GENERAL EQUATION OF A CUBIC SURFACE. 
61 
It appears, therefore, that there is but one line on the surface which does not 
intersect one line at least of a closed quadrilateral on the surface; that there are two 
lines only, forming a non-intersecting duad, which do not intersect one line at least 
of a closed pentagon on the surface ; and that there are three lines only, forming a 
non-intersecting triad, which do not intersect one line at least of a closed hexagon on 
the surface. 
§ 14, Closed Quadrilaterals. — If the lines a, h, c, d, taken in order form a closed 
quadrilateral, it appears from what has gone before that 
when a is given, there are 10 ways of choosing h ; 
when a, h are given, there are 8 ways of choosing c ; and that 
when a, h, c are given, there are 4 ways of choosing d. 
Hence, the number of orders of choosing 4 lines to form a quadrilateral is 
27,10.8.4, and each quadrilateral will appear 4 x 2 or 8 times. 
The total number of closed quadrilaterals therefore is 
27.10.8.4 
4.2 
1080. 
Now we have sliown that there is only one line which does not cut at least one of 
the sides of a closed quadrilateral. 
There must, therefore, be 1080/27 = 40 closed quadrilaterals which each line does 
not cut. 
There are 16 lines which do not cut a given line ; therefore these 40 quadrilaterals 
are formed of 16 lines, and these 16 lines are capable of being divided into sets of 
four quadrilaterals in ten different ways. 
One such set of four quadrilaterals, none of the sides of which cut the given line 
26, is 27, 21, 5, 18 ; 2, 22, 13, 14 ; 9, 24, 12, 7 ; 10, 16, 1, 6. 
§ 15. Closed Pentagons. —If the lines a, h, c, d, e, taken in order form a closed 
pentagon, it appears that 
when a is given, the number of ways of choosing 6 is 10 ; 
when a and h are given, the number of ways of choosing c is 8 ; 
when a, h, and c are given, the number of ways of choosing d is 4 ; 
when a, h, c, and d are given, the number of ways of choosing e is 3. 
Hence the number of ordei’s of choosing five lines to form a closed pentagon is 
27.10.8.4.3, and each pentagon will appear 5 X 2 or 10 times. 
The total number of closed pentagons therefore is 27,8.4.3 = 2592. 
Now we have shown that there are only two lines, forming a non-intersecting duad, 
which do not cut one at least of the sides of a closed pentagon. 
