68 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



must now be generalized to this: "Every net deduced from N(C i& ) 

 can at the same time be deduced from i^ftj." Now the last 

 column but one indicates the name of the corresponding <7 24 -net. 

 So we have e s e k JV(C i6 ) = e x e k N{C. lk ), etc. 



We must remember that the symbols given in Table VII have to 

 be completed by applying the transformations indicated in art. 79. 

 Moreover we fix our attention on the particular form in which 

 the symbols of each constituent appear. Every prismotope g 2 is 

 decomposed as to its vertices into two or three fourdimensional 

 prisms, one of which degenerates in some cases into a regular polygon ; 

 of the fourdimensional prisms g$,g\ the first is determined by its 

 two bases, whilst the latter appears as prismotope (4; 4) or as a 

 combination of prismotopes, etc. 



F. Polarity. 



85. In art. 67 we remarked that in S n any polytope derived 

 by means of the operations e k with or without c from the measure 

 polytope M n can also be derived from the cross polytope C 2 n. In 

 art. 77 we stated this result in the form of theorem LI after having 

 demonstrated it by showing that the total set of symbols of coor- 

 dinates of the group derived from C 2 n is equal to that of the group 

 derived from M n . We have to come back to this result once more 

 here, in order to indicate how it depends on the laws of reciprocity 

 and what is the general relation between the two symbols of expan- 

 sion operations of the same polytope deduced from M n on one 

 hand and from C 2 n on the other, which couples of symbols have 

 been given for n = 3, 4, 5 (compare the foot note in art. 72) in 

 the first and the second column of Table IV. 



It goes without saying that the dependence between theorem LI 

 and the laws of reciprocity merely consists in this that the polar 

 reciprocal polytope of a regular polytope A of JS n with respect to 

 a concentric spherical space is an other regular polytope A' and 

 that in this polarity the vertices, edges, faces, etc. of the one cor- 

 respond to the limits (/) n _ l5 (/) n ._ 2 , (0n-3> etc - °f tne other. So we 

 have still only to deduce the relation between the two symbols of 

 the same polytope. This task can be performed by comparing the 

 first two columns of Table IV with each other and by generalizing 

 for an arbitrary n the outcome of this comparison. So for 

 <C à <C ••• < s < t <C n — 1 we immediately deduce from Table 

 IV the following general laws: 



