40 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
of relations between the exponents of the regular and apparently singular points, 
these points themselves, and the accessory parameters, which will give necessary and 
sufUcient conditions that (46) satisfy (38), — i.e. that the family have a branch (46). 
The mcessary and sufficient condition that the family have a second everywhere 
him unciital branch not a constant multiple of the first can be obtained in the same 
way, or, having found the coefficients of G(x), we may find this condition from the 
ex pro -ion (47). We will satisfy 
with merely indicating the plan of 
at ick ; in subdivision E we will carry the work through for families of order 1. 
The above-indicated method fails in Type IV, since the classes here cannot always be 
distinguished by means of the everywhere fundamental branches, except Class 4, which 
is the only one with two such branches linearly independent. But in every class there 
is an everywhere fundamental branch 
g = ** w (x - ty a 1 (x). 
Remembering that X' - X", ^ - p.", p' - v" are all integers, we see that the integrand 
of (47) is here a rational function, so that we obtain Q" by a mere quadrature of a 
simple kind. Since 0, co , 1 are the only regular singular points of our family 
easy to see that we must have 
(48) Q" = &" {x _ i),» (p {x) + q, ry/ log x + K , log {x __ 1)]? 
where £" (x) is of degree not greater than - (X" + f + „"), and *', K " are not 
functions of x. These numbers, k' and k", can be used, as we have seen in Part I, B, 
§ 4, to characterize by their ratio each class. 1 Substituting in (38) the form of (48) 
which characterizes each class, we shall come out with the conditions we desire. 
C. RELATIONS BETWEEN KINDRED Q FUNCTIONS. 
§ 1. GENERAL FORM OF RELATIONS. 2 
We have already noted, in that section, that the relations of Part I A S 7 take 
parat.vely simple form for families having only regular singular points. We 
now proceed to examine these relations in detail for hypergeometric families ; and we 
.hall here not hm.t our eonsiderations to families all of whose apparently singular 
points have 2 ero for one exponent; our remarks apply equally to eases where the 
exponents of the apparently singular points are any pairs of (unequal) positive integers. 
ftrt ■ .„ fact we h « «, chose, m aotation that * his precisely fte mean . ng here thaj |t has on ^ ^ ^ 
«>« S 6 °' RiemaM ' S ™" IV ' ' OT "* "-*■«* °« •«. object fa the classes of P ba£am tw 
