( 540 ) 



system of coordinates, with one of the vertices; thus this system of 



vertices is indicated in the same symbols by {2ai -\- '1) and we lind 



(P^J = {2ai) + {2ai -j- 1), i. e. (/*,J is the system of the points with 



integer coordinates which are either all even or all odd. 



Finally {P,,) is derived from (P,J by adding to {1\) not the 



whole system of the vertices of the cells C^^\ but only that half 



which is not occupied by the vertices of the inscribed C[Y^'. We 



1 

 express this by means of the equation I\, = {2ai} + ^ (2«/ + !)• 



Here we have to understand by -(2a,-|-l) that system of points of 



w^hich the coordinates are only odd integer numbers under the con- 

 dition that half the sum is either always even or always odd. If in the 

 cell C^^^ which fui-nished us above with the system of coordinates 



8 



a positive C^-}'~^ is inscribed, which for the future w^e shall always 

 suppose, then the point (1, 1, 1, 1) is occupied by a vertex of the 



1 



inscribed 6'v->^-) and so for the non-occupied vertices - {2ai-\-l) half 

 10 2 



the sum of the four quantities ai is odd. 



If we make the connection between the systems of points {Pg), 



(Pie)' [P^d i" ^^^® indicated way, then the number of points of (P, J 



is twice, and the number of points (P^g) is one and a half times as large 



as that of (Ps) and so the fourdimensional volumes of C-^), Cf^^'-^\ C^i^^) 



2 1 

 have to be in the same ratio as the numbers 1 , -, ~ . This can be easily 



o Z 



verified. To make a C'V;'^^' of C^^) we have truncated at eight ver- 



I 2 



tices a rectangular fivecell, which is -— of C^^^ ; so - of C'^.^^-emains. 



24 ^ 3 ^ 



And to make of Cf^ the cell C^^''^'^ contained in the former we 



have halved each of the sixteen parts C^^\ 



5. By the "transformation-view" of each of the nets (C^), (C^) and 

 (C) with respect to a space Sp^ of the bearing space S^)^ as screen 

 we understand the intersection varying every moment, of this non-moving 

 space with the fourdimensional net moving along in the direction 

 normal to this space. If for this movement we interchange the relative 

 and the absolute, we can also take this transformation-view to be gene- 

 rated by the intersection of the non-moving fourdimensional net 

 with a space /S/>»3, moving along in a perpendicular direction and 

 remaining parallel to itself; there we can again assume that this 

 view is observed by one who shares the movement of the space 



