DERIVED PROM THE REGULAR POLYTOPES. 45 



in common; this is immediately evident, if for x x and #? 2 we take 

 in the first symbol the digits q x and 0, in the second r -\~ and 

 ■ — r -J- q ± = — ■ r -\- r. In general these common vertices define a 

 polytope of n — 2 dimensions situated in the space # n _ 2 for which 

 x x = q x , x. 2 — , i.e. the two polytopes are in contact with each 

 other by a limit (/) n _ 2 ; as an y common vertex of the two polytopes 

 lies at equal distances (radius of the circumscribed spherical space) 

 from the centres C and C, this common (/) n _ 2 lies in the space S n _ i 

 normally bisecting C Q C, i. e. this (/) n _ 2 of contact has the midpoint M 

 of C C for centre, i. e. the contact by the (/) H _ 2 is external. But in 

 one exceptional case, in the case q x = 2, q 2 = q 3 = . . . =q n =l 



n — 1 



of the central symmetric polytope (2 11 . . .10), the common limit 



(/) n _ 2 shrinks together into a single point, the midpoint of C C, 



as in that case (q 2 , q 3 , . . . , q n ) becomes a petrified syllable. At any 



rate, for r = q x the net is mixed, as the central polytope and one 



of it adjacent repetitions are not in contact by a limit (/)„_!• 



Tai 

 If for brevity we represent — —- by q the coordinates of C and 



C are 



6 . . . 



9 



q,q,q, ■ • 



• • q 



C . . . 



.r-\-q,- 



-r-\-q,q,q, . . 



.. q 



So, according to formula 1) of art. 1, the distance C C is equal 

 to the period r\ this result will be useful in the treatment of the 

 next case. 



Case r j£ q ± . Let us start from the case r = q x treated above and 

 vary r. As the relation C C=r holds always, this variation of r 

 implies a variation of C C, the effect of a translational motion 

 of the repetition (P) of the central polytope (P)° in the direction 

 C C if r increases, in the opposite direction C C if r decreases. 

 In the first case when C C is enlarged, the polytopes which were 

 either in (/)„_ 2 -contact or in point contact, will become free 

 from each other. In the second case when C C diminishes the 

 midpoint M of the new C C will lie inside both polytopes, i. e. 

 the polytopes will overlap. So the theorem is proved. 



As we cannot use overlapping polytopes we have to discard all 

 the cases r < q x , i. e. we have to consider q { as an inferior limit 

 of r. But if the net symbol — as we suppose — contains all the 

 vertices, there is also a superior limit. For in the case r = ^ -|- /■ 



