( 639 ) 



8 , - - . 



no others. The sections and - being points and therefore not under 



8 8 



consideration, we find as section of the net (Cj ^ threediniensional 

 space-fdiing consisting of' two groundfornis, octahedron and tetra- 

 hedron, where the tetrahedron occurs in two positions of opposite 

 orientation. From a close consideration of this result follows now 

 that the fractional symbols of the intersected cells furnish in general 

 difïerences of ninlti[)les of quarters with that of the central cell and 



13 5 7 

 are thus represented b}' --, -, -, - when the symbol of the central 



8 8 8 8 



3 1 



cell is - or - . We find then again a threediraensional space-filling 



8 8 



consisting of two groundfornis each of which appearing in two 

 oppositely orientated positions, the first semi-regular Archimedian 

 body and the tetrahedron. As we arrive again at eighlcell and 



2 



tetrahedron when starting fi'om the section of the central cell, the 



8 



above-mentioned two cases are for this series the only ones where 

 the threediniensional space-filling consists of two groundforms. In 

 every other case — as e. g. the one answering to the fractions 



1 5 9 13 , . .. 



— , — , — , — — we ünd four difierent groundforms and never more ; 

 16 16 16 16 ^ 



we recommend the designing of the just mentioned quadruplet of 



sections as a good practice. 



If we exchange the infinite sj'Stem of cells O^'^ by a finite block 



of k^ cells CY^ forming together a CY''\ if we divide a diagonal of 



this block into eight equal parts and if we suppose the block to be 

 intersected by a space standing in one of the points of division perpen- 

 dicular to the diagonal, we then find according to circumstances 

 either a finite system of octahedra tK-^-^ and tetrahedra T(-^'i with 

 edges 2 1, -"2, or a finite system of Ai'chimedian bodies A^^^^ and 

 tetrahedra 7'(t 2) ^yith edges 1^2, enclosed in an octahedron, a tetra- 

 hedron or an Archimedian body of greater size, viz., in the section 

 of the block C^-'') with the intersecting space. In connection with 



the notes joined to the pages 15, 16 and 24 of the study "On the 

 sections of a block of eightcells, etc." [VeHiandelinyeii, volume IX, 

 n". 7) we here indicate how large in each of those cases the number 

 of the component parts (X2^/2)^ ^4(1/21^ 7\2i/2)^ j\y2) jg \ye restrict 

 ourselves here to mentioning the results and we only remind the readers 

 that the deduction of these are based on the actual connection 



