36 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
As it is merely a matter of 
terchange arbitrarily amongst each 
other in this symbol the -columns to the right of the vertical bars, and those to the 
left. To the right, the exponents cr/, tr" are interchangeable ; to the left, a similar 
remark holds when a, h, c are ordinary, but is no longer true for a semisingular or 
logarithmic point (except, of course, in the case of equal exponents), since, by the nota- 
tion already adopted in such a case, the upper exponent in the symbol must be the 
larger. 
Let us note here an extension of a property observed by Riemann for P functions: 
If Q is a family with the symbol (32), then 
is a Q family with the symbol 
x — b 
Q 
a b 
(\' +8 p' -a i/ 
Further, Q will evidently have the same apparently singular points with the same 
ponents as Q. But a Q family is by no means completely defined, in general, even by 
33), so that we must still calculate the relations between the remaining constants 
required to define Q and Q ; this we will carry out at the end of the next section for a 
very general class of Q families. In particular we have the relation 
oo 1 
1 .J 
(34) *(*-iyQ(\ * ^=§rMt! *-•- *:. + 
oo 1 
x" yP *-y~ v \x»+a • -«-• •' + 
pparently 
8 and € being any numbers we please ; and Q will have the 
points with the same exponents as Q. 
^ We need not here repeat this work for the apparently singular points, ™ 
is of especial interest. If both cr/ and er/' are different from zero, and o,' > " 
but 
o\ , we 
obtain, by dinding every member by (* - ,,)V, f rom our Q family with its regula 
pomto at 0, co, 1, , new one in which ^ exponent3 of ^ ^ ^ _ ^ 0> 
-~ being changed in its symbol (33) except the exponents of the point 
nothing 
inhnity. I„ „ subsequent work, therefore, we shall discuss only families all of 
I uh TT? T^ P ° intS haTC ZW0 fOT ° ne 1— ■ A .-fold apparently 
Z t 3 17" ? ^^ POh,t3 ' ^ ° f Which » *»P'«> *• * ■- exponents 2, 0. 
5nj r^T? Zr. ^ V° 11Wing "* <* * »* 227), the theorem = 
««*. ' ' % W e2 " fl/ fo * numher "f *s simple apparently singular 
