22 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED REGION. 
Type I. p a ' p b ' pj * 1, p a " Pb ' pa' ± 1, p a ' Rb" pd * h pd Pb Pc" * 1 • 
We observe first that the coefficient /? in 712) cannot vanish for this type; for if 
, all the terms of (16) vanish except the first, which cannot vanish, since no 
product of multipliers in it = 1, and aS — j3y ^ 0, so that aS ^ 0. As a consequence, 
we see that neither a nor b can be semisingular, for any ba-is is fundamental for a semi- 
singular hole, so that if a or & were semisingular we could write yd = yd, and this 
would be the first line of a possible formula (12) ; but here ft — 0, and this we have 
seen to be impossible. Whether we use b or c in this work is only a matter of notation, 
so that it is true for this type that no hole is semisingular. Three possibilities now 
present themselves. 
If the two holes a and b are ordinary, we have C a = C b = 0, and, using the two 
facts that aS - fiy & 0, and that no product of multipliers = 1, we may easily assure 
ourselves, from (16), that a, /3, y, 8 can none of them vanish. On page 2L we have 
that a, p, § can be taken at pleasure, but * ; (16) then uniquely determines y. 
o 
multipliers have a common formula 
■■* 
'O 
Fixing yd, we can choose yd' so that C a has 
Hence in this case all families ha 
(12), and hence are kindred 
If a is logarithmic, while b is ordinary, we have pd = p a ", C a * 0, C b = 0. Us 
the fact that aS - /Sy * 0, so that if a were to vanish y could not, we ded 
from (16) that a cannot vanish. We see here, as on page 21, that a and /3 are at ■ 
choice, except 
any ^reassigned value * 0. Form now a new y.», by adding to the old sue!, a multiple 
of y.' that in the new formula 8 = 0; ft remains unaltered. In lids new formula, 
then, ft, a, and £ are arbitrary, while 8 = 0; y is uniquely determined by (16). 
Hence all families here have a common formula (12), providing they have the same 
multipliers. J 
If both a and b are logarithmic, we may proceed as follows : Choose bases (yj, ?."). 
ly. , y. ), so that ft and ft have any preassigned values * 0. Take a new ?/„", by 
adding to the old such a multiple of y< that a = ; take as a new „.' and y„" such a 
multiple of the old pair that fi will take on an arbitrary yalue * 0,' ft and ft retain- 
V l 1UeS " Finally ' for a last y"> add t0 l| m former such a multiple of 
» that the new 8 = 0. These changes leave ft and ft still unaltered. Then y, for 
o H W ^ miqUely detemillecl ^ ^- A ^» ^ have, for this case, a 
formula possessed by every family with the same J^ 
one and „ 7 ' a ? u,Mtod a11 ca ^ ! a family with given multipliers belongs under 
and only one of these cases, under each of which families with the same multipliers 
mg their former 
