clo d path in 1\. Accordingly, we divide 
all binary families having the same three holes into four typt ..characterized as follows: 
Type 
U 
tt 
u 
I. 
n. 
III. 
Pa! pb pc 
Pa' pb Pc 
Pa Pb Pc' 
IV. Pa 'p b 'p c ' 
*1, 
1, 
1, 
1, 
Pa" Pb' Pc' 
Pa" Pb Pc' 
Pa" Pb' Pc' 
Pa" Pb' Pc' 
*1, 
*1, 
1- 
1, 
p.: p^ pJ 
P„'Pb"Pc' 
Pa Pb" Pc' 
Pa'pb'Pc' 
*1, 
*1, 
*1, 
1, 
Pa' P b' Pc" 
Pa Pb Pc" 
Pa P ' Pc" 
Pa Pb Pc" 
*1. 
*1. 
* 1. 
1. 
Tn each„type the behavior of the remaining four products of multipliers is deter- 
mined by (13). It might seem at first glance that there should be another type 
characterized by the relations 
