42 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
Type II. X' + // + v an integer. 
Class 1. The everywhere fundamental branch is 
«*' (x - ly' G (x), 
where G (x), as in the remainder of this section, stands for some polynom 
Here A 
gative integer, its negative being the degree of 
G 
Class 2. The everywhere fundamental branch is 
**" (x - ly" a (x). 
X + fx + v is zero or a negative integer, its negative being the degree of 
G (x). Since 
we have here 
X' + x" + fi' + p» + v < + v » = i _ nt 
V + jx' + v' > 
n. 
Class 3. The two everywhere fundamental branches are 
z A ' (x - \y G' ( X ), 
**" (x - iy" G" (x) t 
where the degrees of G r (*), G" (x) are - (X' y! + p% - (X" + f + v'% respec- 
tively. We have here 
> V + / + v > < 
w. 
Type III. X' + p* + „', X" + / + v' are integers ; X' - X" is a positive integer 
or zero. 
Class 1. The everywhere fundamental branch is 
a*' (x - 1)*' a (x). 
G (x) is of degree - (X' + ^ + /), We have here 
V + fi' + v' < 0. 
Class 2. The everywhere fundamental branch 
is 
**' (x - 1)"" G (x). 
G (x) is of degree - (V + /*" + */')• We have 
which is equivalent to 
X' + ,*'' + „" < o, 
x + ^ + if > V - V 
w 
