50 CUBTISS. — BINAUY FAMILIES IN A TRIPLY CONNECTED REGION. 
§ 1. CRITERIA FOR EACH CLASS OF P FAMILIES. 
Riemann's definition of a P function includes Type II as well as part of Type I, 
but h<: seems to have overlooked the former in discussing the group. Miss Winston, 1 
following out a suggestion of Klein's, 2 first supplied this deficiency by noting from the 
development in series of the fundamental branches the difference between families of 
('lasses 1 and 2 of Type II. Ritter 3 and Schilling 4 have since obtained essentially 
the criteria which follow for all classes, the latter by geometrical methods, the former 
analytically, but starting from an artificial definition of kindred functions which he 
has not shown to be equivalent to the requirement that they have the same group. 
Our method has its chief claim to interest from the fact that it is a natural extension 
of Riemann's. 
The criteria for the types are given on page 41. For the classes under them the 
necessary relations of page 44, when we put n = 0, become : 
Type II. Class 1. X' + /*' + j/ < 0. 
« 2. X' + y! + v 1 > 0. 
U 
3. Impossible. 
Type IIL Class 1. V + fi f + j/ < 0. 
u 
2. X' + fi' + v ' > X' _ X". 
" 3. X'-V'^X'+^'+^O. 
Type IV. Class 1. X' - X" > V + p' + „' (> ^ _ ^ + J _ ^ 
" 2. /i'-/*">X' + /i ' + "'(>X'-X"+ *'-*")' 
X' - X", 
41 
4. X' + fi' + v> > \fx> - ^ 
p* - v" 
t 
n,e necessary relation X' +/+•>- jj make3 it impossible , iu Type lY> ^ v + 
IV be equal to or less than zero, so that the remaining classes are impossible 
I"" <1-" relation, b..-l,da all possible values „f V + j +. „ % ;lml in ^ t t , 
tually cxc 
Hence, the above relations are, for each class of any 
T * ?# ^ "**** *" a/aw * * ** ^ **»9 to that class 
typ 
and 4 of 
Tv J h TV P ° int , " ' 8 Semi f !nSUlar " CIaSS 8 ° f T ' Vpe ln ' and in Masses 1 ana , 01 
Type IV, and m no other classes. By combining the criteria for these classes we 
obum a necessary and sufficient condition that . be semisingula, This condZ I. 
either X' 4- u! + / or \' + »' 4- > 
i?i^er 0/ ZAe sequence 1, 2 ... X' — X". 5 
1 Math. Ann., Vol. 46, p. I", 1895. . r,, . , . . 
" Pages 413-41 .\ ™ § 2 of h ^ paper. Math. Ann., Vol. 48, 1896. 
Schw 
4 Math. Ann., Vol. 46, p. 538, 1895. 
Reihe eine algebraische Function ihres 
292-335 
