CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 27 
Type III. pj p b ' pj = 1, p a " p b > pj = 1, p a > p b » p< -t 1, pa > pb > p» * 1. 
Here p a ' = p rt ", but b and c are ordinary. 
Class 1. ft and ft must vanish, and y can always be made to vanish, in which 
case none of the other coefficients in (20) = 0. a is logarithmic. There is but 
one branch y a ', and its constant multiples, everywhere fundamental, its multipliers 
being p a ', p b} p \ 
Class 2. a and aj must vanish, and 8 can always be made to vanish, in which 
case none of the other coefficients in (20) = 0. a is logarithmic. Only a branch 
y a % and its constant multiples, is everywhere fundamental, its multipliers being 
/ it it 
Pay pb t pc • 
Class 3. We may take none of coefficients in (20) zero, or as many as four; 
for instance ft ft, y, y x . a is semisingular. Only two linearly independent 
branches, y a ' and y a " 9 and their constant multiples, are everywhere fundamental, 
their multipliers being p a ', p b ' y p c % and p a ' y p b " ', p/', respectively. 
Type IV. pj Pb > Pc ' = l, Pa " Pb ' Pc > = 1, Pa ' Pb " pj = % Pa > Pb < Po » = 1 
No holes are ordinary. Every family under this type has a formula 
yd = yb = y c \ 
yd' = yb" = y c " 
This type has its classes characterized by the invariant 
a 
K 
put it as an equation in homogeneous form, the invariant equation 
(21) 
k> C a - k" C b = 0, 
k" / rc'\ 
where k and k" are never both zero. Each value of — ( or — 1 gives a class, if we 
except the possibility of C a and C b both vanishing, which gives an additional class. 
There are an infinite number of classes under this type. 
Class 1. *" = 0. a is semisingular, b and c logarithmic. In this class, as 
well as in all other classes under this type except Cla s 4, there is but one branch, 
and its constant multiples, everywhere fundamental. 
Class 2. k = 0. & is semisingular, a and c logarithmic. 
Class 3. k + k" = 0. c is semisingular, a and b logarithmic. 
Class 4. (21) is identically satisfied. All holes are semisingular. All 
members are everywhere fundamental. 
