CURTISS. BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 17 
(10) 
y <J> (x) + y ^ (x) + y<P (x) = 0, 
where 4>, <t>, <l> are single valued and analytic throughout T n . Hence we may state 
the th eor em : 
I. Any three kindred binary functions are connected by a linear and homoaaieauB 
relation whose coefficients are functions of the independent variable single valued and 
analytic throughout T n . 
In case all the holes are regular singular points (as defined in § 4), <1>, <]>, <P y may 
be taken polynomials; 1 it will be sufficient here to refer to the more explicit formulae 
of Part II for the case n = 3, from which it is easy to see how to pass to the general 
case n > 3. 
As a particular application of Theorem I we may note that any binary function 
and any derivative of that function are kindred, as can easily be verified from Theorem 
III of § 6. Hence from Theorem I alone we can deduce the differential equation 
(ii) 
♦»£!+*£+**=* 
dx> ' ri dx 
in which the coefficients are single valued and analytic throughout T n . If all the holes 
are regular singular points, these coefficients will be rational, and hence can he taken 
polynomials. Equation (11) is satisfied by every member of the binary family to 
which the function y belongs. Conversely, we know from the theory of differential 
equations that all the solutions of such an equation constitute a binary family. Hence 
we have th e theorem : 
II. A binary family consists of all the solutions of a differential equation (11); 
and every equation (11) defines a binary family. 
B. CLASSIFICATION OF BINARY FAMILIES FOR THE CASE n = 3. 
§ 1. THE PROBLEM. 
The question — when are two binary families kindred? — lias been answered in 
§ 6 of Part I, A, by Theorem III, and it is to be noted that this answer has two parts. 
The first requirement is that the two families have the same multipliers, the xxmd 
that they have a common connecting formula. To be sure we shall find cases where 
the satisfying of the first condition entails as a necessary consequence the satisfying of 
the second, but this is by no means always true. 
Can we so classify binary families for the case n ~ 3 that two families belonging 
to the same class will have the same group when their multipliers are the same, but 
1 See 
Werke, p. 379. 
2 
