( 638 ) 



The second plate indicates the obtained sections in parellel per- 

 spective. The tirst diagram on the lop leftside, represents an eightcell 

 which indicates besides the diameters normal to the different series 

 of parallel intersectii]g spaces a few other lines appearing in the 

 sohition ; for our case (i, 1, 1, 1) to wiiich the four following 

 diagrams refer the axis EE' is this diameter. To characterize 

 this case the mark (J, J, 1,1) is noted down to the right at the 



4 3 2 1 



bottom in the rectangle ; moreover the fractions — , — , -, - placed to 



8 8 8 8 



the left at the top of eacli diagram indicate the part of the axis 

 EE' lying with E on the same side of the intersecting space. It is 

 easj' to follow in these diagrams the changes in form which each 

 face of the regular octahedron forming the central section undergoes, 

 when the point of intersection of the intersecting space with the axis 

 OE moves from to E. Thus the face lying in the upper cube of 

 the eightcell, which is at the same time the visible upper plane of 

 the octahedron regarded by itself, ti-ansforms itself first into a regular 

 hexagon, then into an equilateral triangle of opposite orientation, etc.; 

 if the eightcell is a 0^\ then the sides of the triangles of the first 



and third diagrams are 2^2, those of the hexagons and the triangles 



of the second and fourth diagrams are 1^2, whilst the series closes 



with the transition form consisting of the single vertex E to which 





 the fraction — answers. 

 8 



We now arrive at the question how the remaining eightcells that 

 are likewise cut by the intersecting space are intersected in each of 

 the considered cases. To this end we suppose the above intersected 

 eightcell to be the central one of the net and so we assume the centre 

 of this cell to be the origin of the system of coordinates with respect 

 to which we have determined in the first communication the coordinates 

 of the centres of the remaining cells in the symbolic form (2(2/). The 

 equation of the central space perpendicular to the axis OE^ towards 

 the point (1, 1, 1, 1) is x^ -\- 1\ -\- a\ -\- x^ = ; the length of the 

 normal let down out of the centre {2ai) on to this space is there- 

 fore 2ai. So the eightcell with the centre (2«/) is cut by the central 

 space 2xi = 0, when — 2 < 2£ai < 2, and here the five cases occur 

 where ^ai has one of the values — 2, — 1, 0,1,2. In other words: 



4 

 if with the central cell the central section - makes its appearance, 



8 



■,. n, .02468 



then with the I'emaining cells the sections -, -, -. -, - occur and 



8 8 8 8 8 



