of lines by projection from four dimensions 139 



plane meets a, b, c, d, e, the second plane coincides with the first. 

 It is not necessary for our purpose to prove this, 



§ 4. The theorem that two planes can be drawn from an 

 arbitrary point to meet the lines a, b, c, d is obvious from the 

 theorem in three dimensions that four skew lines have two trans- 

 versals, the proof of which also depends on the fact that two homo- 

 graphic ranges on a line have two common points. For, if we project 

 a, b, c, d from 0, on to an arbitrary threefold space E, the planes 

 joining to the two transversals of the four lines of S so ob- 

 tained, all meet a, b, c, d. And, we now see, e projects into a fifth line 

 meeting these two transversals. When is on e, the projections 

 in 2 of a, b, c, d are all met by the projections in S of a', b', c', d' ; 

 for the plane Od' , for example, meets a, b, c, and meets d because 

 e, d, d' are in the same three dimensional space; thus the pro- 

 jections in S of a, b, c, d are four generators of the same system 

 of a quadric surface of which the projections of a', b', c', d' are 

 generators of the other system. The planes from each meeting 

 a, b, c, d intersect the space S in lines all meeting the projections 

 of a, b, c, d; that is, in lines which are generators of this quadric 

 of the same system as a', b', c', d'. The planes from meeting 

 a', b', c', d' similarly give rise to generators of the system (a, b, c, d). 

 Thus any plane from the point meeting a, b, c, d meets any 

 plane from drawn to meet a', b', c', d' in a line through 0; and 

 every line drawn from in a plane of the former system is the 

 intersection with this plane of a plane of the second system. If 

 be a point of e lying on a plane drawn from a point H, not on e, 

 to meet a, b, c, d (which therefore also meets e), the line HO lies 

 in a definite plane meeting a', b' , c' , d' . Thus either of the two planes 

 of the first system, those meeting a, b, c, d, drawn from a point H, 

 not on e, meets one of the two planes of the second system, those 

 meeting a', b' , c' , d' , drawn from H, in a line; and the two lines so 

 arising intersect the line e. 



§ 5. Hence we can obtain from the four dimensional figure a 

 figure in three dimensions with the characteristics of that used in 

 proving the double six theorem. 



If, in the four dimensions, p, a be the planes drawn from an 

 arbitrary point to meet a' , b' , c' , d' , and p , a' those meeting 

 a, 6, c, d, and if p and a' meet in a line, as also p and a; and if 

 we consider the intersections with an arbitrary threefold space S, 

 of these four planes, and also of the planes joining to a, b, c, d, 

 a', b', c', d' , denoting these twelve lines respectively by (p), ...,{a), ..., 

 then, arranged as follows: 



{a) (b) (c) (d) (p) (a) 



{a') (6') (c') id') ip') {a'), 



