On the 'property of a double-six of lines, etc,. 
475 
On the property of a double- six of lines, and its meaning in 
hypergeometry. By H. W. Richmond, M.A., King’s College, 
Cambridge. 
[Received 25 February 1908.] 
The name Double-Six was given by Schlafli (in Vol. II. of the 
Quarterly Journal of Mathematics, p. 116) to sets of twelve lines 
(«!, a 2 , a 3 , a 4 , a 5 , ay, b u b 2 , b 3 , b 4 , b 5 , b 6 ) having the property that 
every line of the first six intersects every line of the second six, 
except that which has the same suffix ; a r intersects b s provided 
r f s. Such sets of lines appear in connexion with surfaces of the 
third order, but a double-six can be built up and the property 
established independently. 
By interpreting the six coordinates of a line as coordinates of 
a point in space of five dimensions, we arrive at an interesting 
theorem in hypergeometry, essentially equivalent to that of the 
double-six, but presenting the same fact in a more significant form. 
When six points have been taken in space of five dimensions, and 
a plane (i.e. a space of four dimensions) has been determined by 
each set of five of the points, the six points and six planes are 
said to form the vertices and faces of a hexahedron. The theorem 
may be stated as follows: 
If in space of five dimensions a quadric passes through all the 
vertices of a hexahedron and touches five of its faces, it must touch 
the sixth face also. 
Let A x , A 2 , A 3 , A 4 , A 5 , A 6 be the vertices of a hexahedron 
and let a quadric pass through these points and touch the faces 
opposite to A n _, A 3 , A 4 , A 5 , A 6 , the points of contact being 
denoted by B 2 , B 2 , B 4 , B 5 , B 6 . Since A : is on the quadric and lies 
in the tangent plane at B. 2 , B 2 must lie in the tangent plane at A 1} 
and the line A 4 B 2 must lie wholly on the quadric. For a system 
of homogeneous coordinates (x 1} x 2 , x 3 , x 4 , x 5 , x 6 ) let the hexa- 
hedron of reference be that whose vertices are A 1 ,A. 2 ,A 3 ,B i ,B b , By. 
the equation of a quadric passing through these points and such 
that the tangent plane at each of the three last points contains 
the three first is 
A xyc 3 + Bx 3 x x + Cx 2 x 2 + Dx 5 x a + Ex ti x 4 + Fx 4 x 6 = 0. 
By the introduction of suitable real factors into the coordinates 
(a?) this equation may be reduced to one of the forms 
Xyc 3 X 3 X 2 X\X 2 == Hh (x 3 Xg -f- x 3 x 4 F x 4 xf , 
and it will not be necessary to work out more than one case. Let 
the quadric be 
x 3 x 3 + x 3 x 2 + x v x. 2 = x b x 6 + xyx 4 + x 4 x b ( 1 ), 
