86 ANALYTICAL TREATMENT OF THE POLITOPES REGULARLY 



For the deviation indicated we have two reasons. The first 

 is of a didactic cast: it is easier to imagine the motions of the 

 limits of M} 2) than those of HMftV*\ But the second is of 

 more importance: "if the limits of HM^V^) are carried away by 

 the limits of the circumscribed Mjj® which contain them, these 

 latter limits being moved out in the ordinary way, we get precisely 

 those expansion operations which lead to the whole set of poly- 

 topes hmpd. of S n " This advantage is twofold. In the first place: 

 the only expansion of HMJ^Vfy which has no equivalent under 

 the e,. applied to M n {2 \ i. e. the expansion according to the faces, 

 is excluded, and this is right, for we will show afterwards that 

 this expansion is either impossible or it leads to a polytope which 

 can be derived from J/ n (4) . But, what is still more, by adhering 

 to the limits of J/ n (2) we are never at a loss with respect to the 

 question to which group of limits of HM^V^) the expansion is 

 to be applied. So in the case of S 5 the MM 5 admits as limiting 

 bodies tetrahedra only, but they are of tw r o different kinds, i. e. 

 we must distinguish between a T common to two C iQ and a T 

 common to a cell C 16 and a cell Cr 5 ; so, of these two groups the first 

 must undergo the operation <? 3 , if we wish to apply it, as a ^common 

 to two (7 16 is contained in the cube common to the two adjacent 

 eightcells bearing the two C l6 . Moreover we will prove afterwards 

 that the contraction operation always leads to forms deducible from 

 3/ n (4) ; so we have to consider here the operations e k only. 



On the other hand we do not deny that the new definition has 

 a drawback with respect to the operation of expansion according 

 to the edges of M n (V^ a difference in the notation making its 

 appearance there. According to HM n itself this operation ought to 

 be called <>, v HM n \ nevertheless we propose to indicate it by the 

 symbol e 2 HM n . This is still more annoying in S 3 and /S 4 where we 

 have e 2 HM% = e { T = tT and e 2 HM k — e ± C m (see the small table 

 at the end of art. 91). But still we reckon the advantages so pre- 

 vailing that we do not mind of accepting this small disadvantage 

 into the bargain, the more so as it is easily held under control. 



Starting from the new definition we prove: 



Theorem LIX. "The expansion e k ,(k=2,3, . . . , n — 2), applied 



n 



to lLM^VV) changes the symbol of coordinates -£-[11. . 1] of that 



n — /»• 



polytope into -J- [88. .8 11. .1]." 



Proof. If we move the limit HM k &V%) represented by 



cVi = œ, = . . . = x n _ k = 1 , c*? n _/, + 1 , x n _ k+2 , . . . , œ n = i [11 . .. . 1] 



