140 Professor Baker, On a proof of the theorem of a double six 



these form a double six, any one of the lines meeting the five which 

 do not lie in the same row or column with itself. 



§ 6. Conversely we now proceed to show that if 



a^ h^ Cj d^ 



a{ h{ c-l d^ 



be eight lines in three dimensions such that no two of a^, h-^, c^, d^ 

 intersect, while d-l intersects a^, 6^, c^, a{ intersects h-^, Cj, d^, etc., 

 and if one of the two transversals, say I, of a^', hy, c{, d^, intersects 

 one of the two transversals, say m' , of a^, \, Cj, d-^, then these lines 

 may be obtained by projection from four dimensions; namely 

 «j, &!, Cj, d-^, a{, bj', Cj', c?j' are projections of four lines a, b, c, d 

 in space of four dimensions and of the transversals a', b', c', d' of 

 threes of these, respectively, while I and m' are the intersections 

 w^ith the original three dimensional space of planes in four dimen- 

 sions meeting respectively the set a', b' , c' , d' and the set a, b, c, d. 



We give an analytical proof of this. And for this purpose first 

 explain an analytical view of the theorems which have been given 

 in §§ 2, 3, 4, which indeed renders these very obvious. 



It is fundamental that a point may be represented by a single 

 symbol, say P, the same point being equally represented by any 

 numerical multiple of this, say mP, where m is an ordinary 

 number. Then a space of r dimensions is one in which every y + 2 

 points, Pi, P^, ..., Pr+2^ are connected by a sy2ygy, 



w^Pi + m^P^ + ... + m^+2P^+2 = 0, 



where m^, ..., m^+2 are ordinary numbers; thus the space is deter- 

 mined by any r + 1 points of it, themselves not lying in a space 

 of less than r dimensions ; and, in terms of such r -f- 1 points, say 

 ylj, ..., ^y+i, every other point of the space may be represented by 

 a symbol Xj^A-^^ + x^A^ + ... -f x^+i^^+j, where x-^, x^, ..., x^+i are 

 ordinary numbers ; whose ratios may be called the coordinates of 

 this point, relatively to Aj^, ...,Ar^j^. Thus any point of a line 

 determined by two points A, B, is representable by a symbol 

 mA + nB, in which m, n are numbers; and any point of a plane 

 determined by three points A, B, C, is representable by a symbol 

 xA + yB + zC. 



In our figure in four dimensions (§ 2), let the points 



A = {b', c), B = (c', a), C = {a', b), 



A' = {b,c'), B' = {c,a'), C' = {a,b'), 



be regarded as fundamental. Being in four dimensions, they are 



connected by a syzygy ; absorbing proper numerical multipHers in 



the symbols, this may be taken to be 



A + B + C + A' + B' + C = 0. 



