BINARY FAMILIES IN A TRIPLY CONNECTED REGION WITH ESPECIAL 
REFERENCE TO HYPERGEOMETRIC FAMILIES. 
i 
are 
INTRODUCTION. 
It was Riemann who, in his celebrated paper on hypergeometric functions, 2 first 
emphasized the importance of the group of substitutions which such a function under- 
goes when continued along all possible paths which do not pass through a singular 
point. Again, in another paper, 3 he discusses from this same point of view families 
composed of the solutions of homogeneous linear differential equations with algebraic 
coefficients. In the present memoir his methods are applied, and extended, where 
necessary, to the study of the properties of Unary families (a term to be explained 
presently), particularly those whose members are analytic in a triply connected region. 
In Part I the properties of the most general families of this sort are discussed, while 
Part II is devoted to the special case of binary families whose only singularities 
three regular singular points, i. e. hypergeometric families. 
Since the solutions of a homogeneous linear differential equation of the second 
order with single valued coefficients constitute a binary family, it would, from 
point of view, have been most natural to make this discussion a chapter in the theory 
of such equations ; but we have preferred to follow the methods of Riemann. It is 
important, however, to note that all our work may be stated in terms of the differ- 
ential equation ; in particular the classification of binary families in a triply connected 
region given in the last half of Part I may be regarded as a classification of the 
sub-groups of differential equations of the above kind, obtained from the general group 
of the equation by so introducing cross-cuts in the region of definition of the 
coefficients as to make it triply connected. 
« This paper was accepted in June, 1903, by the Faculty of Arte and Sciences of Harvard University in 
satisfaction of the requirement of a thesis for the degree of Doctor of Philosophy. 
one 
Werke, p. 67. 

(18 
p. 379. 
8 Zwei allgemeine Siitze iiber lineare Differentialgleichungen mit algebraischen Coefficienten. (1857). Werke, 
