44 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION 
Class 3. The everywhere fundamental branch is 
a*' (x - 1)"" G (a), 
where G (x) is of degree — (X' + f/ + v"). Hence we have 
\'+p'+v"< 0, 
which is equivalent to 
i/ - v" > V + uf + i/. 
This necessitates the relation 
X' + ff + • > V - X" + M ' - ^' - w . 
Class 4. Every branch is here everywhere fundamental. We have 
&V = a? (x - 1)*" G (x), 
where G(x) is of degree < - (X' + p" + /'). Similarly considering QW and # ( "'>, 
we have 
\< + M " -f- v " < 0, X" + fi! + *" < 0, V + /' + i/ < 0. 
These conditions are equivalent to 
X' - X" - «, 
V + fi' + v' > \u> ■■" 
v' - v" - n. 
Remaining Classes. The everywhere fundamental branch is 
«*' (x - iy G {x\ 
where G (x) is of degree - (X' + / + v). Hence we have 
X' + ^' + y' < o. 
These necessary conditions we now give in the following condensed form 
Type II. Class 1. X' + /*' + i/ < 0. 
" 2. V + /*' + ,/> 
n. 
u 
TO. 
to. 
3. >X' + f t'+j/ > 
Type III. Class 1. X' + /*' + v ' < 0. 
- 2. X' + p' + p^x'-X" 
3. X'-X">X' + M ' + ,/ > 
Type IV. Class 1. X' - X" > X' + ^ + v > (> /_,,»+,/_ „>' 
- 2. Ai'-/*">V+^+ • (> V - X" + 1/ _ ,/' 
It 
u Q .j 
u 
3. • -^>V+y + ^(>V-.X" + //-M"-«) 
X' - X" - n, 
4. V + fx' + v' > Iff,' _ ^" _ w 
v' -v" 
Remaining classes 
to. 
0. 
