14 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
From the theorem stated at the end of § 5, we see that Theorem II can be given 
in terms of the connecting formula ; in fact, as we shall proceed to prove, the follow- 
ing gives us a criterion that two families be kindred : 
III. In order that two binary families belonging to T n be I I red, it is both necessary 
and sufficient that the families have the same multipliers, and connecting formulae in 
which the corresponding coefficients and the associated constants C m , C b , C c , . . . , C if are 
the same. 
From the preceding theorems the sufficiency of this condition is obvious. To show 
its necessity, we first note the fact that if a fundamental basis for a given hole with 
multipliers p, p", has at that hole the generating substitution f p , (y „ ), a corresponding 
basis of the other family must, by definition, have the same generating substitution 
that hole. Forming the characteristic equation (2), we see that the multipliers of 
the second family at the given hole are also p', p" , as was to be proved. M 
the possession of the generating substitution ( ° n „ ) about a hole for which the 


multipliers are p, p", characterizes a fundamental basis at that hole. We have thus 
proved that if (/, y") is a basis of the one family fundamental for a given hole, every 
corresponding basis of the other family must be fundamental for the same hole, and 
must besides have the same associated constant C as (//, y 
Let (Va, yd') and (y a f , y a ") be corresponding bases of the two families; a connectin 
formula of the first family is 
(7) <*'■ ^«") = B W> *") - C <jr/, y e ") = . . . = L (*', y!'). 
Denoting by B~ x the inverse of B, we have 
B-* (y.', y a ") = (y b \ y b »), 
°~ l (y.'. y.") = (y e \ y.")> 
L~ x Ofa', y m ") = fo', yiy 
Since the basis B * (i Ja ' } y a ") must correspond to the basis B~* (y a ' 9 y m "), it follows 
from what we have just seen that B~* &', y a ») is a basis «; j*r, f undame ntal at b 
and having the same associated constant C b as (y/, y >"). Proceeding similarly, we 
ha\ 
c- 1 «/, ya") = (#, $J% 
(y.'» y.") = (#, y,"> 
so 
that for the second family we have a connecting formula 
