* 
10 CURTISS. — BINARY FAMILIES IN A TRIPLY CONN B< TED REGION. 
where, in general, none of the coefficients vanish. But we can always choose bases 
which have a simpler generating substitution. Consider the characteristic equation 
(2) 
m 1 — p n 
i 
™ 2 w 2 - p 

This equation is readily shown to be invariant under any linear substitution that 
may be performed upon (y l9 y 2 ), so that it is the same for every basis, and its prop- 
erties are therefore characteristic of the family. Designate by p< /, p a ", the roots 
of (2); these we call the multipliers of the family for the hole a, the i .son for this 
terminology appearing presently.- Neither of these multipliers can u.ish, since 
m x n % — m.^! ± 0. 
Two cases present themselves : 
!• Pa * p a ". This we designate the ordinary case 
Let 
(3) 
ft" '=«*& + (Pa" -«i)y a , 
N 
it being understood that if either n x or m, vanishes, we take *' = m,, Pn » = «, With 
this convention it is easily seen that y a >, y» form a basis. It may be readily verified 
for this basis that its generating substitution for the hole a is 
to/. *">«(*' ° 

Thus 
P 
substZ^ T e We . haV ! ^^ " '"* fr« *") * W <* undergoes a MA**** 
suosmution when continued alnno- a f,,^ i - i 
term m n lfmU. f ° ,, S * fundamen ^ P»th about a; hence the use of the 
term multipliers for n ' o " Tf z.' z." „ 
nndero-op, L •* ' ' are any constan ts * 0, the bask (/ V, *V 
undergoes the same generating substitution. li/ ^ 
This, however, is true of no other bases 
variant. 
te W h il tt fr ° m the &Ct that the ^racteristic eqnation (2) is to- 
the Me « andt h ^ f ° f *" f ° rm *•'• ^JteJLrf trlL for 
uue llole a > and a basis formed frnm ti>„ m s , 
II p ' = " H -7 MdamenM basil. 
ently ' (3) rivet" tat niT "T* % d<> "^ ^ Vanish > a case to be «te«d pr 
-J^lf!" ^ ° ne branCh and its ™Wp'es * which unde,-,n tta «Ji 
substitution for the hoi 
hich undergo the generating 
t „.t 
This branch 
y = Pa y 
about 
multiple of y 
designate by y/, and by y a " we wi n denote anj branch not a 
ien a basis (y a ', y ») undergoes the generating substitution 
(ft', ft") = ( ^ 0> j ( V t v ffS 
