47 6 Mr Richmond , On the property of a double-six of lines, 
and consider now the coordinates of B 2 and B 3 . The former lies 
in the tangent planes at A 1 and A 3 , which are x 2 + x 3 = 0 and 
x x + x 2 = 0 ; we may take therefore the following coordinates : — 
For B 2 , 1, - 1, 1, f g, h, 
where by (1) gh + hf+fg + 1 = 0, 
or (/+ 1) (# + 1) (/> + 1) = (/- 1) (9 -l){h- 1). 
For B 3 , 1, 1, —1, l, m, n, 
where ( l + 1) {in + 1) (n + 1) = (/ - 1) (m - 1) ( n — 1). 
The coordinates of A 4> A s , A 6 may now be found; A 4 lies in 
the tangent planes of B S) B 6 , B 2 and B 3 , and from these we deduce 
the coordinates : — 
For A 4 , a, f l, - 1, 1, 1. 
For A 5 , b, g, m, 1, - 1, 1. 
For A 6 , c, h, n, 1, 1, — 1, 
where (a+l)(/ + l)(Z + 1) = {a — 1) (/- 1) (l —1), 
(6 + 1) (g + 1) (m + 1) = (6 — 1) (g - l)(m- 1), 
(c + l)(A + l)(n + l) = (c-l)(A-l)(n -1). 
From these relations in a, b, c, f, g, li, l, in, n it is clear that 
(a + 1) (6 + 1) (c + 1) = (a - 1) (6 - 1) ( (c - 1), 
or bc + ca+ ab + 1 = 0. 
Hence the point B x whose coordinates are 
— 1, 1, 1, a, b, c 
lies on the quadric (1), and its tangent plane passes through 
A 2 , A s , A 4 , As, A 6 ; that is to say, the quadric touches the sixth 
face of the hexahedron. q.e.d. 
It is useful to construct this figure with a quadric in space of 
five dimensions as the starting-point. Any point A x may be 
taken on the quadric, and five points B 2 , B 3 , B 4 , B 5 , B 6 lying on 
the quadric and in the tangent plane at A 4 may also be taken 
arbitrarily ; the rest of the figure may then be obtained uniquely. 
For of the simply infinite system of planes which contain four 
points B 3 , B 4 , Bs, B 6 , two touch the quadric : one of these has 
A x for its point of contact and contains B 6 ; the other is there- 
fore determinate. Five points A s , A 3 , A 4 , A 5 , A 6 are thus found 
whose tangent planes contain four of the points B 2> B 3 , B it B 5 , B 6 ; 
and these tangent planes, by the theorem proved above, will inter- 
sect in a point B x on the quadric. The six coordinates connected 
by a quadratic relation which define the twelve points 
A 1} A,... As, B 1} B 2 ... Bs 
