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



also homographic with (Pj). Thence the ranges {P'), (P), on d, are 

 homographic ; and, if not coincident, they will have two common 

 points, which may coalesce. When P has a position in which it 

 coincides with P', there is a single plane containing 0, Pj, P2, P3, P, 

 where P3 is the point of c on the plane OP, P' . Thus through the 



point can be drawn, either an 

 infinity of planes meeting all of 

 a, b, c, d, or else two, which may 

 however coincide. 



When is on the line e, the 

 plane Od' meets a, b, c, and it 

 also meets d because, as we have 

 shown, e, d, d' are in a three 

 dimensioned space. Equally 

 the planes Oa', Ob', Oc' meet 

 a, b, c, d. As there are thus 

 more than two planes through 

 meeting a, b, c, d, it follows, 

 by what we have shown, that 

 there is an infinity; this is 

 when is anywhere on the line e. The aggregate of planes so 

 obtained, by taking to be every point of e, is identical with the 

 aggregate of all planes meeting a, b, c, d, namely any plane meeting 

 a, b, c, d can be identified with one of these; for taking on e, 

 and P on d, this P determines Pj, P^, respectively on a, b, when 

 regarded as belonging to one of the coincident ranges on e, and 

 determines Pj, Pg, respectively on a, c, when regarded as belonging 

 to the other range on e. Thus every plane meeting a, b, c, d also 

 meets e, or more generally five associated lines are such that every 

 plane meeting four of them also meets thefifth"^. 



In general, as we have seen, from any point on a plane meeting 

 a, b, c, d (and e), there can be drawn another such plane. If the 

 point be on the conic through the five points in which the first 



* The reader may compare the proofs of this result given by vSegre, Circolo 

 Mat., Palermo, n, 1888, 45, Alcune considerazioni....The elementary theorems 

 here given for the geometry of four dimensions are of course well known; but I 

 have thought that it was necessary for the purpose of this Note to supply demon- 

 strations. The reader may consult Bertini, Introduzione alia geometria projettiva 

 degli iperspazi, Pisa, 1907, a volume of 400 pages, p. 177. In English there is 

 Mr Richmond's paper On the figure of six points in four dimensions, Quart. Journ., 

 XXXI, 1899; Math. Annal., Lm, 1900 (see also Trans. Camb. Phil. Soc, xv, 1894, 

 267), which deals with a diagram intimately related with that of the text, and 

 CooMdge, A treatise on the Circle and Sphere, Oxford, 1916, p. 482, etc., where the 

 lines of four dimensions are replaced by spheres. The origin of the five associated 

 lines seems to be a result given by Stephanos, Compt. rendus, xcm, 1881, p. 578. 

 I have not seen it formally remarked that the property of the double six follows 

 from the geometry of four dimensions ; indeed the argument given in § 1 was invented 

 in ignorance of this. The fifteen points and lines of our figure (Fig. 2) are the 

 diagonal points and transversal lines of the figure considered by Mr Richmond. 

 See also Hudson, Kummer's Quartic Surface (1905), Chap. xii. 



