54 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
and here (j> (x) is always of degree 
i, 
2 
on account of the lemma of the preceding section. Formula (51) we can use with 
merely a change of notation. 
4. THE DIFFERENTIAL EQUATION. 
The differential equation of a P family is easily written down from the more 
general equations of Part II, A, § 3. Putting n = 0, we have from (39), if the regular 
singular points of our family are 0, oo, 1, 
<PP 
dx* 
rl-V-X" l-v'-v'HdP t T X'X" , „, i>'v"l P ft 
x(x 
and from (44) we obtain for the case where the singular points are a, b, c the equation 
first given by Papperitz, 1 
(53) ^P + r i-X'-V' + l-M'-V' + l-*'-*'H 
da? |_ x — a x—b x — c J 
dx 
( \'X"(a-5)(a-c) ^ fi" (b - a)(b - c) v' v" (c - a) (c - b)~\ P 
x — a x — b x — c J {x — a) (x — b) (x 

The differential equation (53) affords an existence proof for a P family ivith any 
three singular joints and any exponents whose sum is 1. It therefore gives an 
page 50 
proof for all the classes we have characterized in terms of the expon 
E. Q FAMILIES OF ORDER 1. 
§ 1. THE DIFFERENTIAL EQUATION. THE FAMILIES Q AND Q. 
In the first three sections of the paper of Ritter which we have so often cited, are 
given a discussion of the differential equation of the general Q family of order 1, 
applications to a classification of P families according to their groups, and an example 
of the expression of a Q function linearly in terms of P functions. For many points 
of mterest the reader may refer to that paper; here we shall only treat the problem of 
classifying Q families of order 1, and add a few remarks about the expression of Q 
tactions of order 1 in terms of P functions. For our purposes, it is sufficient to take 
the regular smgular points at 0, oo, 1. The apparently singular point . (which must 
be simple ) is not to coincide with any of these. 
1 &*<*>■ Ami., Vol. 25, 1885. 
