GENERAL EQUATION OF A CUBIC SURFACE. 
67 
Each set of three poiuts in a small square in Figure B gives three points, which are the 
intersections of the sides of two triangles. Each set, thm’efore, lies on a straight 
line. 
§ 22. Each pair of triangles which do not have a common liiie, such as 1, 2, 3 and 
6, 8, 10, gives three complete schemes for a pair of tetrahedrons in perspective, viz.: 
,.123 
j 
1 2 3 
and 
1 2 3 
! 6 . 4 5 
6 . 23 24 
6 . 22 25 
i 8 9 . 7 
' 
8 18 .27 
8 19 . 26 
JO 11 12 
10 17 20 . 
10 16 21 
and each pair of tetrahedrons gives four pairs of such triangles. 
Therefore the number of pairs of tetrahedrons 
= f X the number of pairs of such triangles 
= 1.45 . 32/2 = 3 . 45.4 = 6 . 90 = 540. 
It follows that the line, which is the intersection of the planes 1, 2, 3 and 6, 8, lO, 
lies in a plane with the intersections of the pairs of planes 
4 , 5 , 6 and 1 , 9,11 
7 , 8 , 9 and 2 , 4,12 
and 10 , 11 , 12 and 3 , 5 , 7. 
There are, therefore, three distinct planes of perspective passing through each of 
the 720 lines, and each perspective plane passes through four of the lines. 
§ 23. From a closed quadrilateral, such as 
' 2 3 
4 5 
we can, by choosing the four lines which cut two consecutive sides of the quadri¬ 
lateral, obtain the figure 
. 7 . 
L 2 3. 
4 5 
. 12 . . 
K 2 
