( 695 ) 



(2h) 



which express that K can be built up either out of the four forms 



(Ik) (k) 



Ti or out of the live forms // can serve to control. 



We shall now indicate at full length how the obtained relations 

 will serve to get us over the entire difticulty of the determination 

 of the demanded numbers. To this end we notice that the vertices 



(-) (2A:) 



of the k^ uieasure poly topes Mr, forming together a block J^/5 

 project themselves on a diagonal of that block except in the ends 

 in the 5 k — 1 points dividing this diagonal into hk equal parts. If 

 we indicate (diagram 3) the 5/; -f- i points obtained in this way on the 

 diagonal by A^, A^, A^, . . . , A^,]^, then the segment A^A^ beai's the 



(2) 



projection of a single i/5 , the segment A^A^ that of a group of 

 five, the segment A^A. that of a group of fifteen measure-poljtopes, 



Fig. 3. 



etc., where the numbers 1, 5, 15, etc. of the measure polytopes with 

 the same projection are the coefficients a,, of the terms xp in 

 (1 + X -{- .c' 4- . . . -|- ,i'/^-i)' for /> =0, 1, 2, etc. When determining 



1 (2/,-; 



the section - Mr, we find that the intersecting space Sp^ hits the 

 diagonal of projection in the point of division Ai, from which ensues 



(2) 



that the groups of polytopes M ), corresponding to the coefficients 

 a^,a^,...(ik~:y are not yet cut, the groups corresponding to the coeffi- 

 cients a]-, a\.j^\, . . . (t:A—; are no more cut, so that we have but to 

 deal with the four groups shown to the right of the diagram : 



7/-'^ 7.(2) 7.(2) ,2) 



Now for the coefficients a^, the particularity appears that for ^9 <^ k 

 they can be represented as binominal coefficients viz. by the equation 



whilst for greater values of p they are "gnawed" binominal coeffi- 

 cients. So we find here immediately 



■•2k) (2k) (-2) (2) (2) ro') 



an(jl in quite the same way 



