CURTISS. 
BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
37 
§3. THE DIFFERENTIAL EQUATION. ACCESSORY PARAMETERS. 
To complete the data necessary to determine a Q family, we turn to the differential 
quation of the family 
W 
have 
Part I, A, § 7, that if (/, y 
ft 
any ba 
of 
binary family, every member of that family satisli 
differential equation of 
the form 
<P y 
dx 
z + P 
dy 
dx 
+ 9!/ 
0: 
and from the work given in that section we easily obtain the equations 
P 
d_ 
dx 
D 
9 
D 
i 
ir 
d 
dx 
log A 
where D denotes the functional determinant 
and jy the determinant 
.., 
dx 
dy' 
y 
dx 
f 
dy" 
dx 
dy' 
d?y' 
dx 
dx* 
dy" 
d*y" 
dx 2 I 
For a Q family of order n whose symbol is 
o 
00 
35) 
Q 
1 
v' 
v" 
»i 
** 
a 
! 4- 1 <r 2 + 1 • 


we have, using the notation of the preceding section 
(36) 
1 
X' 
\ 
rr 
V 
x 
+ 
1 
v 
f 
v' 
t 
cr 
<r* + l 

x 
i 
a 
2 
a 
x 
1 
x 
8 
X 
8 
2 
X 
8 
The calculation of q for such a family is, perhaps, best performed as follows : 
Obviously the deri\ 
ol the members of the family whose syml 
themselves constitute a Q family 1 fo 
hich u 
th 
functional determinant 
(35) 
jy 
» We must note one exception, namely where one branch of the family whose symbol is (35) is a constant. 
dQ. 
Here the family -s~ is not a binary family, but q 
0, a result which agrees with our subsequent formulae 
