CUKTISS. BINARY FAM 
41 
B. CRITERIA FOR THE CLASSES OF PART I, B. 
§ 1. NECESSARY CONDITIONS FOR EACH CLASS IN TERMS OF THE 
EXPONENTS ALONE. 1 
In Part I, B, we first divided all possible binary families in a triply connected region 
four types by means of 
multipliers. Remem 
bering the connection between multipliers and exponents, we may 
for Q families of any order, as follows : 
Type I. V + /i' + v\ X" + y! + „', \> + tf + ^ X f + fi' + A are none of them integers. 
" II. X' + u! + v' is an integer : X" 4- u! 4- J. V _l „" _i_ J \' _i_ ../ i ,.n „„„ „„*. :..,. 
u 
t; 
III. 
IV. 
/v -r ^ -r ^, /v f /* -t- v, a' -t- /i" + z/, a/ + /x' + */' 
X' + /*' + i/ is an integer ; V + /*' + j/, X' + ^t" + 1/ 
X' + /*' + */, X" + /*' + i/', are integers ; X' + /*" + j/, 
X' + ^' + ^ X" + /*' + •, X' + fi" + i/ f V + n' + ^ 
V + fi' + 
+ *>", are all of 
v'\ are not integers. 
We hardly need repeat here what we have already noted for the more general case, 
that by a proper choice of notation every Q family will come under one, and but one, 
of these types. The conditions given are sufficient as well as necessary for each type. 
We have seen in Part I, B, § 4, that in every class other than the one of Type I 
each family has an everywhere fundamental branch. Let us now examine the form of 
such a branch with exponents X, /*, v, corresponding to multipliers p a , p 6 , p e , for a Q 
family of order n whose regular singular points we take at 0, co , 1. Such a funda- 
mental branch has, in fact, the form 
#*> = x k (x-lya (a), 
(45) 
■ 
where X stands for either X' or X", vfor either v or v", and O (x) is a polynomial. For 
Gr (x) is single valued, and is analytic, except in the point co , where it lias the exponent 
X + fi + p 9 which must be an integer, since co is not a branch point. Hence the only 
singularity of O (x) is a possible pole at infinity, and G (x) must therefore be a poly- 
nomial whose degree, the negative of its exponent at co , is — (X + /* + v). We have, 
then, the necessary relation 
X + n -f- v = either zero or a negative integer. 
From this result, combined with the table of Part I, B, § 4, which gives us the mul- 
tipliers of the everywhere fundamental branches, and hence their exponents, we are 
enabled to give certain relations among the exponents of the regular singular points 
which must hold for the families of each class. We proceed to take these up in order. 
Type I. There is but one class here, characterized in terms of the exponents 
above. 
1 Compare Ritter's classification of P families in §2 of his paper already cited. His numbering of classes 
differs from our scheme, though his types are the same. His criteria are easily reduced to ours. 
