( 62 ) 



We indicate the i)rojet'tioii of the regiUai' *S'(5) obtained in fig. 1 

 as the "pentagonal projection" of that poiytope, and we try to show 

 in tlie following pages how easily the corresponding projections of 



has parallel projections on either of the two planes, i. e. that the five lines at 

 infinity cutting these pairs of non intersecting edges have the lines at infinity of 

 the two planes of projection for common transversals. 



Now there are altogether fifteen pairs of non intersecting edges and therefore 

 also fifteen lines at infinity each of which cuts a pair of non intersecting edges. 

 Moreover it can he shown easily that these tilleen line.~ at infin'ty lie on a cuhic 

 surface. For, in barycentric coordinates with respect to the regular fivecell as sim- 



.') 

 plex of coordinates these fifteen lines at infinity, for which the relation S Xi = 1 



.") 

 changes into S xi = 0, are represented by the equations 

 1=1 



Xi + Xk = 0, XI + Xm = 0, Xn — 0, 



where i,k,l,m,n stands for any permutation of 1,2,3,4,5, and these relations 



.") 

 satisfy the equation s .r,''^ = of the diagonal surface of Clebsch. So the Schlafli 



double six completing the fifteen lines mentioned above lo the 27 lines of that 



.5 



surface S Xi^=^0 consists of the lines at infinity of six [)airs of planes O(A^iA'o) 



and 0{X:iX^) corresponding to the six pairs of circular permutations 

 (12345) I (l^Xvi) j (124:5:)) j (12453) j (12534) J (12513) j 



(13524) i ' (I3'.25) i ' (14523) \ ' (14325) ( ' (15423) j ' (15324) ( 



with the property that in each pair any digit has in the two constituents diffeient 

 adjacent digits. Each of these six pairs consists of two reciprocal polars with 



respect to the sphere £ Xr = at infinity common to all the spherical spaces of 

 i—\ 



Si, as the two planes of each pair are perfectly normal to each other. According 

 to a known property, found for the first time by F. Schur, the six pairs of lines 

 of a Schlafli double six are really always reciprocal polars with respect to a 

 quadratic surface (compare Th. Reve "Beziehungen der allgemeinen Fliiche drifter 

 Ordnung zu einer covarianten Flache dritter Glasse", Math. Annalen, vol. 55, p. 257, 

 and G. Kohn "Ueber einige Eigenschaften der allgemeinen Flache dritter Ordnung", 

 Wiener Sitziingsberichte, vol. 117, p. 66). 



If we deduce in the ordinary way the projection OCA'oA'g) iVom the piojections 

 OiXiXo), OtX^Xi) after having rotated each of the two regular pentagons over an 

 arbitrary angle, we obtain the projection of the fivecell on any plane the line at 

 infinity of which cuts the lines at infinity of OiXiX^) , 0(X:,Xi). This shows that the 

 projection on an arbitrary plane can only be got in two tempos, i. e. by passing 

 first to two arbitrary projections 0{X:,Xs} , 0{XiXi) and by deducing a new pro- 

 jection O(XiXo) after having rotated each of the projections 0(XoA'';5), OiX^X-^) 

 over an arbitrary angle. Or otherwise: if 1,1' are the lines at infinity of the planes 

 OiXiXo) , 0{X-iXi) and m, m tliose of an other pair of planes perfectly normal to 

 each other, there are always two real lines n, n' intersecting I, I', m, m' and repre- 



