DERIVED FROM THE REGULAR POLYTOPES. 39 



Let (P') b n and {P')\ represent farther more the two x ) poly topes con- 

 gruent to (Pf a and concentric to {F) a n which admit of a zero symbol 

 with respect to the simplex of coordinates. Then we have only to 

 prove that either (P')n or (^7* is equipollent to ÇPf n . Now from 

 the fact that (P)l and (Pf n have (P) n _ 4 in common it follows that 

 (P) b n and {P") b n must admit a set of limits congruent and therefore 

 of the same or of complementary import with (P) n _ 4 of (P)£; so 

 one of these limits of (P')\ — say (P'), t _ 4 — and one of these 

 limits of (P'X — say (P")n_ 4 — must lie in spaces S' )l _ 1 and S" n _ i 

 parallel to /S^i and on both sides at the same distance from the 

 centre O of (P) a n . Of these spaces S' n _ t and #" n _ 4 let S' n _ x be 

 that one which lies on opposite sides with respect to O with /S^'iV 

 Then it will be possible to bring {P') b n into coincidence with {P) b 

 by means of a translational motion; for, if by such a motion the 

 limit (P') w _ 4 is brought into coincidence with (P) n _ 4 , the polytopes 

 will coincide, as this is the case not only with the limits mentioned 

 but also with their centres. So (P) b ( is orientated with respect to 

 the simplex of coordinates. 



From the lemma to the theorem in view we have onlv to take 

 one step more. The lemma immediately shows that, if the net in 

 S n consists exclusively of polytopes derived from the simplex, all 

 the polytopes are orientated with respect to the simplex of coor- 

 dinates , as soon as this is the case with one of them ; for we can 

 always consider any two polytopes of the net as the first and the 

 last of a series of polytopes any two adjacent ones of which are 

 in n — 1 -dimensional contact. So with respect to the simplex from 

 which the central polytope (P)° has been derived all the polytopes 

 of the net are orientated. But this includes that by passing from 

 any vertex of the net to an adjacent one the coordinates chnnge 

 by integers and as we can reach any vertex of the net by means 

 of a set of these motions — starting from a determinate vertex of 

 (P)° — the coordinates of any vertex of the net must be integers. 



So we have shown now that the property of admitting vertices 

 with integer coordinates only belongs to all the nets , the polytopes 

 of which are exclusively of simplex extraction. This very general 

 result brings us in contact with the two following questions : 



a). Can the result be expressed by saying that any net with 

 the assigned property of its constituents admits a net symbol with 

 integer coordinates only? 



This question must be answered negatively. We cannot pass to 



*) In the particular case of' a central symmetric (P), t these two positions coincide, etc. 



