CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
Class 3. The two everywhere fundamental branches are 
43 
x K ° (x - I)"" G" (x) , 
for this notation obviously includes the cases where the exponent of either branch 
at the point is X', when G' (*), or G" (*), has a factor &-*\ A similar remark 
applies in the first four classes of Type IV. The degrees of G' (z) and G" (x) are 
+ ft + v) and - (X" + ^ + v ") respectively. We have here 
X" + »> + v > < o, \" + M " + v "<0. 
These two conditions are equivalent to 
x' - X" > X' + fi' + V > 
n. 
Type IV. All exponent sums X 4- ^ + v are integers. X' - X", // - //', / 
are all positive integers or zero, so that we have always 
V 
X' + / + V> * 
Class 1. The everywhere fundamental branch is 
x K " (x - ly G (x\ 
f 
where G (x) is of degree - (X" + // + v). We have 
which is equivalent to 
X" + y! + v' < 0, 
X' - X" > V + y' + i/. 
This carries as a necessary consequence the relation 
\ f + p' + i/ > n f — p" + v' — v" — n. 
Class 2. The everywhere fundamental branch is 
x*'(x-iy'G(x), 
» 
where G (x) is of degree - (X' + //' + v'). Hence we have 
which is equivalent to 
X' + /*" + i/ < 0, 
A 4 
' - /' > X' + / + •. 
This necessitates the relation 
V + / + •> x' - x" + •-*"- * 
