( 693 ) 



into I*R, which nieaiis a decrease of QS = QP' , and this is 



5 



repeated every time a cell is taken further to tlie I'ight. If we exchange 



the central cell bj an other one of which the projection PoP\QiQo 



covers for three fourths that of the central one, then Öó' passes into 



1 

 Q^R', again a decrease ot -, and this too is repeated every time 



5 



the projection moves onward in the direction PP' to an amonnt of 



PP^. So here too we find five different symbols jjM^, of which 



the fractions graduallv increase with - . With the aid of the above 



5 



table this result of the notation pM^ can be transformed into that 

 of the rhombotope symbols. 



We have now answered the question put at the commencement. 

 If we wish to fill Sp^ with C's and a single other groundform, then the 

 form (10, 30, 30, 10) with the same length of edges can do service ; 

 both forms appear then in two oppositely orientated positions. If by 

 the side of C\ ^ve allow two other groundforms to fill Sp^ , we can make 

 use of the forms (20, 40, 30, 10) and (30, 60, 40, 10) of the same 

 length of edges; if we take into consideration difference in orien- 

 tation, then this sj)ace-filling demands five forms. And if one does 

 not object to connecting more than two really different groundforms 

 we can take the five forms 



1 5 9 13 17 



20 ' 20 ' 20 * 20 ' 20 ' 



^TqJ' Vie'TeJ' (ïö'ïej' (lë'ïöj' (ïë'^ 



of which the first is a 6V*^ -); these appear in only one position. 



4. Before passing on to the general case of Spn we indicate the 

 shortest way, by which one can calculate the number of component 

 parts when filling a fourdimensional bloclv of one of the found forms 

 but of ;(-times larger linear dimension. To }n-epare the general case 

 of an arbitrary n we introduce a simpler notation. We distinguish 

 the transition fV)rms and the intermediary forms by the letters 7' and / 

 and then indicate by exponent — this, to avoid i-ootsigns, in K2 as 

 new unit — the size, by a footindex the place of the section. We 

 then represent the polytope, formed by truncating regularly a regular 

 fivecell with a length of edges 7^1/2 at the five corners to the frac- 

 tion q of the edge by the symbol qS^^'X Thus each of the five dif« 

 fercut forms is i-epresentcd by four different signs as follows: 



1. e. 



