CURTISS. — BINARY FAMILIES IX A TRIPLY CONNTCTED REGION. 23 
are always kindred. Hence Type I furnishes but one class. -Stated in other words 
under Type I the group Is alw< is completely det* rmined by the multipliers. 
Type 1 1. p a ' p b ' p c ' = 1, p a " pi! pj * 1, Pa' Pb" Pc * 1, Pa Pb Pc" ± 1 • 
Every hole is here ordinary ; if, for instance, p rt ' = p/', we should have p a "p>'pc' = 1, 
contrary to hypothesis. Equation (16) reduces to /3y = 0. We have, then, the 
following three cases: 
1. /3 = 0, 7*0. 
2. £ * 0, 7 = 0. 
3. yg = 0, 7 = 0. 
In all these cases neither a nor 8 can vanish, since aS — fiy -A 0. In each case 
the coefficients of (12) that do not vanish can take on any value * hy an appropriate 
choice of fundamental bases, but it is impossible to prevent the specified coefficients 
from vanishing, or to make any others vanish. Every connecting formula can be 
reduced to one of three special ones, marked by the respective cases, but these three 
cannot be carried over into each other. Families of Type II, with the same multi- 
pliers, may fall under any one of these three cases, so that we have here three classes ; 
these we will designate Classes 1, 2, 3, respectively. 
Type III. Pa ' Pb ' P J = 1, p rl " Pb' Pc' = 1, Pa Pb" Pc' * 1, Pa Pb' Pc" * 1. 
rr 
Here pj = p B ", but p b ' * p b ", pj * p"; hence C b vanishes, and (10) reduces to 
C a a ft = 0. Since a and /3 cannot both vanish, on account of the relation a 8 — /?y ^ 0, 
and since when C a = we can always take J3 = 0, on account of the fact that (y b ', y b 
is then fundamental at a, we have only the following three cases to consider : 
1. C a * 0, a * 0. Here /3 always vanishes, from (16), while 8 never vanishes 
since a 8 — $y i=- 0. Given a formula (12), we may first so change it by a new 
of y b and y" that a and 8 have any values ^ that we please. The 
'-> 
formula we can again change by using as a new y a " the old y a " (which we can suppose 
to have been so chosen that C a has any value we please ^ 0), plus any multiple we please 
of y a '. This leaves C a , a, 8 unchanged, but we can make the new y anything we 
please, including zero. Hence the formulas (12) of all families with the same multi- 
pliers under this case can be taken the same (including (7 a ). 
2. C a & 0, a = 0. The discussion here is much the same as for the previous case. 
/3 can never vanish 
we can take /3, y, C a anything we please except zero, 
while to 8 we can give any value including zero, so that the formulas (12) of all 
families under this case can be taken the same. 
