CURTISS. 
INARY FAMILIES IN" A TRIPLY CONNECTKD REGION. 
35 
oo , 1, when these are the regular singular points. In ollior words, every non-singular 
point of a P family has the exponents 0, 1. Every Q family (of order > 
has at 
least one branch which vanishes to an order higher than the first in some non-angular 
point 
This we may show by an examination of the polynomial </> (x), win h 
(30). We have 
(31) 
*(*) 
k(x 
°i 
» 2 ) T « . . . (r 
o 
where k is a constant * 0, and eacli of the exponents cr is a positive integer, their sum, 
the degree of <f> (x), being n for a family of order n. The points t„ s 2 , . . . , s, are here 
all different from 0, oo , lj each is a vanishing point of D, and in fact these are t lie 
only such points other than 0, co, 1. We will call them apparently singular p< tints ' 
(Poincare); in particular, referring to (31), we shall call 4 a (refold apparently sin- 
gular point. From (29) we see that t is an apparently singular point when, and only 
when, t' + t" > 1, i. e. when at least one exponent of t is greater than 1. Hence, an 
apparently singular point is one at which some branch of the family vanishes to an order 
higher than 1. Since <j> (x) is a constant for P families, they can have no apparently 
gular points. 
Let us designate the exponents of 
(29) we may then say: 
integral exponents cr/ and 
To 
tr 
cr 
1, 2, . . . , #c) by 
cr rf old apparently singular pc 
hose siim is a, < + 1. 
cr/'. Referring \ 
belong two positi 
Each point s { introduces 
parameter into the family; it will turn out that 
only the groups of P families are in all cases completely determined by the exponent 
and the regular singular points. 2 
It will be useful to introduce here a notation analogous to that used by Rieman 
for P functions. Accordingly we will designate by the symbol 
(32) 
Q 
ft 
b 
f 
ff 
V 
1/ 
any Q family with the regular singular points and exponents indicated. If we wish to 
specialize still further by indicating the apparently singular points and their ex- 
ponents, we will use the symbol 
b 
(33) 
Q 
1/ 
1/ 
h 
h 
1 
a 
a 
1 
l 
1 
a 
2 
n 
a 
t 
11 
2 
: 3 •) 
1 Cf. Klein's term " Nebenpunkte," p. 225. 
gular points 
carry these into any other three 
e. and we can make no direct c< 
poi 
sxpressed 
assume throughout this paper 
