438 Dr Burnside, Convex Solids in Higher Space 



There are just |(w+l)(w+2) points of intersection and 

 \n {n+ 1) {n + 2) lines of intersection of a general set of (w— 1)- 

 spreads, n + 2 in number, in n-dimensional space. The hnes pass n 

 by n through the points and the points he 3 by 3 on the hnes. If 

 each {n— l)-spread is denoted by a single symbol i, the point in 

 which all the {n — l)-spreads except i and j meet may be denoted 

 by ij and the hne of intersection of all the {n— l)-spreads except 

 i, j and k by ijk. The three points ij, ik, jk he on the hne ijk. If this 

 configuration is projected from an arbitrary point of the n-dimen- 

 sional space upon an arbitrary {n- l)-spread in it, the configuration 

 becomes a hke one in (n- l)-dimensional space. If the points of 

 the original configuration are all finite points, the projection may 

 clearly be carried out so that if ij is between ik and jk in the 

 original configuration, the same is true after projection. Taking 

 again an arbitrary point and an arbitrary {n— 2)-spread in the 

 {n - l)-dimensional space, the configuration may be projected into 

 a like one in {n- 2)-dimensional space; and the process may be 

 continued. Moreover if all the points of the original configuration 

 are finite points (so that for each set of three such as ij, ik, jk one 

 IS actually between the other two) and if each projection is'carried 

 out as suggested above, then in the final two-dimensional figure ij 

 will be between ik and jk if it was so in the original configura- 

 tion. ^ ^ 



It will be said that ij and ik are opposite or adjacent according 

 as jk is or is not between them. If ij, ik are adjacent and also ij 

 and il, then tk and il are adjacent. Hence, with m single symbols, 



the m - 1 points il, i2, im maj be divided into two sets such 



that all those of either set are adjacent, while any two taken one 

 from each set are opposite. A suitable symbol to indicate the 

 separation is 12 13 | 14 15 16, all those on either side of the bar 

 being adjacent. 



It will now be shown that, apart from permutation of the single 

 synibols, there is just one scheme for the separations of the set 

 of 2 m (m- 1) points arising from m symbols. 



When n is odd, 7 for example, the "typical separation is 

 I 12 13 14 15 16 17 

 I 21 23 24 25 26 27 

 13 I 23 34 35 36 37 



24 I 14 34 45 46 47 (i), 



15 35 I 25 45 56 57 

 26 46 I 16 36 56 67 

 17 37 57 I 27 47 67 

 which is associated in an obvious way with the symbol 



{12} {34} {56} {7}. 



