CURTISS. BINARY FAMILIES IN A TRIPLY CONNECTED REGION 
53 
where <£ (x) is everywhere on the sphere a single valued analytic function, and hence is 
a constant. Further, since the exponent at oo of <j> (x) is 
V + X" + fit + ft" + v ' + v" 
1, 
</> (x) vanishes at oo, and therefore is identically zero. From this result, and the com- 
mon connecting formula of the two families, we deduce, with Riemann, the equations 
(52) 
p(0 
p(A") 
W 7 ) 
poo 
poo 
poo 
poo 
p(o . 
Now p^jr is clearly single valued in the neighborhood of the point 0, and similarly 
for 
p( M ') p{v>) 
9 
pUO' i>(0 
?: ? 
in the neighborhood of oo, 1, respectively. Hence from equations (52) 
P<o . 
we see that —^ is single valued and analytic on the entire sphere, provided that i** 
and P (A "> do not both vanish at any point other than 0, go, 1. To prove that this is the 
case, we need only remember that the functional determinant 
D 
poo 
p(K") 
dPM 
dx 
d P^ 
d x 
is equal to ic v + x "- 1 (x - iy'+""-i <f>( x ) (see page 34), <£ (x) being for P families a constant, 
and as observed in the footnote to page 34, <f) (x) 
0. From this it is obvious that i** , 
-P (A,,) cannot vanish together in any point other than 0, oo, 1. 
It is evident, therefore, that 
p(x 
■m*i • being sing. 
valued and 
aly 
over 
the entire sphere, must be a constant. In consequence, the families 
Cj P^ + c 2 P^ and 5, P^ + H P K '\ 
must be identical, and this establishes our theorem 
Note that this proof fails for Q families of order > ; in fact the theorem itself 
is not true. 
§ 
We may take our formula here bodily from Part II, C, § 1- We have 
P P 
pn pi 
£*+* (X 
iy+ v '4> (*) 
