412] 
A MEMOIR ON CUBIC SURFACES. 
375 
40. Dr Hart arranges the 27 lines, cubically, thus: 
A 
B x c\ 
a x 
b x 
Ci 
«1 
A 
7i 
a 2 
B 2 c 2 
a. 2 
h 
c 2 
a 2 
A 
72 
A 3 
B 3 c 3 
Cl 3 
b 3 
C 3 
«3 
A 
7s 
where letters of the same alphabet denote lines in the same plane, if only the letters 
are the same or the suffixes the same; thus A l} A.,, A 3 lie in a plane A X A 2 A 3 ; 
A x , B 1} C x lie in a plane A 1 B 1 C\. Letters of different alphabets denote lines which meet 
according to the Table 
where the letter in the centre of the square denotes a line lying in the same plane 
with the lines denoted by the letters of each vertical pair in the same square. Thus 
A x lies in the planes A x q^cl x , A x b 2 ^ 2 , A x c 3 ^ 3 (and in the before-mentioned two planes 
A X A,A 3 , A X B X C X ). 
41. I find that one way in which this may be identified with the double-sixer 
notation is to represent the above arrangement by 
1, 
2', 
12 
3', 
4, 
34 
13, 
24, 
56 
14, 
25, 
36 
2, 
6', 
26 
A 
16, 
6 
4', 
5, 
45 
23, 
46, 
15 
3, 
35, 
5' 
and then the identification may apparently be effected in (720 x 36=) 25920 ways, viz. 
we may first in any way permute the r, s'? o'* by this means not altering 
the double-sixer r \ 3' i' 5' 6'. and then upon the arrangements so obtained make any of 
the substitutions which permute inter se the 36 double-sixers. 
42. The equations of the 45 planes are obtained in my paper last referred to, 
viz. taking the equation of the surface to be 
W (1, 1, 1, 1, mn + , nl + , Ini + , l 4- -j, 7n-{- — , n + Y, Z, TL) 2 + kXZY = 0, 
Im 
l 
_ p 2 -/3 2 
2(p-a)’ 
a = Imn + 
Imn ’ 
/3 = Imn — 
1 
Imn ’ 
where