( 382 ) 



With reference lo the system of coordinates 0{Y X , F, , Y' % , zj 

 of the four diagonals AA', BB' , CC', DD' already used in the first 

 communication the five series of parallel sections are characterized 

 by the symbols (1,0,0,0), (1,1,0,0), (1, 1, 1, 0), (1,1,1,1), (3,1,1,4) 



mentioned above, from which can be deduced that the octuple of 

 vertices of the sixteencell projects itself on the chosen axis of projec- 

 tion in these five cases in ar i 'a i icemen ts indicated by (1, 6, 1), 

 (2,4,2), (3,2,3), (4,4), (1,3,3,1). Of these live cases the first and 

 the fourth are evident by themselves; so the parts of plate I bearing 

 the headings (1, 0, 0, 0) OE ig and (1,1,1,1)0/2,, can be understood 

 immediately. The three other cases can be explained by the three 

 following diagrams, a common characteristic of which is that the 

 eight vertices of the sixteencell have been obtained by starting from 

 the cube that is found by intersecting the eightcell of fig. 1 by the 

 central space normal to AA' , by splitting up the eight vertices of 

 that cube into the two sets of vertices <»f bodily inscribed tetrahedra 

 and by erecting normals on the space bearing that cube - - i. e. by 

 drawing in the diagram in parallel perspective lines parallel to AA" 

 the length of which is equal to \ AA", in the vertices of one 

 of the tetrahedra to one side and in the vertices of the other tetra- 

 hedron to the opposite side. This representation of the cube with the 

 two quadruples of points ABCD, A'B'C'D' has been repeated in 

 three different positions by a motion parallel to itself from left to 

 right and from above to below over the same distance, which gives 

 rise to the three diagrams 2, 3, 4 which we will now examine one 

 after another. 



In tig. 2 the eight vertices of the sixteencell have been projected 

 on to the line F S F' 8 , joining the midpoints of two opposite edges 

 of the cube and forming therefore an axis (JF % of the eightcell ; on 

 this axis the vertices A, B project themselves in F e , the vertices 

 C, D, C', D' in O and the vertices A' , B' in F' 8 . We find again 

 here that the axis OF s of the eightcell is at the same time an axis 

 OK 18 of the bodily inscribed sixteencell, as F B is the midpoint of AB, 

 and deduce now from the arrangement (2, 4, 2) of the projections of 

 the vertices all that is indicated on plate I under the heading 

 (1, 1, 0, 0) OK 16 . 



In tig. 3 the centres of gravity F 19 ,F' ie of the opposite faces 

 ABC, A'B'C' have been determined, and the axis OF l6 joining these 

 points forms the axis of projection. Then the three vertices A, B, C 

 project themselves in F 1S , the two vertices JJ, D' in (J, the three 

 vertices A', B' , C' in F ie . From the arrangement (3, 2, 3) can then be 

 deduced what appears in plate I under the heading (1, 1, 1, 0) OF l9 . 



