DERIVED FROM THE REGULAR POLYTOPES. 



29 



or — which conies to the same — by the condition that the 

 coordinates of A' satisfy the law stated in theorem I, that the 

 difference of any two different adjacent values must be unity. 



We now set to work and select for the /; -f- 1 vertices the ver- 

 tices A A , A 2 , . . ., A k+i of jS {i> (n -\- 1) and for A of fig. 3 the vertex 

 A v . According to this choice the coordinates of M are 



X^ X.y . . . X fc I ^ 



£+1 



) œ k + 2 



œ k + 3 



— ... — œ n . ! — U 



So the coordinates of the three points 0, A, M satisfy the equations 



c6.i 



Xo 



lb I. 



l °k+i » «*/.• + 



X i. 



X 



fc + 3 



œ 



H+l» 



but then these relations hold for any point of the plane OAM , as 

 the n — 2 equations represent a plane. As moreover 



AA' = MM'= (A — 1) OM, 



or in the notation of vector analysis 



A' — A = M' — M eee (A — 1) (M — 0), 



we have successively for the mentioned coordinates in true values 

 and for the mentioned differences of coordinates: 







M 



M — 



A'— A 



A 

 A' 



i 



n + l 



1 



A + l 



1 1 



k + l~~n + 1 



(*-«(ïTï-ïtl) 



1 



n + l 



1 

 fc + 1 



1 1 



k + 1~~ n + 1 







n + l 

 



1 



~ n + 1 



(a--i: 







n + 1 



-(*--«^+i 



i. e. the difference 



A 



of the coordinates x k + i and a? /c+2 of A' 



n ~r~ 1 



is either unity or zero. But if we make it zero we get A = 1 , 

 i.e. we find back the original S (i) (n -\~ 1). So for the expanded 



7\ 1 



poly tope we have to take - — : — - = 1 or ?\ = k-\- 2, giving for A' 



k -f- 1 



the coordinates 



