372 
A MEMOIR ON CUBIC SURFACES. 
[412 
it is so complicated that it can hardly be considered as at all putting in evidence 
the relations of the lines and planes; that of Dr Hart (Salmon, “ On the Triple 
Tangent Planes of a Surface of the Third Order,” same volume, pp. 252—260), 
depending on an arrangement of the 27 lines according to a cube of 3 each way, is 
a singularly elegant one, and will be presently reproduced. 
36. But the most convenient one is Schlafli’s, starting from a double-sixer; viz. 
we can (and that in 36 different ways) select out of the 27 lines two systems each 
of six lines, such that no two lines of the same system intersect, but that each line 
of the one system intersects all but the corresponding line of the other system; or, 
say, if the lines are 
1, 2, 3, 4, 5, 6 
1\ 2', 3', 4', 5', 6', 
then these have the thirty intersections 
1', 2', 3', 4', 5', 6' 
1 7 7 7 "7 7 
2 • .... 
3 . . ... 
4 . . . . . 
5 . 
6 
Any two lines such as 1, 2' lie in a plane which may be called 12'; similarly the 
lines 1', 2 lie in a plane which may be called 1'2; these two planes meet in a 
line 12; and any three lines such as 12, 34, 56 meet in pairs, lying in a plane 
12.34.56. We have thus the entire system of the 27 lines and 45 planes, as in 
effect completely explained by what has been stated, but which is exhibited in full in 
the diagram. 
37. The diagram of the lines and planes is