(492) 



now imagine this rhombotope, for the special case that the projection 

 of the intersecting space >S'/^_i — so also the projection of the 

 rhombotope itself — falls along OA and let iis truncate it by the two 

 spaces Spn—2 standing normal to the plane of projection in the ends 

 A, A' of the segment AA' of that projection lying inside the rectangle 

 and cutting the axis of the rhombotope therefore at right angles ; we then 

 find the required section, to be indicated according to the number of 

 its dimensions by i)„ [ . We directly determine the length of the axis 

 of the untruncated rhombotope and of Dn-\ , but before this we 

 shall deduce some general theorems easy to find. 



5. The edges of Mn project themselves on the assumed plane either 

 along one of the n lines FQ, P,Q,, P^Q^, . . ■ Pn-^Qn-i , P' Q' , or 

 as parts of PP' or QQ' . Because the vertices of Z)„_i must be 

 vertices of Mn or points of intersection with edges of i/„, these 

 points project themselves — compare fig. 1 for n^S and for 

 72 = 9 — for even n exclusively in the ends A, A', for odd 7z 

 exclusively in those ends and in the centre 0. 



From this ensues for ?z = 2n' the general theorem : 



"The section B^n'-i of ^hn' is a 2/?/ — i-dimensional prismoid 

 with respect to each pair of opposite bounding spaces Sp-m'—i «'^nd 

 so in 2n' ways". 



Here follow two theorems holding for arbitrary n : 



"Each line through the centre normal to two opposite bounding 

 spaces Spn-i is axis of D„-\ with the period n — 1." 



"Each space Spn~2 through parallel to a bounding speice Sp, 1—2 

 divides A.— 1 into two congruent 11 — 1-dimensional prismoids." 



In the demonstration of these three theorems the entire equivalence 

 of a pair of opposite bounding spaces Spn—2 with any other pair 

 has the chief part; moreover the third causes us to inquire how 

 the space >S/?«_2 through the centre parallel to a bounding space 

 intersects 2)„— i- We prove as follows that this section is a Z)„_o. 



If the projection / of the intersecting space Spn—\ revolves round 



0, the /S/^i-2 normal to the plane of projection in remains in 



its place and Spn-\ thus describes a pencil with this Spn--» as axial 

 space. Therefore then the varying section keeps going through the 



section of Sp^il^ with M„ . We can easily know the nature of this 

 section of n — 2 dimensions by regarding the case in which /coincides 

 with /o- Then our Dn-\ is an M„-\ and this measure-polytope 

 projecting itself along /„ is intersected according to a Dn-2 by the 

 space /S/i-o, which is in normal to the plane of projection and 



