6 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



we will indicate the system of points (2^, I-5-, 3-, — |-, — If) of 

 jS 4 for which the difference of any pair adjacent values is unity 

 by (4, 3, 2, 1, 0). Indeed, it is easy to return from (4, 3, 2, 1, 0), 

 where the sum of the values is 10 and therefore too large by 9, 

 to the real coordinate values by subtracting -| from each; so this 

 important simplification — of a temporary character — can do 

 no harm. 



We indicate the entire system of points obtained by taking the 

 values 4,3,2,1,0 in all possible orders of succession by putting 

 these values in decreasing order between round brackets; this sym- 

 bol (43210) will be called the "zero symbol" in distinction with 

 the symbol of true coordinates. 



3. The simplification alluded to above allows us to indicate in 

 a very transparent manner the sets of values furnishing the polytopes 

 with one kind of vertex and one length of edge (equal to the edge 

 of the simplex of coordinates) found by M rs . Stott. So we have for 

 u = 2, 3, 4, 5 successivily in the symbols explained in the memoir 

 of M rs . Stott : 



