34 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION 
if t denotes such a point, with exponents t, t" (which must be positive integ 
zero, and unequal), we have about t the developments analogous to (27) 
v = 4*°° 
< 29 > 2>-(.-U"K-i 2* <*-<)% <7o*<> 
► -o 
If t is the point infinity, we have about it a development analogous to (28). 
Let us now take our regular singular points at 0, oo , 1, as we may always do by 
means of transformation (26). We have the equation 
i 
I 
(30) 
D 
^'+*"-»(a:-lr'^'-'0( 7 ), 
where the function <f> (x) has obviously no singularities in the finite region of the plane, 
since the only singular points of D are 0, oo , 1, at winch developments (27) and 
28) hold. At the point oo , <£(*) has the exponent of D in that point increased b~ 
(X + \ - 1) + („' + /' - 1); i. e . the exponent 
(V + V - 1) + {fA > + f + X) + („' + * _ i). 
Since the exponent sum X' + X" + J + ^ + „< f „" j, al an m we 8ee that 
* <*) can have at most a pole in the point infinity. Hence * (*) must be a poly- 
norma and * degree is obviously the negative of its exponent at infinity ; i ,, it is 
1 - (V + X" + M ' + „» + j + ^ 
-ZS ^: e l mUS * ^ a """ ***« « zero. Hence tke e^onent sun, 
can have only the following values : 
+ 1, 0, -1, - 2, . . . 
In particular + » « c— ^ ^ ^ ^ ^ ^^ ^ . + ^ ^ 
Families whose exponent sum is + 1 
shall call P families, using 
which Riemann adopted in W. «U V/ /«es, using 
1-ted to familieltr ^ Ttlf T' v ^ ^^ * *" 1S 
■om is 1 - *, we will J™t ^ \ ° f f ° rdlna ^ A family whose exponent 
Riemann. 'w^Jllill'^* "** * « bei »* 
Functions belonging to these famil 
used by 
/«»rfiom respectively W e ° « , designate P /„„ctfons and £ 
namely those of a J' " * ** P'^' regard P famili <* as special families, 
namely those of order 
can be 
The distinction between P anH n ^ t 
ktmdl o/ a P « w *W,» , Can be S iven in another w ay : iVi 
We shonld observe here that <h z' 
absurdity for , and r are linearly itd^^ ^ f" J" - W ° UM make *> ™ ish ^"tically, 
In the paper: Ueber die *■£■» Jl T. . See § 4 of the first-mentioned mnpr «f PS*™,-- 
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