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24 CURTISS. — BINARY FAMILIES IX A TRULY CONN'ECTKD 
3. C a — 0. Since every basis is fundamental at a, we can here ah ivs take for (12) 
y a " m |>» 
A family of Type III, with given multipliers, may belong to any one of these three 
cases, and the formulae of different cases (including the values of C a ) can obviously not 
be carried over into each other. Hence we have here three classes which we number 
in correspondence with the above three cases. 
Type IV. Pa ' P ' Pc ' = 1, Pa » Pb > Pc > = l, pa ' Pb " Pc < m l, p j pb < pr « = l. 
■ 
We may deduce from the relations characterizing this type the equations p a ' = p a ", 
i it i n 
Pb = pb , pc = pc • Equation (16) reduces to C a C b 8 = 
It will be convenient, though not necessary, to make use of C c here ; we may then 
consider the following five cases : 
1. C a = 0, C b ±0, C c ±0. 
2. C a * 0, C b = 0, C e * 0. 
3. C a * 0, C b ,t 0, C c = 0. 
4. C a = y C b = 0, C C = Q. 
o. C a * 0, C b ± 0, C c ± 0. 
These exhaust all possible cases, for if two holes are Bemisingular, it is obvious that the 
third will also be semisingular. The groups corresponding to each case are evidently 
different. In all cases we may take for (12) 
(19) 
Va = y h \ 
for in cases 1, 2, 4 every basis is fundamental either at a or 6, while in cases 3 and 5 
equation (16) gives = 0, and since b is logarithmic and 8 * 0, a * (on account of 
Py * 0), we are at liberty to rename ay b andyy/ 4- $y", which are 
fundamental bases for b, y b \ and y b " respectively. To find whether there is but one 
class under each case, we must see whether in (19) C a and C b can be given the same 
set of values for every family under each case. 
1. C B == 0. C b can always be given any value we please * 0, since *»" is at our 
choice. There is but one class here. 
2, 3 There can obviously be but one class in each of these cases, from their resem- 
blance to Case 1. 
t w = 1 C r,°' j alWayS ' ^ tWs ° aSe - There is but one c1 ^ here. 
0. We shall find an infinite number of classes under this case, for wnen we have 
chosen our bases ln (19) so that one of the two quantities n. ft is fixed, the other is 
aj ^b 
