440 Dr Burnside, Convex Solids in Higher Space 



The general statement is that if, in forming a scheme for '2n 

 from the standard scheme for 2n— 1, the first of the new symbols 

 that enters to the left of the bar is 2r + 1, 2n, then the permutation 



(2w, 2r+ 1, 2r+ 3, ,2n- 1) 



will change the resulting scheme for 2w into the standard scheme 

 for 2w: and that if 2r, 2n is the first that enters to the left of the 

 bar, the permutation 



(2n, 2r, 2r+2,....,2n- 2) 



has a similar effect. There is no difficulty in estabhshing similar 

 results for building up the scheme for 2n+ 1 from that for 2n. 



Returning now to the configuration of n + 2 (n— l)-spreads in 

 space of n dimensions, of general position, whose | (n + 1) (n + 2) 

 points of intersection are all finite points, the separation 



ia, ib, ic,. . . .\ id, ie,. . . . 



implies that of the n + 1 points which do not lie on the *th {n— 1)- 

 spread, the set ia, ib, ic,. . .are separated by the ith. {n— l)-spread 

 from the set id, ie,. . .. 



In particular for five planes in ordinary space the scheme is 



I 12 13 14 15 



The first line imphes that plane 1 does not divide the tetrahedron 

 2345 into two parts ; and the second line implies that plane 2 does 

 not divide the tetrahedron 1345 into two parts. 



The third line implies that plane 3 does divide the tetrahedron 

 1245 into two parts, one of which is the tetrahedron 2345, while 

 the other is a polyhedron with vertices 12, 14, 15, 23, 34, 45 

 bounded by each of the five planes. Similarly plane 4 divides the 

 tetrahedron 1235 into the tetrahedron 1345 and a polyhedron with 

 vertices 12, 13, 25, 14, 35, 45 bounded by each of the five planes. 

 Plane 5 divides the tetrahedron 1234 into two polyhedra, each of 

 which is bounded by all the five planes, 12, 14, 23, 34, 15, 35 being 

 the vertices of one and 12, 14, 23, 34, 25, 45 those of the other. 



Hence any five planes of general position in space, whose points 

 of intersection are all finite, form the five faces of just two distinct 

 convex polyhedra each of which has six vertices. The scheme for 

 six 3-spreads in 4-dimensional space may be similarly dealt with. 

 Any five of the 3-spreads bound a convex polyhedron, and the 

 scheme shows how this polyhedron is divided by the remaining 

 3-spread. The result shows that there are just three distinct convex 



