05 
m of 
also 
not 
being 
three 
with 
ange- 
, 07, 
duad 
Combining AC with 8, 9, we have the triads 8 (24, 37, 56) and 9 (24, 36, 57), 
that is, the triads 
824, 837, 856: 924, 936, 957: 
which, with the original three triads 123, 145, 167, form a system of nine triads; 
8 and 9 might, of course, be interchanged, but no essentially distinct system would 
arise thereby. Hence we have a system of nine triads by combining the original three 
triads 123, 145, 167, with any one of the sixteen pairs AC, AE, &c. But it is 
sufficient to consider the combinations of the three triads with each of the pairs 
AC, AE, AF, AH; in fact, these are the only systems which contain the triad 824; 
and since there is no distinction between the two pairs 4, 5 and 6, 7, or between 
the two numbers of the same pair, it is allowable to take 824 as a triad of the system. 
Hence— 
Second Case.—The system consists of the three triads combined with AE; viz. it is 
123, 145, 167 : 824, 837, 856: 926, 935, 947 : 
which, it is to be observed, consists of three triads of triads, each triad of triads 
containing all the nine numbers; viz. the system is 
123, 479, 568: 145, 269, 378 : 167, 248, 359. 
Cor.—We may have nine points such that the points belonging to the same triad 
lie in lined, viz. the figure is that of Pascal’s hexagon when the conic is a line-pair. 
Third Case.—Combining the three triads with AC, AF, or AH, it is readily seen 
that we obtain in each case a system of the form 
Aolol ', 
Afiy, 
AP'y, 
B(3&, 
ByOL , 
By'a! , 
Gyy , 
Cap, 
GaP', 
viz. in the case where the pair is A C; that is, the system is 
123, 145, 167 : 824, 837, 856: 925, 936, 947 ; 
and in the cases where the pair is AF or AH, the identifications may be taken to be 
A 
В 
C 
a 
P 
7 
a 
P' 
7 
9, 
3, 
4, 
5, 
2 * 
7, 
6, 
3 
9, 
8, 
i; 
2, 
3, 
4; 
6, 
7, 
5 
9, 
8, 
i; 
5, 
4, 
6; 
3, 
2, 
7