DERIVED FROM THE REGULAR POLYTOPES. 105 



hmpd. be the cross polytope HM^ of # 4 in which case all the 

 spaces /S 3 pass through the centre." 



The simple geometrical proof of this theorem can be copied from 

 that of theorem XL (see art. 66). 



110. We have to add a single word about the reciprocation of 

 the hmpd. nets. The results obtained here run parallel to those of 

 art. 68. 



In general the system of vertices found by polarizing an hmpd. 

 net is the combination of several groups of limits M k (2p) of the 

 measure poly topes of the net JV(M n (2p) ), p being the period. These 

 groups are formed by the centres of the constituents B , C, A, 

 A^,. . ., A^- % \ i.e. 



n 



the even vertices of N(M, t ®fy s represented by \%p&± , . . . , %pa u ], Z«i even, 



l 

 n 



n odd n ii rt , // // // , X«i odd, 



l 

 // centres of the M n of N(M H W), 



if n h n limiting M„ _ i of the M n of N(M,S 2 i )V ), 



ii ii it ii n Mn — 2 '/ " " " " , 



or 



B \ 



// 



a 



// 



A 



// 



AM 



// 



A&) 



// 



^(»-3) 



II II II J/g // // // 



In the case of the net NH n itself only the first and the third 

 group are present; so in 8^ we find then the net N(C 2i ). In all 

 other cases we have to deal with at least three groups, the first 

 three. As we already remarked in art. 68 an other paper, also 

 destined to complement art. 39, will contain more ample develop- 

 ments about these reciprocal nets. 



G. 8ymmetry , considerations of the theory of groups , regularity. 



111. We first determine the spaces of symmetry 8y n _ ± of HM n 

 itself and afterwards those of any hmpd. derived from it. 



Case of HM n . — We have to investigate here how the reasoning 

 which led us to the spaces of symmetry of the measure polytope 

 is affected by the alternate truncation. 



In the case of M n we found two possibilities under which the 

 space 8 n _ ± bisecting orthogonally the join A x A 2 of two vertices 

 A i9 A 2 is a space 8y n _ i of the polytope, i.e. that A ± A 2 is either 

 an edge or the diagonal of a face; in the first case we got the n 

 spaces cT l = , in the second the n(n — 1) spaces x i + x k = 0. Now 

 on the one hand it is immediately evident that the alternate trun- 



