106 ANALYTICAL TREATMENT OF THE TOLYTOTES IIEGULAHLY 



cation behaves itself differently with respect to these two groups 

 of spaces: it destroys the symmetry property of the first and pre- 

 serves that of the second. But on the other hand we have to 

 examine whether the alternate truncation does not enervate the 

 force of the argument by means of which we excluded the cases 

 that A ± A 1 was a diagonal of a limiting M k of the M n for k >• 2, 

 i. e. that the projections of the two regular simplexes S(k) of the 

 vertices of M k adjacent to A x and to A 2 on the space normal to 

 ./, A 2 are of opposite orientation. Indeed this argumentation has 

 to be revised, as the two simplexes 8(k) disappear altogether by 

 applying the truncation and are replaced as groups of vertices of 

 HM k adjacent to A x and to A 2 by the two sets of -^ k {Je — 1) ver- 

 tices of 31,. lying in the following layers jS n _ ± normal to A i A 2 . 

 But the two poly topes 1 ) determined by these groups of vertices are 

 neither central symmetric and maintain the property of the diffe- 

 rently orientated projections, unless they coincide in the space S rir _ i 

 normally bisecting A ± A 2 for k = 4. So any space orthogonally bisec- 

 ting a diagonal of a limiting sixteencell of HM n is an Sy n _ i and 

 therefore HM n also admits two groups of spaces Sy n _ u the spaces 

 ®i i x k — and the spaces x ( + %k i œ i i x m == 0* The number 

 of the former is always n(n — 1), whilst that of the latter is 

 ± n (n — 1) (n — 2) {n — 3) for n > 4 and four for ^ = 4. 



Case of the limpd. derived from HM n . — From the structure 

 of the hmpd. it is immediately evident that a space S n _ i is an 

 <%n_i for an hmpd. if and only if it is an Sy n __ x for the HM n from 

 which the hmpd. has been dirived. So we have proved the 



Theorem LXX. "Auy hmpd. of S n admits two groups of 

 spaces Sy n _ i3 viz. the n (n — 1) spaces cc t + x k = and the 

 \ n (n — 1) 0—2) {n — 3) spaces x { +_ x k + œ L + œ m = 0". 



112. From theorem XLIII we deduce: 



Theorem LXXI. "The order of the group of anallagmatic 

 displacements of HM n and of the hmpd. derived from it is 2 n ~ 2 n\ 

 for n > 4". 



'The order of the extended group of anallagmatic displacements 

 of these polytopes, reflexions with respect to spaces Sy n _ x included. 

 is :2" ' n\. In this extended group the first group of order 2 n_1 n\ 

 forms a perfect subgroup". 



l ) Compare for these polytoj-.es: "The sections of the measure polytope M n of space 

 Sp n with a central space Sp n _ i perpendicular to a diagonal", Proceedings of Auister 

 dam, vol. X, p. 4 ( J.">. 



