( 426 ) 



moreover by a.i the unit corresponding to the polytope itself. So, 

 if the polytope belongs to the group of the EuLERian polytopes, 

 what we suppose to be the case, the relation holds 



what we express by saying that the EuLERian form A, i.e. the left 

 hand side of the equation mentioned, is equal to unity. 



For the different constituents (P)„,, {P)„^, ... , {P)n^^ of the prismo- 

 tope, all of them supposed to belong to the EuLERian group, we 

 introduce for n and .4 different letters a, />,..., /> and A,B,...,F. 



3. We now prove the following lemma: 



"The number of limiting elements {J\ of 7 dimensions of the 

 prismotope (P,,^ ; I\; • • • ; Pv) is equal to the sum of the terms out 

 of the product AB . . . P of the EuLERian forms of the constituents, 

 for which the sura of the subscripts is equal to </, with the j)Ositive 

 or the negative sign according to q being even or odd.". 



Example. In the case of three pentagons as constituents we find 

 by developing {ci,—a,-^a,){h^—h,-\-h;){(',—c,-^c,) where 



a, = ^'o = t'o = a, = ij = i\ = 5, a, = h.^ = c, = 1, 



if (/o' ?i' • • • i'öfer to the prismotope, 



q^ = 125, q, = 375, q^ = 450, q, = 27b q, = HU, q^ =: 15, q, = 1 



and therefore a sixdimensional polytope with the symbol 

 (125, 375, 450, 275, 90, J 5) 



of characteristic numbers; this symbol satisfies the law of Euler. 



Proof of the lemma. Let us represent the prismotope {P,,^ ; P„„ ; ... ; P„^) 

 under consideration by Ft for short and let pt = {a,^ ; ||,., ; . . . ; ^t^^ 

 represent a new prismotope, the constituents of which are definite 

 limiting elements «,, of {P)„,,.ii^, of (P)„, , . . . , rr^^^ of (P)„^, where 

 it is allowed to take vertices for some of these limiting elements in 

 which case the corresponding dimension number si is zero and the cor- 

 responding {P„.) inactive in the formation of pt. Then pt will be a 



p 

 limit {l)g of Pt under the condition ^s; = q. Reversely each element 



{l),j of Pt can be generated in this manner. So the number of limits 

 {l)q of Pt is equal to the number of the different ways in which 

 we can gather a set of limits a., , i^,, , •••, ^s , the numbers 5/ satis- 



p 



fying the condition ^.Si = q, which last number is represented evidently 



