32 CURTISS. — BINARY FAMILIES IX A TRIPLY CONMCTKD REGION. 
when a is semisingular, our notation is so chosen that \' — \" > 0, while, for a log- 
arithmic, X' — X" > 0. Equal exponents, then, occur only when a is logarithmic. 
A new notation will be found useful to indicate a basis fundamental at a with 
exponents X', X"; such a basis we designate by (y (A '\ y {k,) ). This coincides with the 
notation (y a f , y") except in the case of a semisingular point. We have, then, the 
formula applicable to all cases 
-{-co 
(25) 
tfV) = (x- af'^ gj (x - a)% 
v =o 
tn =(*- «riV(* - ay + J*.. />•) log (x - «), 
v-0 
2>rri 
where g ' * 0; g " * when X' * X", but we may have g Q " = when X' = X", in 
which case a is logarithmic and C a * 0. 
It 
also true that a binary family has two exponents at every point in T t 
olved are of 
family being analytic at such points, these exponents must be positive 
The cases where the holes for a family are regular singular point 
number, are of comparatively little interest, for the functions there ii 
very elementary form. We shall be concerned hereafter with the special class of 
binary families whose holes are all regular singular points, three in number. Such 
families we call Hyper geometric Families} At the three regular singular points a, b, c, 
such a family will have exponents X', X"; /, p r l „', „", respectiv ly, and formula* 
analogous to (25) will exist for each point. 
§ 2. THE EXPONENT 
The exponents are a first essential in the definition of a hypergeometric family ; 
they are, however, not entirely independent of each other, since they are subject to 
relation 
V + X" + fj + f + v> +v » = &n integer 
The positions of the regular singular points «, b, c are, from one point of 
non- 
they can be carried over by a linear transformation into any other 
any 
must remember that we have then 
three without changing the exponents, but 
new independent variable. In particular „ h , « u • i • n 1 
resrwt - v , .. particular, a, b, c can be carried over into 0, co , 1 
respectively, by the transformation 
(26) 
x 
We Jf £S J« " S-«! f ^nctione „ - which he ultimate]y extcndg to ^ 
cases here considered. 
-~ V m ^, s ^ xtlat aU niem u er f , . -c^tj cAii^iius iu an cases ueieuuuoiuoiv-. 
hypergeometric series. ** tamilles are expressible in terms of elementary functions and 
/ 
