( 38Ó ) 



Mathematics. "On fourdimensumal nets and their sections by 

 spaces." (Third part. By Prof. P. II. Schoute. 



The net (C 16 ). 



1. In the first part of this investigation we have found that the 



net (C\ 9 ) of cells C\e ~ is formed out of three equally strongly 



developed groups of nomothetic cells C?^ , one group of erect cells 



6'ic polarly inscribed in eightcells C$ and two groups of inclined 



cells 6'ic and Cii ' bodily inscribed in eightcells G's" , namely 

 the positive group and the negative one. If we restrict ourselves 

 once more, with respect to this net, to the sections by spaces normal 

 to one of the four different axes of one of the sixteencells, it is 

 evident from the table of connections between these different axes 

 given in the iirst part (p. 544) - if we bear in mind that the three 

 groups of cells of the net are equivalent - that we have only to 

 consider three series of parallel intersecting -pace-, viz. those normal 

 to one of the axes OE 6 , 0F e) 0K S of the circumscribed eightcells. 

 In these three cases, corresponding successively to the fifth, the fourth, 

 and the third line of the quoted table, we find indeed for the erect 

 sixteencells series of spaces normal to OJS le , 0K it and 0F lt , whilst the 

 first of the three cases provides us, for the inclined sixteencells, with 

 a series of spaces normal to 0R lt . So all in all we have to bring 

 to light three different threedimensional space-tillings and their trans- 

 formation connected with a parallel motion of the intersecting space. 

 But in order to do this we have to consider more than the four 

 usual series of intersecting spaces respectively normal to an axis 

 0E lt , OK 16 , OF lt , OR xt \ tor the spaces normal to 0K 8 presenting 

 themselves in the last of the three cases are not normal to any one 

 of the four axes of the inclined sixteencells but to the line connecting 

 the centre of one of these cells with the point characterized by the 

 coordinates (3,1,1,1) with respect to the system of coordinates witli 

 the four axes 0E U of that cell as axes. So all in all we have to deal 

 with live series of parallel intersecting spaces which may be charac- 

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

 as they are always normal to the diameter of the cell passing 

 through the point the coordinates of which with respect to the 

 axes 0E lt of the cell are given by the corresponding symbol. 



2. In the same way as we have done this in our second com- 



