DERIVED FROM THE UUGULAli POLYTÖPES. 69 



polytope in true value coordinates, this symbol also represents all 

 the limiting spaces ^ n-1 of the new polytope in space coordi- 

 nates, i. e. that these spaces S n _ i are represented by the equations 

 b ± œ ± + h n 2 + • - • + K+i œ n + ± = °> where ô , b l9 . . . , 6 n+i stands 

 for any permutation of the n -j- 1 digits a t . *) 



Finally it is easy to see in what manner the process of trun- 

 cation is transformed by inversion. As we have no intention of 

 studying the new system of semiregular polytopes for itself, it may 

 suffice here to remark that truncation at a limit (l) p , which implies 

 the determination of the intersection of a definite space S n _ ± with 

 the limits (/) p + 1 passing through that (/) p , is transformed into the 

 assumption of a point in the line joining the centre of a limit 

 (/) n _p_i of the new polytope to the centre O of that polytope, which 

 implies that this point is joined to all the limits (/) n _ p _ 2 °f that 

 (/) n _p_i by new limits (l) n _ p _ i replacing the chosen one, etc. 



37. We now prove the theorem: 



Theorem XXII. "Any polytope {P) n of simplex descent in /S lt has 

 the proporty that the vertices V { adjacent to any arbitrary vertex 



V lie in the same space S a _ x normal to the line joining that vertex 



V to the centre O of the polytope. The system of the spaces /S n-1 

 corresponding in this way to the different vertices V of {P) n include 

 an other polytope (P)' n , the reciprocal polar of (P) n with respect 

 to a certain spherical space with O as centre". 



In order to prove the theorem we consider the polytope with the 

 zero symbol (a i , a 2 , ... a n+i ) and in connection with it the linear 

 expression 



This expression assumes the value 



a\ -j- a\ -\- . . . -\- a 9 



H + l 



for the pattern vertex V and the same value diminished by unity 

 for each of the points V i adjacent to V. For we pass from the 

 pattern vertex V to any vertex V i adjacent to it by making two 

 digits p and p — 1 interchange places and by this process the sum 



n + l 



P 1 ~\~ (P — I) 2 contained in Sa^ is replaced by p{p — Y)-\-{p — \)p 



i 



= 2p 2 — 2p. So the coordinates of the points V t adjacent to the 



l ) Compare "Nieuiv Archief voor Wiskunde", vol IX, p. 138—141. 



