28 CURTISS. — BINARY FAMILIES IN A TRIPLY CONNECTED RFMON 
The remaining classes are infinite in number, each corresponding to a value 
of -r other than 0, oo , — 1. In all of them all holes are logarithmic. 
In conclusion, we deduce from this table the following theorems : 
I. The condition that some product p a p b p c — 1 is sufficient, as well as necessary, that a 
family have an everywhere fundamental branch. 
II. A necessary, but obviously not a sufficient condition that a hole he semi singular is 
that at least four products p a p b p c = \. Theorem I shows that a family which has a 
semisingular hole must have an everywhere fundamental branch. 
III. The classes of Types II and III are completely characterized by the multipliers be- 
longing to their every where fundamental branches. With Type IV this is no longer true. 
§ 
RELATION >VS 6 S a =l. 
Let S c S b S a = f J M. Then the equation S c S b S a - 1 is readily seen to be in 
to. n 2 
reality equivalent to the four 
m x = 1, Wj = 0, m 2 = 0, w„ = 1. 
The constants m x , n l9 m?, n 2 , may be obtained directly in terms of the coefficients of a 
connecting formula, the associated constants C, and the multipliers, by performing the 
substitutions indicated. We are thus led to four equations connecting these latter 
quantities, which will be found equivalent to three such equations besides the relation 
' " ' ff t n 
Pbpb PcPc = 1. Other methods more convenient than using the substitution 
product 8 C S b S a suggest themselves ; for instance, to use the equation in the form 
St S a = S7 1 . Without going into the details of this computation, we here append, under 
four cases, the system of equations sought for, giving them in as simple and symmet- 
rical form as possible. With the first set it will be interesting to compare the equations 
given by Riemann for the class of hypergeometric families which he considers. 1 
!• Pa * Pa", Pb f * p 6 ", pJ * p c ". 
(Pa' Pb' Pc' ~ 1) a, B = (p a » pb > p c '-l)fi y v 
(Pa' Pb" P J - 1) a x 7 = (p a » p b " pj -\)a y v 
(Pa' Pb' Pc" - 1) ft 8 = ( Pa » p b ' p c » -1)0$!, 
(Pa' Pb" Pc" ~ 1) ft 7 = (/>„» Pb " Pc " - 1) a B v 
2 - Pa = Pa", p b ' * p b '\ pJ * p 
n 
(P^PbPc'- 1) K 8 - /3 7l ) = O a a x £ Pa 'p b >p c >, 
{Pa Pb Pc ~ 1) (a, 7 - a 7l ) =C a a ia p a > p b » p c \ 
(Pa Pb'pc"- 1) (ft « - y = Q a ft $ Pa r Pb > 9 » 
(Pa Pb p c " - 1) (ft 7 - a SJ = <7 a a ft p a > p b " p c ' 
1 Werke, p. 73. 
