CURTISS. BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 45 
§2. SUFFICIENT AS WELL 
In the 
■ 
case of P families (where n = 0), the necessary conditions of the preceding 
section are also sufficient that a family be of the designated class, but this is no long* 
true when n > 0. To be sure we can in many cases give sufficient conditions; thus, in 
Type II, the exponent sums determine the class of a family, if X' + // + v does not 
lie between and 1 - n inclusive ; in Type III, the same is true, if X' + // + v do. 
not lie between X' — X" and 1 — n inclusive ; and we miirht also name cases where 
the class is determined by the exponent sums in Type IV. The cases where thtv 
means for determining the class of a family fail, we may conveniently refer to as 
the doubtful cases. In these doubtful cases we must refer to the differential equation 
terms of 
of the family for sufficient conditions for the classes in 
define a family, — i.e., the exponents, the apparently singular points, and the acces- 
sory parameters. 
It will be useful in this connection to obtain a general expression for a basis in 
Types II, III, IV. Let an everywhere fundamental branch (there must always be 
least one for families of these types) be 
(46) 
Q'=-.x*(x-iy G{x\ 
where G(x) is of degree - (X + fi + v). We have 
<r d * 
D 
dx 
Q» dQ 
11 
^'+a"-i ( x _ iy+*"-i $ (Z), 
dx 
where <f>(x) has the value given by (31). This gives for <?" the expression 
(47) . <r-*[/ffp'' + \l 
Types II and III have their classes characterized by the everywhere fundamental 
branches which have been given in the preceding section for each class. We get a 
necessary and sufficient condition that a family have an everywhere fundamental 
branch (46), with exponents X, ^ v, by substituting (46), the coefficients of G(x) being 
undetermined, in the differential equation (38) of the family. In addition to equa- 
tions determining the coefficients of G (x), we shall in this way also obtain a system 
