46 
MR. H. M. TAYLOR ON A SPECIAL FORM OF THE 
h ^ d ‘ 
H14), 
which meets (2), (5), (8), and (11) ; and 
X 
a 
which meets (1), (6), (7), and (12). 
§ 3. It is well known that every plane section of a cubic surface is a cubic curve. 
If, therefore, two straight lines be part of such a section, the remaining part of the 
section is a third straight line. If three straight lines form the section of a cubic 
surface by a plane, every other straight line on the surface must ‘meet one of these 
Imes and only one. We must, therefore, be able to construct all the remaining 
straight lines on the surface, by drawing all the straight lines which intersect each 
of the four triangles formed by the four sets of straight lines 1, 2, 3 ; 4, 5, 6 ; 
7, 8, 9 ; and 10, 11, 12. 
Now, since the twelve lines make triangles when taken also in the groups 6, 8, 10; 
1, 9, 11 ; 2, 4, 12; and 3, 5, 7, it follows that every straight line on the surface 
must intersect one and only one from each of these groups. 
Every remaining straight line on the surface must therefore intersect one line in 
each row, and one line in each column in the scheme 
1 2 3 
6 4 5 
8 9 7 
10 11 12 ^ 
There are nine ways in which we can select one from each row and one from each 
column, viz. :— 
