( 697 ) 



If we resli-ict ourselves to these tonus and if again we do not take 

 the transition form consisting of a sijigle vertex into consideration, 

 we have in both cases to deal with n dilferent forms, namely for n 



even with - n transition and - n intermediary forms, for n odd 



2 2 -^ 



with - {n — 1) transition and - {n -j- 1) intermediary forms. Thus we 



get again in Spn—x two more or less regular s[)ace-tillings in which 

 the regular simplex of that space shares. 



In connection with the symbols ^S^^' the relations hold here 



(1) (3) (I) 



h ^= I\ — (n)i I\ , 



(2) (4) (2) 



T^ =^ I\ — (n)i /i , 



(1) (5) (3) (1) 



Jo = A - (n)i li + (n)2 ii , 



(2) (6) (4) (2) 



which leads 

 for n even to 



(•2) ill) v"— 2) ('i— 4) (kn—\) 



for n odd to 



(I) («) hi—l) n— 4 K«— 1} 



^è(n+l)= ^1 — (w)l^l +('02^1 —•••• + (—1) (w)K«-1)^1> 



whilst the ratios of volume are determined by 



CO (2) n) (2) (1) 



il ^Ti ^ l2 ^ T, /3 



1 2"-' ~ 3"-» - (?i)j ~ 4"-i-(n),2"-i "" 5"-'- (/i)j3" i +7T, ~ ^^^' 

 Farthermore the formulae of reduction hold : 



(2^-) (2) (2) (2) 1 



I, =:(X'-Hn-2),._iTi + (/:+^*-3)„-i7r + .... + (/:)„_! 7'_; 

 f2i-+i) (1) (1) ml' • '^^^ 



which enable us to calculate the number of the parts of different 

 kinds, into which a block of {2kY or {2k -\- 1)" measure-poly topes 



(2) 



AIn can be cut up. 



As an example, which gives something to calculate, we consider 

 the case of the middle section perpendicular to the diagonal of a 



10 (2) 



block of 10 measure-poly topes i/ 10. We then find in connection 

 with the relations 



