76] 
SURFACES OF THE THIRD ORDER. 
453 
There is great difficulty in conceiving the complete figure formed by the twenty- 
seven lines, indeed this can hardly I think be accomplished until a more perfect 
notation is discovered. In the mean time it is easy to find theorems which partially 
exhibit the properties of the system. For instance, any two lines, a x , b 2 , which do 
not meet are intersected by five other lines, a 2 , b x , a B , a 7 , a 9 , (no two of which meet). 
Any four of these last-mentioned lines are intersected by the lines a x , b 2 and no other 
lines, but any three of them, e.g. a 5 , a 7 , a 9 , are intersected by the lines a x , b 2 , and by 
some third line (in the case in question the line c 3 ). Or generally any three lines, no 
two of which meet, are intersected by three other lines, no two of which meet. 
Again, the lines which do not meet any one of the lines a B , Oy, a 9 , are a 2 , a 3 , b 3 , b x , c 1; c 2 : 
these lines form a hexagon, the pairs of the opposite sides of which, a 2 , b x ; a 3 , c x ; 
b 3 , c 2 , are met by the pairs a x , b 2 ; c 3 , a x and a x , b,, respectively, viz. by pairs out of the 
system of three lines intersecting the system a 5 , a 7 , a 9 . And the lines a 5 , a 7 , a 9 may 
be considered as representing any three lines no two of which meet. Again, consider 
three lines in the same triple tangent plane, e.g. a x , b X) c x , and the hexahedron formed 
by any six triple tangent planes passing two and two through these lines, e.g. the 
planes x, y, z, y, £. These planes contain (independently of the lines a x , b x , c x ) the 
twelve lines a 2 , a 3 , a i} a s , b 2 , b 3 , b i} b 5 , c 2 , c 3 , c 4 , c 5 . Consider three contiguous faces of 
the hexahedron, e.g. x, y, z, the lines in these planes, viz. a 4 , b 5 , c 4 , a 5 , 6 4 , c 5 , form a 
hexagon the opposite sides of which intersect in a point, or in other words these six 
lines are generating lines of a hyperboloid. The same property holds for the systems 
x, y, g, y, £, y, z. But for the system £, y, £", the six lines are a 2 , b 2 , c 2 , and 
a 3 , b„ C 3 , which form two triangles, and similarly for the systems g, y, z; x, y, z; and 
x, y, so that the twelve lines form four hexagons (the opposite sides of which inter 
sect) circumscribed round four of the angles of the hexahedron, and four pairs of triangles 
about the opposite four angles of the hexahedron. The number of such theorems might 
be multiplied indefinitely, and the number of different combinations of lines or planes 
to which each theorem applies is also very considerable. 
Consider the four planes x, £, x, x, and represent for a moment the equations of 
these planes by x + Aw = 0, x + Bw = 0, x + Gw = 0, x + Dw — 0, so that 
A= 0, 
C- 
l(p — a) + 2mn 
P+ß 
D = 
V p 
«) + 
mn 
P~ß 
By the assistance of 
B-G= ^ ^ [In (p-a) + 2m] [Im (p - a) + 2 n], 
it is easy to obtain 
(A — G) (D — B) _ , p — ß [l (p — a.) + 2mn] [m (p — a) + 2nl] [n(p — a) + 2lni\ 
(A — D){B — G) mn p q. ß [pm (p — a) + 21] [nl (p — a) + 2m] [Im (p — a) + 2n] ’