CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 47 
We shall find a slight complication of our results possible for families having a 
semisingular point; accordingly we will at first confine ourselves to families of the 
classes where a semisingular point is impossible. Here we have § (V) = Q a ' Q K ") 
Let (Q', Q"), (Q\ Q"), denote any pair of corresponding bases of any two kindred 
Q families which belong to one of the classes above indicated. Let the two families 
have the notation schemes 
oo 1 oo 1 
Q U" • »» * j and 5 (x" ? ;» * 
and let Q be of order n, Q of order w. Corresponding exponents of the two fami 
(and these are to be the ones similarly situated in the two symbols) differ by inn _ 
since corresp onding multipliers must be equal (see Theorem III, Part I, A, § 6). If 
designate by X + X, the smaller in real part of the two sums, X' + X" and X' + X", o 
a similar notation with the other pairs of exponents, we obtain, by a process so cl 
in details to the work of page 16 of Part I that the result can easily be written at o; 
by comparison with equation (9), the relation : 
6 
(480 
T? 
Of Q 
Q 
u 
a^+ A (x - !)"+*<£ («), 
<l> (x) being single valued and analytic everywhere except in the point oo , where it has 
a pole of order 
(49) 
(\+\ + P + fi + v+v), 
and is therefore a polynomial of degree given by (49). 
If we write, using the ordinary notation for the absolute value of a number 
AX = |(X'-X")-(X'-X")|, 
A/* = |(/*'-/*")-(A?-?')|. 
we have the equations 
Av = |(j/ - v")-^(v' - v") 
AX X' + X" + X' + X" 
(X + X) = -5 2 
2 
A/z /*' + /' + ?+ ? 
0* + /») = -o 2 
2 
Ar i/ + y" + v' + V 
» 
(" + *) = -o- 2 
» 
so that the degree of <£ (%) is 
(50) 
AX + A/i + Ay+w + yl^-, 
2 
