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PROFESSOR CAYLEY ON CUBIC SURFACES. 
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 
r, 2', 3', 4', 5', 6', 
then these have the thirty intersections 
1', 2', 3', 4', 5', 6' 
1 r~: : r~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 
