DERIVED EROM THE REGULAR ROLYTOPES. 35 



this vertex V to the centre O of the polytope. The system of the 

 spaces S n _ i corresponding in this way to the different vertices V 

 of (P) H include an other polytope (P)' n , the reciprocal polar of (P) n 

 with respect to a certain concentric spherical space, unless (P) n be 

 the cross polytope ce n _ i M a in which special case all the spaces 

 jS n _ i pass through the centre 0.' 



After the first section of this memoir had been published we 

 perceived that the analytical proof of the corresponding theorem XXII 

 might have been replaced by a much simpler geometrical one 1 ), 

 applicable to any polytope (P) n deduced from a regular polytope, 

 whether simplex or not, by the operations e k and c. 



This simple geometrical proof runs as follows: 



All the vertices V- t adjacent to V lie on two spherical spaces 

 (with oo" _1 points), the circumscribed one with centre O and an 

 other with centre V and radius VV- ( equal to the edge. So they 

 lie in the spherical space (with co"~ 2 points) common to these two 

 spherical spaces and therefore in the space S n _ 1 normal to VO 

 containing this intersection. If this S n _ i cuts VO in P we have 



2 VP. PO — W { from which it ensues that the distance PO is the 

 same for all the vertices V, i.e. that the spaces S tl _\ are the polar 

 spaces of the vertices V with respect to a definite spherical space 

 (with ex" -1 points) round O as centre. 



Moreover the special case of the cross polytope, where P coin- 

 cides with 0, is self evident. 



67. In the section concerned with the simplex we have explained 

 by the laws of reciprocity why it may happen that two different 

 groups of operations of expansion applied to the simplex produce 

 under circumstances either two poly topes equal and concentric 

 but of opposite orientation , or the same polytope. What corres- , 

 ponds to this here is that any polytope derived from M n can also 

 be derived from the cross polytope C 2 n of S n which is the reci- 

 procal polar of M n . As we had already occasion to remark in 

 art. 48 we shall have to come back to this assertion in the third 

 section . 



But the state of affairs with respect to equal measure polytope 

 nets with different expansion symbols is a quite different one. In 

 a joint paper of M rs . Stott and myself published two years ago 2 ) 

 it is shown geometrically that we have in general the relations: 



x ) To some of the free copies at my disposal I added a post-scriptum, containing 

 this remark, on page 69. 



2 ) Compare the second foot note of art. 38 of Section I. 



S* 



