116 
[401 
401. 
A NOTATION OF THE POINTS AND LINES IN PASCAL’S 
THEOREM. 
[From the Quarterly Journal of Pitre and Applied Mathematics, vol. ix. (1868), pp. 268—274.] 
Taking six points 1, 2, 3,4, 5, 6 on a conic; let A, B, C, D, E, F, G, H, I, J, K, L, M, N, 0 
denote each a combination of three lines, thus 
12.34.56 = A 
13.45.62 = B 
14.56.23 = G 
15.62.34 = D 
16.23.45 = # 
12.35.64 = # 
13.46.25 = G 
14.52.36 =H 
15.63.42 = 1 
16 . 24.53 = J 
12.36.45 = # 
13.42.56 = # 
14.53.62 = if 
15.64.23 = N 
16.25.34 = 0 
then any hexagon formed with the six points may be represented by a combination of 
some two of the letters A, B, &c., viz. the three alternate sides are the lines repre 
sented by one letter, and the other three alternate sides the lines represented by the 
other letter: for example, the hexagon 123456 is AE; and so for the other hexagons. 
Any duad AE thus representing a hexagon may be termed a hexagonal duad; the 
number of such duads is sixty. Each Pascalian line may be denoted by the symbol 
of the hexagon to which it belongs ; thus, the line which belongs to the hexagon AE, 
is the line AE. 
I form the following combinations: 
I MO . DHJ each involving all the duads 12, &c. except those of 123.456, 
„ „ „ 124.356, 
DEG .BN 0 
ELM.BGJ 
HLN. CGI 
EFI.JKN 
A EH. GKO 
AMN.GDF 
AGJ. ELO 
ABI.DKL 
GKM.BFH 
125.346, 
126.345, 
134.256, 
135.246, 
136.245, 
145.236, 
146.235, 
156.234,