12 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
and the pair of multipliers p? 9 p". Proceeding in the same way for each hole, we 
have for a, b, e, . . ., I, fundamental bases (y/, y a "), (y b % y 6 "), (y/, y c "), . . ., (y/, y,"), 
and corresponding pairs of multipliers p a f , p a " ; p b ', p b " ; p/, p c " ; . . . ; p{, p{'. A 
given hole may be ordinary, logarithmic, or semisbujidar ; in each case there are 
certain degrees of freedom in the choice of a fundamental basis, as noted in § 4. 
Every fundamental basis for a can be put into the form (4), where <f> a ' (x) and <f> a "(x) 
are single valued and analytic in any doubly connected region we may form from 
T n by suitable cross-cuts which do not meet the boundary of a. Similarly for each of 
the other holes, with an appropriate change of notation in (4). 
Let us note here an important relation between the multipliers of a family. From 
equation (1) we deduce at once the equation 
Det S a • Det S b • Det S c . . . Det Si = 1, 
where Det S stands for the determinant of the substitution S. But from the invari- 
ance of (2) it follows that for every generating substitution about a we have 
Det S a = P J Pa '\ 
and analogously for Det S b , Det S c , . . . , Det 8 t . Hence we have the relation 
( 5 > Pa' Pa" Pb' Pb" Pa' Pe" . • • *>' ' » 1. 
By definition (2) of § 2, any set of fundamental bases is connected by a relation 
r „, ' , a „. » 
(6 ) ». = » li + » it = «, ,.' + £, y« = . . . = „„_, „/ + ff ,,» 
I. =m'+« it = 7, li + «,*."=...= y^ y', + S„_ 2 y«. 
This we will call a connecting formula} Obviously, from the degree of arbitrariness 
that enters into the choice of fundamental bases, some of the coefficients in a con- 
necting formula will be more or less at our disposal. One of the chief questions 
wh.ch will concern us is the determination of the remaining coefficients after those 
disposal have been fixed by 
of bases in (6). As a matter of 
fact we shall see later that the pre-assigni ng of the multipliers will, in general, though 
not in all special cases, determine these remaining coefficients when « = 3. It i 
known that m general f or » > 3 additional conditions are needed 
The importance of the connecting formula is evident from the following theorem, 
which is at once seen to be true : 
The group of a famihj is completely determined in case we know its multiplier,, the 
coeffivents of a connecting formula, and the constants C , C b C 
•»to the generating suhstitutions of the oases in that connecting'), 
&. which 
1 Klein: Zusammenhangsformel. 
