CURTISS. 
BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
21 
(P ?**) 
(9 Ta") 
11 {r iv) + ^ <. fa 
We can always choose the constants^;, q, r> s, so as to satisfy the equations 
ap 
r 
a 
Bp 
8 
A 
E 
<2 
9 
s, 
and we .shall then have 
7 
r 
Sa_0 
Our new formula which we now substitute for (17) is 
Va 
y<* 
a y,! + y b ", 
VT^y* +By '" 
bap 
where (y a % yj% (y/, ?/„"), stand for the fundamental bases (p T a ', q Y"), (r IV, s r 6 "), 
respectively. Obviously a necessary and sufficient condition that the two families 
(which have the same multipliers, and for both of which C a = C b 
0) be kindred 
-Bafi 
7 
7 
And to find out whether this is true, we need only make use of equation (16) ; but this 
we defer for the present. 
The work in this case has been given at some length in order to make the plan of 
procedure clear. If a hole is semisinuular, we must remember that any basis whatever 
is fundamental at that hole; if logarithmic, we may 
basis to another by taking any multiple of the 
change from one 
fundamental 
first member of the old for the fi 
member of the new, the second member of the new basis being any linearly independent 
member of the family ; but the associated constant C willjn general suffer a change. 
So much, by way of generalities, will suffice. The method of classification will 
become clearer as we proceed, case by case, in the following pages. The general pro- 
cedure is to observe for each type the possible forms of (12), for families with the 
same multipliers, which cannot be carried over into each other by a change of funda- 
mental bases; to each.such form will belong a class of families. Such a plan of classi- 
fication is evidently of the sort we wish ; for families with the same multipliers will 
be kindred when, and only when, they belong to the same class. 
We now proceed to an examination of each type. 
