( 496 ) 



Spn—2 of Z)„— 1 in such a way in turns white and black, that two 

 opposite bounding spaces Spn-i have a different colour. The octa- 

 hedron is really the only one of the regular bodies that allows this 

 operation. ^) 



9. If the simplex *S/, expands from a point to an >S',, (n|/2) and 

 at the same time S'n contracts from an ;S"„(wl^2) to a point, then 

 Sn lies at the beginning of the process within S'„ and at the end 

 inversely S'n lies within Sn- Gradually first the vertices, then the edges, 

 then the faces, etc. of Sn have passed outward. We shall now in- 

 vestigate when that takes place. 



From the diagrams of the expanding plate given in the first part 

 it is evident, that the section of Mn with a space Spn—i changes its 

 nature when the point of intersection P of that space Spn—i with 

 the diagonal A A' passes one of the 7i — J points of division 

 Ai, A^, ... As the nature of the section of course also changes when 

 bounding elements of S'n lying inside Sn pass outward, the latter 

 must take place at those moments when those points of division of 

 the diagonal AA' of the moving J/„ pass through the fixed space 

 Spn—\ . This theorem then really holds : 



"In the translation of Mn in the direction of A A' through the 

 space Spn—\ in succession the vertices, the edges, the faces, bounding 

 bodies, etc. of Sn come entirely outside S'n at those moments that 

 the point of intersection P of the diagonal AA' with the space 

 Spn—\ coincides successively with the points of division .4^, A^, A^, 

 A„ etc." 



We regard — in order to prove this theorem — the arbitrary 

 stadium of the simplexes Sn{APy2n) and S'n{A' P\/1n), divide 

 the n vertices of Sn in an arbitrary way into two groups /? and y 

 of p and n — p points, and indicate by /i' and ;' the groups of the 

 p and ?2— j9 corresponding vertices of S'n, by B, C, B' , C' (fig. 2) 

 the centres of gravity of the point-groups ^, y, /i', y' — i.e. the 



C' B P B' C 



— — • » ) 1 — ♦ » 



Fig. 2. 



1) In contradiction to this seems that for n = 5 through each 'edge three faces 

 pass and thus three bounding bodies (12, 18, 8) lie around it. This contradiction 

 however is only apparent; it is annulled by the remark that two bounding bodies 

 (12, 18, 8) having a face in common agree or differ in colour according to the 

 face being triangular or hexagonal. Of the tliree faces one is triangular, two are 

 hexagonal; the bounding bodies to which the two hexagonal faces belong, differ 

 in colour from the two others, these agreeing in colour. 



