DERIVED FROM THE REGULAR POLYTOPES. 5 5 



is already excluded (art. 26) — this remark excludes overlapping- 

 of any two poly topes, as we can derive from it that not a single 

 vertex can lie inside any polytope of any group of constituents. 

 Case r=l. In this case the enumeration of the zero symbols 



n n — 1 p n — p + 1 n 



(1 00 . . 0), (11 00 . . 0), . . ., (11 . . 1 00 . . 0), . . ., (11 . . 1 0) 



of the n groups of constituents is much simpler. Moreover the 

 polytopes of the first group and those of the last admit only one 

 kind of limits (/) n _i viz. simplexes, those of any other group only 



two, limits (/) n _! with respect to (1 00 . . 0) of the lowest and 

 of the highest import. 



Here overlapping is also excluded, as can be shown by means 

 of the same remark used above. Here the polytope (P) a can be 

 represented by 



fa -fï, «2 + 1» > '*p+-I> <V+i+Q, , fl» + i + 0), 



its limit (l)\"-'\ lying in the space S n _ i with the equation a» 1 =« 1 -|-l by 



K+ 1) K + l» > a p -\-\, «p + i + 0, , a„ + 1 + 0), 



the second polytope (P) b of the list containing also this limit by 

 («i + 1+0, a 2 -\-\, , a p ^-l,a p + i~\-0 } ,^ + i + 0). 



For the rest the proof can be copied from that given above. 



Now that theorem XV has been proved we go back to the 

 polytopes (P), t and (P) b in contact with each other by a common 

 limit (/) u _i in order to indicate a relation between the import of 

 that common limit with respect to (P) (/ and (P) b on one hand and 

 the places of the groups of constituents, to which (P) a and (P) b belong, 

 in the list of polytopes of the general case r > 1 on the other. 

 To that end we indicate by G it G 2 , . . . , G n \ G n + i successively the 

 kinds of polytopes represented by the first, the second, . . . the 

 last but one, the last line of the list of polytopes and — as on 

 page 17 — by the symbols g , ^i,y 2 , . • • >y n — i in relation to any n- 

 dimensional polytope limits (/) n _! of vertex import, edge import, 

 face import, . . . , the highest import of that polytope. Then we find: 



Theorem XVI. "If two polytopes of the net, (P) a of group G k 

 and (P) b of group G k _ v , are in (/)n-i contact, the common limit 

 is a g v _, for (P) a and a g n _ v for (P) b ". 



The proof of this theorem lies in the remark that (P) ö , according to 

 the subscript {a ± -)- 1 , a 2 -f- 1 , . . . , a v -f- 1, a v+i> a v+2 , . . . , a n + 4 ) 



