CURTIS S. 
BINARY FAMILIES IX A TRIPLY CONNECTED KKGION. 
33 
We have seen that the exponent sum must be an integer; we may go further and 
state that this integer must be 1, 0, or negative. This result is obtain- 1 from a. study 
of the functional determinant 
D 
y 
n 
dy<_ 
d x 
dx 
If we compute 
d yW 
dx 
and 
dy( 
from (25), we ha\ 
d_y^ 
dx 
dy 
(x 
A'-l 
foo' *' + 0i' (*' + *) (*-«) + -"L 
(A") 
dx 
(* 
<O x " _1 W *>" + 9x O" + !)(•-«)+...-] 
+ 
C. 
2tt/ 
I ■ 
./■ 
ay- 1 2 ^' (* 
v = 
«r 
As we have already remarked, p. 17. a binary function and its derivative are always 
kindred. Accordingly we have, using the relation of page 15, 
I) 
T)<$tA 
y 
(*') 
y 
(A") 
dyM 
dx 
dy 
dx 
(A") 
<* 
«) x ' + *"- , W^o r '(^-^+^,(* 
a) + . . . } 
Detvl • 
+ 
C a 
2irt 
j(* 
«) 2X '~ 1 W + ft (* 
a) + , . . } 
H 
Z) can have no logarithmic terms. If a is not logarithmic, C ( 
and 
r _ // 
f/o £7o 
^/ 
) ^ 0, so that we have the development about 
+ • 
(27) 
D 
(x 
a) 
A'+A"-l 
S^-c* 
a/, <7 * 
o 
If a is logarithmic, and X 
If A' 
so that (2 
then g ' g 
rf 
ft 
holds. In all 
X", the same formula holds, since X' 
* 0, but g ' 2 C7./2 vi*0, while 2 X 
cases, therefore, (27) gives the 
X 
1 
rr 
teg 
\' + X 
// 
1 
development of D 
about a when a is finite. If a is the point infinity, the reader may easily verify the 
development 
(28) 
D 

In the preceding paragraph we have discussed only 
of a regular 
point, but obviously analogous developments must hold at non-singular points ; in fact 
i 
See Klein, p. 222 et seq , where only ordinary singular points are considered. 
3 
