CURTISS. BINARY" FAMILIES IN A TRIPLY CONNECTED REOION. 61 
Since our criteria, both for types and classes, are entirely in terms of the exponents, 
P families with the same exponents at the same points are kindred. So much we 
conclude from this section : in the next section we show that a 
determined by its exponents 
F family is in f;iet 
§ 2. THE P FAMILY DETERMINED BY ITS EXPONENTS. 
From the differential equation of the P family, we could infer that its only para- 
meters are the exponents, but it will be interesting to follow Riemann here. We 
wish to prove that but one P family, with given singular points, can have a given 
set of exponents. 
We shall need here the following lemma, which is true for P families but not for 
all Q families : 
If two P families are kindred, they have a common connecting formula [including th 
r rr 
associated constants C) in which the fundamental bases have exponents X , X ; ^ , fi ; 
r rr 
v , v , respectively. 
This follows, as a matter of course, from Theorem III of Part I, A, § 6, when a, b 
are none of them semisingular. But when, for instance, a is semisingular, every ba$ 
is fundamental at a, and hence the two branches of a basis may both have exponents A 
We proceed, therefore, to show the truth of our lemma for each class where sen 
singular points enter. 
Type III, Class 3. We have, from Part I, B, § 4, 
P a > =aP b r = a x iV, 
P a " = 8 P b " = B, P c ". 
The exponents of P a ' and P a " must both be X", since the fact that the sum of the th 
exponents of an everywhere fundamental branch must be less than or equal to ion 
readily seen to bar out any other possibilities. Hence we have 
P(a') = a POO + £ pb") = a x P^ + & &>t 
Here none of the coefficients can vanish, and in any two families of this da I we can 
take a, /3, 8 the same; then if one family has the above formula, any kindred family 
must have the same formula, since a basis (/**>, /*?) of the first faunly wneepoode 
to a basis (i**>, JPWJ) of the kindred family; for connecting formulae involving these 
bases must exist which are the same for the two families ; and to complete our proot 
