( iil ) 



I)}- 2£<(^^ A,., . . . />., lor 2£si = (j, iuul iii its lui']i tliis expression represents 



' /=;; 



the absolute value of the number indicated in the statement of the 

 lemma, Avhere it is atfeeted by the sign ( — 1)7. 



4. If we bring tlie poljtope {P)„ with the symbol (a,, a^, a„_, ..., *7„_i) 

 of eharactei'istic nnmbers in relation with the polynominm 



rt„ + n^.v + a„.^•'' + . . . + a„_M-"-i + .v" 

 which may be called the "EuLERian function" of the polytope, the 

 connection between the characteristic numbers of the prismotope and 

 those of its constituents can be expressed Ncry simply by : 



Theorem I. "The EuLKRian function of a prismotoi)e is the product 

 of the EuLERian functions of its constituents". 



Corollaries. "The Eii,ERiau function of I he simplex AS^n-J-T; of ^S„ is 

 (n + l), + (n f-1), X + ('^ + 1)3 '^"- + ... + (» + !), .v;"-i + .7;", 



for which can be written - - |(.t'-|-l)"+' — Ij." 



,1; 



"The EiLERian function of the simplotope (compare p. 45 of vol. II 



ot my textbook "Mehrdimensionale Geometrie", Leipsic, Göschen, 



1905) with the constituents S {ui -f 1), {1 = i,'2,. . . , p) is represented by 



1 {(,;+l)'.+ l _ J I |(..+ 1)-+' _lj..J(,,+l)",+ '_l|." 

 ,vl' 



"The EuLERian function of a prism of rank /(.■ is divisible by (2-[-.t')^". 

 So i2-\-d'f' is the EuLERian function of an n-dimensional parallelotope 

 (I.e., p. 39, vol. II and as for the characteristic numbers line Bn ■ ■ • 

 of p. 245, vol. II)." 



So the characteristic numbers of the parallelotope of >S'„ are con- 

 nected in a simple way with the digits of the number representing 

 the ?i''' power of 21, if in this evolution the ordinary reduction 

 to higher uiiits is suppressed, i.e. if we write 21^ = 8(12)61, 

 2r = (16)(32)(24)81, etc. 



E.vawple. We consider the sixdimensional prismotope {tCO: tCO) 

 with two tCO (see the last polyhedron of lig. 55, l.c, p. J89, vol. IH 

 for constituents; as the symbol of characteristic numbers of tCO is 

 (48, 72, 26) we tind 



as EuLERian function and therefore (2304,6912,7680,3840,820,52) 



as symbol of characteristic numbers. 



Remark. Theorem I is not reversible, i.e. the decomposability of 



the EuLERian function into polynomia which i-epresent EuLERian 



functions of polylupes willi sinaller numbers ol' dimensions does not 



29 

 Proceedings Koyal Ac.td. Amsterdam. Vol. XIV. 



