DERIVED FROM THE REGULAR POLYTOPES. 81 



In this list we have availed ourselves of the occurrence of the 

 three zeros in order to represent the 17 polytopes by nine symbols. 

 So the second line represents three different polytopes which can 

 be obtained by putting seccessively — 3 -|- 3 under each of the 

 three zeros. For each line the number at the right indicates how 

 many different polytopes through the point 321000 correspond to 

 that line. 



This list shows that we may find here two kinds of edges though 

 we have three groups, edges (32) 1000 common to 9 polytopes, 

 edges 3(21)000 common to 16 polytopes, edges 32(10)00 common 

 to 9 polytopes, as in the first and the last group the sets of 9 

 polytopes are both 4 e ± e 2 , 3 ce i e 2 e 3 , e it e, t . By investigating the 

 fourdimensional contact between the nine polytopes of each set 

 can be found whether the edges (32) and (10) belong to the same 

 kind or not. 



From the numbers of differently shaped limits the fraction of 

 regularity has been deduced ; it is given in the last column of 

 Table II. 



45. We finish this part of our memoir concerned with the off- 

 spring of the simplex by a remark about what may be called the 

 "circnmpolytope" of a net. This polytope, which has for vertices 

 the vertices of the net joined by edges to any arbitrarily chosen 

 vertex of the net, is by its form a criterion for the regularity of 

 the net. If the net admits one kind of edge the circnmpolytope 

 must admit one kind of vertex, etc. This circumpolytope is in the 

 cases of the threedimensional nets : 



3, . . . a CO , 



Sjj ... a prismoid limited by two equilateral and six equal 

 isosceles triangles , 



3 in • • • a prismoid limited by two squares and eight equal 

 isosceles triangles, 



S IV . . . a pyramid on a rectangular base , 



3 F ... a tetrahedron limited by four equal isosceles triangles; 

 of these five polyhedra only the fourth has vertices of two diffe- 

 rent kinds. 



The theories developed here enable us to find the circum- 

 poly topes corresponding to the different nets of simplex extraction 

 in # 4 and S 5 . But instead of deducing these polytopes here we 

 conclude by the following general problem, for the proof of which 

 we refer to the dissertation of M r . Elte: 



