DEBITED FROM THE IIEGULAE POLYTOPES. 43 



quantity r, which is the same for the different frames of the same 



net, may vary from net to net. We call it the period of the net. 



All the simplex nets proper have similar frames, as their images 



of points V are similar. l ) If, as in art. 1, our space 8 n is the 



n+ 1 



space 2 x t = 1 lying in a space of operation S n + 1 and deter- 



mining on the axes of a given system 0(X i ,X 2 , .... ; I n + 1 ) of 

 coordinates equal intersepts OA n we can say that the nets of jS u , 

 the vertices of which admit with respect to the simplex of coor- 

 dinates J i A 2 . . . A n + 1 integer coordinates only, always admit frames 

 projecting themselves normally on any of the n- dimensional spaces 

 B' n of coordinates of JS n + i as sets of vertices of systems of measure 

 poly topes of that JS' U . 



c) In the third place we prove that each net of S n admits a 

 net symbol. 



By combining the zero symbol (q u q 2 , . . .,q H , 0) of the central 

 polytope with the frame symbol (rô ± -, rb 2 , . . . , rb n , rb n + 4 ), 2 b. { == 

 of the net we obtain the symbol 



(rb l -f- g u rb 2 -\- q,, , rb n + q ni rb n + ,), N) 



where the q i and r are given integers, whilst for the b t we can 

 take any system of integers with sum zero. As this symbol contains 

 the coordinates of all 2 ) the vertices of the net, it is the net symbol. 

 If Ave write this symbol in the form 



0A + </i, rb 2 -f q 2 , rb n + q nt rb n + , + 0) 



we have got a symbol representing the net decomposed into the 

 repetitions of the central polytope. 



*) We remark already here that later on cases will present themselves which are at 

 variance with this simple result. We will treat these cases — and explain why they 

 appear as exceptions — as soon as they turn up. 



2 ) This is only true, if each vertex of the net is also vertex of a repetition of the 

 central polytope in the same orientation. So, if the net contains a non central symme- 

 tric constituent in two opposite orientations and in each vertex only one of these two 

 differently orientated constituents concurs, the net symbol corresponding to one of these 

 constituents as central polytope would only contain half the number of vertices of the 

 net and would have to be completed by a second symbol giving the other half. In that 

 particular case the system of vertices breaks up into two equivalent parts P and Q with 

 the property that the net is equipollent to itself for any two points of the same half 

 as homologous but congruent with opposite orientation to itself for any two vertices of 

 different halves as homologous. This particularity presents itself in the plane in the 

 case of the net of triangles and dodecagons (fig. 8), already discarded above for an 

 other reason. Here we exclude, also provisionally, all the eventually possible nets 

 where this particularity of the division of the system of vertices into two equivalent 

 systems might present itself. 



