( 3(^8 ) 



A'I. "The six liiijidred vertices of a l\^^^ lonii iii Iwo did'ci-eiit 



N\avs the vertices of lixc /'ooo • l" lil<<' ^vav / ooo ,üiv<'>^ ni tw (» 



\vaA s llie liiiiiliiiLi- sjtaces ot fixe / 120 



Willi the aid of these theorems it is easv to ari-ive at the reiiiainiiiu- 

 |)ossihle cciiti-ic dccoiiiposit ions of the foiir-diiiieiisioiial |)olvlo|t('s. 



Ill spaces with a ureater iiiiiiilxM of (Hiii<'ii>ioii> it is known that 

 hut three reuidar pidvtopes are to he found, i c in S,^ the siinph'X 

 /-*,; 1 1 . the polvtopc of ineasiii-c Z'^,, and the |»oivlo|>e Z^^" '■^''■'l"'"*'"'^"^ 



ndatcd to the pi-ceedini»-. With respect t(. the-^e there is an extension 

 f(»r theorem I and theorem III (Mil\ , namelv I for n^'l — 1 and 



111 for ii-=^'l*'. These extensions run as follows 



\'/.l-l li-ix 111 rl /'' 



VII. -'in space S^,,_^ the T vertices of a l\j.^>_^. hn'in the 



T ' siimtlexe> /'^ . In like wav a I' „ 



Aerlices of T simplexen /';^, 



^'-1 



Uixe.s 



>.'' r 1 



accordinu- to the liniitinii- spaces of '2'' 2 dimensions 2' simplexes 



0/' 



\ 111. "In spare S ^, the 2' \-ertice> (»f a /' ^ , j l'i"<'i" tlieNcrtices 



of -y'-i''^ /''-^-" 'J. In like \va\- a /' \, -ives aecordim.' p, the 



«M w '0/' ■ •>' 



limitiiiLi- spaces of '1'' — 1 dimeii^ioii^ 2" I* ~ 



.V'+l 



In the meeliiiii' of .Inne 2.. r.M):5 i'rok J. M. van BivM-MKI,fa com- 

 mnnicaled a |)a|)er: "'On (ihsuriitnnicoin jioii iids trjiidi niiiij cIiiiihji', 

 into chciiiicdl com jiiiiiikIs or siiliifiiiii . 



('rhi> [laper will not he pni)lislie<l in these Proceed in,i>-s.) 



(December 23, 1903). 



