572 
PROBLEMS AND SOLUTIONS. 
[705 
as one out of many ways of effecting the identification. Observe that there is not in 
the system any triad of triads containing all the numbers. It thus appears that 8, 9, 
a duad, gives only a single form of the system. 
Cor.—It is possible to find in a plane nine points such that the points belonging 
to the same triad lie in lined. The nine points are, in fact, on a cubic curve; and 
the figure is that belonging to a theorem of Prof. Sylvester’s, according to which it 
is possible to find on a cubic curve a system of points 1, 2, 4, 5, 7, 8, &c., (a series of 
7 A 
numbers not divisible by 3), such that for any triad (such as 145) where the sum of 
the numbers, one taken negatively, = 0, the three points are in lined; and so also 
that, if two of the points become identical, in the figure 13 = 14, then there is not 
any new point, but the preceding points are indefinitely repeated; thus, 2, 14, 16 being 
in lined, and 14 being =13, 16 must be =11, and so on. 
Second and Third Cases.—8 and 9 do not form a duad. There are thus three 
triads composed of 8 with (2, 3; 4, 5; 6, 7), and three triads composed of 9 with 
(2, 3; 4, 5; 6, 7). If with these numbers (2, 3; 4, 5; 6, 7) we form all the arrange 
ments of three duads other than those which contain all or any of the duads 23, 45, 67, 
there are the eight arrangements 
A = 24, 37, 56, E = 26, 35, 47, 
B = 24, 36, 57, F = 26, 34, 57, 
C = 25, 36, 47, G = 27, 34, 56, 
2) = 25, 37, 46, 
H = 27, 35, 46, 
where A has a duad in common with B, with D, and with G: but it has no duad 
in common with C, E, F, or H. We have thus the sixteen pairs 
AC, 
AE, 
AF, 
AH, 
BD, 
BE, 
BG, 
BH, 
CF, 
CG, 
CH, 
DE, 
DF, 
DG, 
EG, 
FH, 
rent 
duads.