( 63f> ) 



Mathematics. — "On fonrdiiiii'.ii.swiKil iwls and tJudr sections hy 

 spaces" (Second part). By Prof. P. H. Schoute. 

 (Communicated in the meeting of February 29, 1908). 



The net {C,). 



1. The problem to determine the section of the net (6'J with a 

 given space can be naturally divided into two parts. The first part 

 occupies itself with the question, how a series of spaces parallel to 

 the given one intersects an eightcell; in the second is indicated, how 

 the section of each of the eightcells intersected by the given space 

 can be deduced from that section which determines this space in 

 the eightcell assumed in the first part. Of course the four series of 

 parallel spaces normal to an axis of the eightcell come here to the 

 fore and then in the tirst part of the problem are investigated in 

 the first place the so-called "transition forms" where the intersecting 

 space contains one or more vertices of the eightcell, whilst between 

 each pair of transition forms adjacent to each other a single intermediary 

 form is introduced, namely that one by the space which bisects the 

 distance between the two spaces bearing those transition forms. 

 Generally this is sufficient for our end; moreover it is not difficult 

 to interpolate where necessary other intermediary forms. 



In the preceding communication of the same title we have packed 

 up each of the cells C,, of the net (6\J and each of the cells t\, 

 of the net (C^J in the smallest possible eightcell with edges parallel 

 to the axes of coordinates, with the intention to connect the spacial 

 sections of the nets (6\,) and (C,,) with those of the net ((7,) by cutting 

 with each 6\g and each C,, also the case C^ enclosing these cells. 

 With a view to this application we add to the above indicated 

 four series of parallel intersecting spaces two others, viz. those 

 normal to one of the two lines connecting the origin of coordi- 

 nates with the point (3,1,1,1) and the point (2,1,1,0); indeed, 

 these lines are — see the last table of the preceding communication 

 — axes of one or more of the cells 6'j, and C^, enclosed in a cell 

 C . Also for these two new series we restrict ourselves to the forms 

 of transition and the intermediate forms lying in the middle between 

 two adjacent forms of transition. 



In order to simplify the survey of the sections appearing in the 

 six series of parallel spaces we give the results to which the first 

 part — the determination of the section with one Cj — leads in two 

 different ways. In the tirst place we project all vertices, edges, faces, 

 bounding bodies of the cell 6'^ on the axis normal to each of the 



