54 
MR. H. M. TAYLOR ON A SPECIAL FORM OP THE 
which is identically true, as is at once seen by dividing the equations giving g, by 
he, ad respectively, and subtracting. This verifies the fact that each of the lines in 
the pair i. intersects 14. 
Similarly it can be proved that each of the pair v. intersects 14, and that 13 inter¬ 
sects each of the lines in the pairs ii., iv., and 15 intersects each of the lines in the 
pairs iii. and vi. 
We have now proved that of the tw^enty-seven lines on the cubic surface, each 
cuts ten of the others; furthermore we have shown which line cuts which others. 
Now we might represent all the twenty-seven lines by their projections on a plane, 
where we should have to distinguish between the projection of the actual intersection 
of a pair of lines and the apparent intersection of the projections of two non-inter¬ 
secting lines. We might from such a figure deduce many of the I’elations which exist 
between the lines ; but the figure would be complicated, and the deductions would be 
attended with some difficulty. 
§ 10. Now instead of this we will represent each line by one of a series of 
straight lines in a plane, and we will then assume the figure turned round through a 
right angle, so that we have two lines representing each of the twenty-seven lines on 
the surface. 
The intersection of two lines in the figure which represent the same line on the 
surface we mark with a zero. 
The intersection of two lines, which represent two intersecting lines on the surface, 
we mark with a star, and the intersection of two lines, which represent two non¬ 
intersecting lines on the surface, is marked with a dot. 
With this convention all the intersections of the tw'enty-seven lines on the surface 
are represented in Figure (A), in which each line is denoted by the number by w’hich 
it has been known in the preceding investigation. 
Of course it must be possible from such a figure to deduce all the relations which 
exist among the lines; but it will be found in actual practice that different trans¬ 
formations of the figure are more useful for different purposes. 
§ 11. We will next point out the geometrical properties implied by certain combi¬ 
nations of the stars and dots which may occur in the figure. 
Such a combination as 
h \ .^ ,, 
a 0 * 
a l> 
implies that two lines intersect. 
Here the rows and columns must represent the same lines. 
