PBOEESSOB CAYLEY OUST CUBIC SUBPACES. 
247 
40. Dr. Hart arranges the 27 lines, cubically, thus : 
Aj Bj C, 
A 2 B 2 C 2 
a 3 b 3 c 3 
d x b x Cj 
^2 ^2 ^2 
«3 ^3 <?3 
«1 Pi Yl 
a 2 P 2 Y 2 
K Z P 3 Yz 
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 15 A 2 , A 3 lie in a plane AjAgAg ; A n B 1; C, 
lie in a plane AiB^. Letters of different alphabets denote lines which meet accord- 
ing to the Table 
a x b 2 c 3 
b x c 2 a 3 
Gy a 2 b 3 
A, 
By 
Cy 
K 1 P 2 Y3 
Pi Y 2 «3 
Yl a 2 Pz 
c 2 a 3 b x 
a 2 b 3 Gy 
b 2 C 3 dy 
a 2 
b 2 
C 2 
Pz Yl «2 
Yz «1 P 2 
«3 Pi Ya 
63 Cy a 2 
C 3 dy b 2 
a 3 by c 2 
a 3 
b 3 
C 3 
Y2 a 3 Pi 
«2 Pz Yl 
P 2 y 3 ccy 
where the letter in the centre of the square denotes a line lying in the same plane with 
the lb .as denoted by the letters of each vertical pair in the same square. Thus Aj lies 
in the planes A A,§ 2 /3 2 , A 3 c 3 y 3 (and in the before-mentioned 'two planes A,A 2 A 3 , 
AjBjCj). 
41. I find that one way in which this may be identified with the double-sixer nota- 
tion is to represent the above arrangement by 
1 , 2 ', 12 
14, 25, 36 
4', 5, 45 
3', 4, 34 
2, 6', 26 
23, 46, 15 
13, 24, 56 
1', 16, 6 
3 , 35, 5' 
and then the identification may apparently be effected in (720x36 = ) 25920 ways, viz, 
we may first in any way permute the {,, §,, £,, f ,, by this means not altering the 
double-sixer \, |, 5 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(l, 1, 1, 1, mn+~< nl+ r, lm+1-, l+), »+L rc+^X, Y, Z, W)’+*XZY=0, 
