DERIVED FROM THE REGULAR POLYTOPES. 71 



ce A <? 3 N(C S ) and ce i e 2 e 3 N(C S ), the constituent C 8 forming at the same 

 time the g x of the former couple and the g 2 of the latter. We shall 

 have occasion to profit by this coincidence in the next article. 



G. Symmetry, considerations of the theory of groups, regularity. 



89. On account of the fact that the offspring of the cross polytope 

 is identical with that of the measure polytope, the theorems XLII 

 and XL1II may be applied to any form of cross polytope descent. 



So we have only to add a few lines with respect to the regularity 

 and for the same reason this task has to be performed with respect 

 to the nets deduced from iV((7 16 ) only. 



If Ave individualize the 31 nets of Table VII by an N bearing 

 the rank number as subscript we can say that the nets N i7 7V 17 , iV 24 

 are regular and that the degree of regularity of the nets JV 3 , iV 5 

 with two equal extreme constituents is known, as these nets are 

 at the same time measure polytope nets. As moreover each of the 

 26 remaining nets admits faces of at least two different shapes, the 

 degree of regularity of each of these nets is either T 4 ^ or T 3 ^, according 

 to its having only one kind or more than one kind of edge. But 

 now it is immediately clear that any net admitting a constituent 

 ç s has at least two different kinds of edges, as the erect edges of 

 the fourdimensional g 3 , characterized by the property that the same 

 coordinates of the two end points differ by unity, cannot be at the 

 same time edges of the groundform in any of its three orientations. 

 So we have still to treat the twelve cases iV 7 2 , N±, N 6 , N l9 iV 8 , N is , 

 iV 19 , iV 20 , 7V 21 , iV 22 , N 23 , N. n forming two different groups, one of the 

 nets iV 18 , iV 19 , N. n Avith groundforms admitting only one kind of edge 

 and one containing the other nine not characterized by this property. 

 Now we can decide the question with respect to any of the nets 

 of these tAvo groups with the least amount of trouble by means of 

 the following general problem , where G is the groundform given 

 in Table VII, P the pattern vertex obtained by omitting the square 

 brackets of G, whilst Q, and Q\ represent the vertices of the net 

 adjacent to P, of which Q t are and Q! i are not vertices of G: 



"Determine the repetitions r of G (in its three orientations) 

 Avith P as vertex and examine whether or not all the vertices 

 Q, and Q\ are vertices of the same number of these repetitions 

 {G included)". 



The first case must present itself for the three nets iY T 18 , iV 19 , N 21 . 

 For the groundform of each of these nets admits one kind of edge 

 and its repetitions containing P are grouped regularly round P ; 



