( 538 ) 



into 16 equal and similarly placed cells 6'^' ^ by means of four spaces 

 through the centre parallel to the pairs of bounding spaces. Then 

 divide each of the sixteen parts 6'^') (tig. 1) by the space in the 

 midpoint of the diagonal concurring in the centre of C^^) normal 

 to this line into two equal halves ; here the section as is known is 

 a regular octahedron A^^ A^^ . . . A^^ . We now direct our attention 

 first to the half cells f .') surrounding the point 0; they form a C^^^^- 

 Of the 24 bounding octahedra sixteen are furnished by the sections 

 ^j, -4j3 . . . ^34 , whilst the eight remaining ones are obtained by joining 



Fie. 1. 



in each of the eight ends of the chords along the four axes OX^ , 

 OX^, OX^, OX^ through 0, e. g. in X^ , the eight rectangular tetra- 

 hedra Xj {A^^ A^^ -"^ia), where it is clear that in X^ eight of those 

 tetraheda really meet, because we can reverse the direction of each 

 of the segments X^A^^, X^A^^, X^A^^. Furthermore we observe that 

 around an arbitrary vertex A of the original cell also 16 half 

 cells C<^') are lying and that these form in exactly the same way a 

 C^^~\ Bv this the net of the C^'^^ has been transformed into a net 

 of cells Ci,^^), where the centres and the vertices of the cells C^^ 

 form the centres of the cells Ci^-'> placed in the same way. 



If we add to the considered sixteenth part C^'^ (fig. 1) the three 

 parts generated by reversing the sign of one of the two axes OX^ 

 and OX^ or of both, it is immediately evident that A^^ is the centre 

 of a face of the original cell C^^l From this is evident to the eye 



