PROCEEDINGS 



OF THE 



Camk&g^ ^]^Hai50p]^kaI Somtg. 



On a configuration of twenty-seven hyper-planes in four- 

 dimensional space. By Professor W. Burnside, F.R.S. 



[Beceived 23 January 1909.] 

 [Bead 9 February 1909.] 



Clifford has shewn the existence of a plane figure con- 

 sisting of 2** circles and 2** points such that each circle passes 

 through n + 1 of the points and each point lies on n-\-l of the 

 circles*. If this figure be inverted with respect to a point outside 

 its plane, the circles become plane sections of the sphere into 

 which the original plane inverts. The configuration may then be 

 specified as one of 2'* planes and 2** points, such that in each plane 

 lie n + 1 of the points and through each point pass w + 1 of the 

 planes. The set of points is, however, restricted to lie on a sphere, 

 or, if the configuration is modified by a projective transformation, 

 on a quadric. 



That such a configuration exists, apart from the restriction of 

 the points to lie on a quadric, is true ; and I believe a proof of 

 the fact has been published though I cannot give a reference 

 to it. 



In Mr Grace's memoir a proof is given of the existence of some 

 very remarkable configurations of spheres and points in three- 

 dimensional space (pp. 182 — 188). If one of these, regarded as 

 existing in four-dimensional space, be inverted with respect to 

 a point, not in the three-dimensional space of the configuration, 



* Clifford, Collected Papers, pp. 51, 52. See also J. H. Grace, " On circles, 

 spheres and linear complexes" (Camb. Phil. Trans. Vol. xvi. pp. 153-190), with 

 which memoir this note is more directly connected. 



VOL. XV. PT. II. 6 



