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



may denote by a, namely that common to the two spaces (6, c) 

 and {a, d). We see how much more naturally this arises than the 

 statement, to which it is evidently analogous, in the three dimen- 

 sional figure considered in § 1. It follows that the lines A A' and PP' 

 intersect one another, say in L. Similarly the plane, ^, of inter- 

 section of the spaces (c, a), {b, d), contains the lines BB' and QQ', 

 which then intersect, say in M; and the plane, y, of intersection of 

 the spaces {a, b), {c, d), contains the lines CC and RR' , intersecting, 

 say, in N. The points L, M, N are however all in each of the spaces 

 [a, a'), {b, b'), (c, c'), and so in a line, the intersection of these 

 spaces. For instance the line A A' joins a point {A') of the line b, 

 to a point (A) of the line b', and so is in (6, b') ; and joins a point (A) 

 of the line c, to a point (A') of the line c', and so is in (c, c'); 

 thus L, on the line AA', is in the spaces {b, b'), (c, c'). But the 

 line PP' joins a point (P) of the line a to a point (P') of the line a'; 

 thus L is equally in the space {a, a'). Similarly both M and N 

 are in the line of intersection of the spaces {a, a'), {b, b') and (c, c'). 

 Thus the space {d, d') passes through the line of intersection of 

 the spaces {a, a'), (6, b'), (c, c'); for we similarly show that each of 

 L, M, N is in the space {d, d'). We denote this line by e; evidently 

 its relation to the lines a', b', c', d' is exactly similar with its relation 

 to the lines a, b, c, d; the plane, a, for example, defined as that 

 common to the spaces (6, c), (a, d), is equally the plane common 

 to the spaces {¥, c'), {a', d'); and so on. It is usual to speak of e 

 as the line associated with a, b, c, d; examination of the figure of 

 fifteen lines and fifteen points which we have constructed will show 

 that there is entire symmetry of mutual relation, and that we may 

 speak equally well of any one of the five lines a, b, c, d, e as being 

 associated with the other four; further e is also associated with 

 a', b', c', d'; and indeed, taking any line of the figure, the eight 

 lines of the figure which do not intersect it, consist of a set of 

 four skew lines and their transversals, and the line in question is 

 associated with either of these two sets of four. There are then 

 15'2 ^5 = 6 ways of regarding the figure as depending upon a set 

 of five associated lines. 



§ 3. Consider now what planes exist meeting the lines a, b, c, d. 

 In four dimensions an arbitrary plane does not meet an arbitrary 

 line; two such elements which meet lie in a threefold space. It 

 can be shown that a plane meeting a, b, c, d can be drawn through 

 two arbitrary points, one on each of any two of these four lines, 

 so that there are oo ^ such planes. Further that every such plane 

 also meets the associated line e. Further that two planes meeting 

 a, b, c, d can be drawn through an arbitrary point of the four 

 dimensional space, and, for instance, an infinity of such planes 

 can be drawn through any point of the Une e. Also, if the two 



