DERIVED FROM THE REGULAR POLYTOPES. 



29 



[a+a-a k - Si ,a+a-a k - 8i ^ it . . .,«+^-a 1 ]K_ Si+1> 



a 



/t-s,+2> 



• • ,*k) 



nn qn nn 



^le—St+l > %k-s 1 +2> •■ • • i &k f 



L^A+1 J a k+2> • • • 5 ^n-1 > ^/J I 

 01 on t> 



m nr> nr> 



n-+l) t6 /i+2> • • • > ^n 



[a-\-a—a k ,a+a—a k _i,. . . 9 a+a— a±] 



%\> <%2> j #7c 



(%+l ? %+2 J > a k+s.J L^A+s^+l J ö /r+s 2 +2> • • • » #n-l > ö nJ 



t> in m mm m 



ct k+l> (L k+2> j^A-.+Sî ^/c+Sj+l? ^k+So+o' • ■ • > *« 



which two limits (/) n _i admit as centres the points 



k — s l 



Si 



n — k 



aa. . .a t i t 1 . . . t x 00 . . .0 



/c {f 2 n — A— s 2 



aa. . .a t^t^. . .t 2 00 ... ] 

 t ± and 4 being determined by tlie relations 



k k+s. t 



Si t A = Z^ tt, , S 2 H == ^ a i t 



i=k— s x +l i=fe+l 



showing that we have <C ^ , < a for e = 1, 2. So the centres of 

 these two {l) a _ x lie on the boundary of the measure polytope 

 M n (+a) and therefore the (/)„ _i themselves lie partially within that 

 measure polytope. 



Now for each of the two cases there is only one constituent 

 passing through the chosen limit (l) n _ iy viz. 



[tt-hCl—llk—s^ iï+Cb-Wk—Si—ii • • • > a-\-(Z— (t\\ \_a k _ Si+ i, a k _ Si+2 > • • • > a n— 1> a n\ 



m m m mm m 



'M j ^2 ? • • • J Uy k—rS ï lV k— s,+l > ^fc— Si+2 > • • • » ^u 



t??! , a? 2 > • • -J ^fc-fSj ^/c+s.,+1 > ^&+s 2 +2 ? • • • > <^n 



So, all we have to do yet is to investigate the position of the 

 centres. If we indicate these points by the letters G a , G bi , G, h , G ab , 

 G ubz and we remark that for these five points we have 



0C\ 



CCc) 



,v 



m ■ • m m 



k-ss <6 A— s,+l lt k—s l +2 • • ^kf 



G, 



G 

 G, 

 G 

 G 



f', 



ab l 

 a h- 



m m m m m — 



^A'+l oL k+2 • • ^k+s.,' ^k+So+l ^fc+Sjs+2 — 



the following list of coordinates 



= . . - - a? n , 





■ / 1 j . . . 

 a 



^k — .<?!-(- 1 ' * * * 



a 



®k ■ + 1 > • • • 







^k + s. + 1 » • • • 









a 

















a 



a 



a 









a 



k 













a 



a 



k 







According to this list of the two triples (G a , G bi G ab ^), (G a , G b> G ab J 

 of collinear points G ab lies between G a , G bi , and G ab between 



