CURTISS. — BINARY FAMILIES IN A TRITLY CONNECTED REGION. 11 
From the invariance of equation (2), it follows that n = p a '. m, however, will either 
assume any given value except zero, according to our choice of a basis, or else vanish 
for all bases. If m ^ 0, we shall speak of the hole a hx logarithmic for the family; 
if m = 0, a will be called semisingular } (yd, yd') is again called a fundamental basi-. 
but we should note that by this definition this term is, in the case before us, to be 
applied to any basis whatever in which the first member is a branch undergoing a 
multiplicative generating substitution about a. In case a is semisingular, all members 
of the family have the generating substitution about a, 
V = Pa y, 
so that every basis is fundamental ; but in the logarithmic case, a particular branch 
y a ' and its constant multiples are the only members undergoing such a substitution 
about a. 
If m? = ?h = 0, (3) is invalid, but we have evidently the semisingular case. 
If a and a" are any quantities that satisfy the relations 
e — p a , e — p a , 
while a x is some point of the hole a, we shall have, provided a t is not the point infinity, 
(4) 
yd = (x-a i Y' <p a ' (*), 
O a 
yd' = (*- «i)°" <l>d' (*) + 2^. yd log (x - a,), 
where <j>d (x) and <pd' (x) are single valued and analytic in T 2 . If a x is the point 
infinity, x — a x is to be replaced by - in (4). Formula (4) will represent a fundamental 
basis in all cases; for the ordinary and semisingular cases C a = 0, while for the 
logarithmic case C a i 1 0. 
In particular, the hole a may be a single point. In this case we have Laurent's 
developments for </>d and </» a " in a properly chosen circle about a, and these develop- 
ments may have, for certain families, only a finite number of terms with negative 
exponents (positive exponents if a is the point infinity), a is then said to be a regular 
singular point of the family. 
§ 5. FUNDAMENTAL BASES FOR AN n-TUPLY CONNECTED REGION. 
CONNECTING FORMULAE. 
Passing now to the general case of n holes, we see that if we join together by 
n — 2 cross-cuts all the holes except t, leaving that hole free, we have a doubly con- 
nected region, so that from § 4 we deduce the existence of fundamental bases (y/ 9 y") 
1 These terms are used by Klein, as well as lengthier ones which are less convenient 
) 
