CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 49 
* 
Cl a* (a? - 1)"' G> (x) + c 2 aF (g - 1)"" tf" {x) . 
in the classes of Type IV, other than 1, 2, 3, 4, the family is 
Cl a? (x - 1)"' & (x) + c 2 {^" (x - 1)"" G" (x) + X* (x - 1)"' Q> (x) [*» log z + */ log (* - 1)]} ; 
where, in both cases, 67' (a), G" (x\ are polynomials, while k" and / are non-vanish- 
ing constants. 1 It is a familiar fact that P functions are expressible in terms of 
elementary functions and hypergeometric series, so that we have established the result 
mentioned in the footnote to page 32 : All members of hypergeometric families are expres- 
sible in terms of elementary functions and hypergeometric series. 
The question naturally arises as to how simply a Q function can be expressed in 
terms of P functions. Thus Klein, 2 after showing that we can form a Q function of 
any order k by taking a linear combination of suitable P functions, says : " Um eine 
Q Function in allgemeinster Weise zu bilden, bei der k irgend einen vorgeschriebenen 
Wert hat, muss ich k — 1 verwandte P Functionen . . . mit Hiilfe constanter CoetH- 
cienten linear zusammensetzen." Whether this means that every Q function of order 
k is so expressible is not clear. We must certainly except functions belonging to 
classes impossible for P families. From relation (50) we have no clue to this result; 
as we proceed step by step from a Q function to its expression in terms of the P func- 
tions indicated, the degrees of the coefficients would seem to increase, but common 
factors may, of course, occur. Looking at the question from another point of view, 
we have at our disposal the k ratios of the undetermined coefficients of the expression 
Klein mentions, with which we may hope to form a Q function with the desired appar- 
ently singular points j but can we also so determine the Q function we have formed 
that it will have any of the possible sets of accessory parameters ? This question 
the writer has not been able to answer, in general ; in the case of Q functions of 
order 1, we shall see that, with the exceptions noted, the answer to the above question 
is affirmative. 
D. P FAMILIES. 
We are now in a position to develop the properties of P families as a corollary of 
the more general results of the preceding sections. Riemann, in his celebrated paper, 
confined himself to the case where the difference of no exponent pair is an integer ; 
we shall here be able to extend his results to all cases. 
1 In a footnote on pages 30 and 31 of Ritter's paper (Math. Ann., Vol. 48, 1896), Schilling has shown how we 
may pass from a Q function expressed in terms of P functions of Type I, Class 2 to , . ,Q func »»^ C J"^^ 
of order 1, by allowing coefficients to approach certain limits, and it seems likely that this process can be used 
other cases. 
a Page 229. 
4 
) 
1912 
