56 ANALYTICAL TREATMENT OF THE POLYTOPES REGULAELY 



of the symbol representing it, gives for Ha a value surpassing the 

 corresponding sum for (P) a by v, on account of the units added 

 to the digits a if a 2 , . . . a v . A.s Ha diminishes by a unit if we go 

 down one line in the list, our (P) b belongs to group G k _ „ if 

 (P) a belongs to group G k . Now we know that the sum of the v 

 definite coordinates is maximum for (P) a and minimum for (P) b in 

 the space bearing the common limit, which proves that this limit 

 is a ^v_i for (P) a and a y u _ v for (P) b . 



By means of this theorem we can indicate the group to which 

 belong the polytopes touching a given poly tope along its limits of 

 a given import; if (P) a belongs to group G k and it has limits g hi 

 it is touched along these limits by polytopes (P) b belonging to 

 group G n _ h _ ± . 



The theorem also holds for the case r == 1 , where the zero 

 symbols of the successive groups G i9 G 2 , . . . , G p , . . . , G n are 



n n — 1 p n — p + 1 n 



(100 . . 0), (110Ö . . 0), . ., (11 . . 100 . . 0), ..., (11 ... 10). 



There we can state it in this form: "Any polytope [P) a of a net 

 for which r = 1 is touched along its limits of vertex import by 

 polytopes of the preceding, along its limits of highest import by 

 polytopes of the following group". 



30. We now apply the theorem XV to the cases n = 2, 3, 4, 5 

 and put the results on record in the second table added at the end 

 of this memoir. 



First one word about the general plan of this table. Horizontally 

 it is divided into four parts, corresponding successively to the cases 

 n = 2, 3, 4, 5. Vertically it breaks up into seven columns with the 

 first ûve of which we are concerned here. The first column, indi- 

 cating the rank number of the net, enables us to individualize 

 each net by a very short symbol, consisting of the value of n in 

 italian figures, bearing at the right a roman rank index, 2 Iir in- 

 dicating the net of hexagons in the plane. The second column gives 

 the value of the period r from 1 to n -f- 1 upward. The third 

 column contains the partition cycle, represented by that permutation 

 in which the first digit is as small as possible. The fourth column 

 brings the net symbol corresponding to that cyclical permutation of 

 the partition cycle, whilst the fifth is concerned with the zero sym- 

 bols of the different groups of constituents. With respect to these 

 columns — the others will be explained in part G — we have 

 to insert a few remarks. 



