( ^9 ) 



(431, 432, 435) and (521, 523, 524), (251,253,254) represent two 

 P, of edge import, the prisms of fig. 8 and 9 upside down. 



The vertices with the triplets of digits where the 1 is foremost 

 form the vertices of a limiting body of vertex import, a CO in the 

 position of tig. 6. 



Six wreaths of ten vertices, semiregular decagons. 



e,e,e, ^(5) — (120, 240, 150, 30) — (5^0, lOP,, lOP,, 5^6»). 



This most inflated of the polytopes >S(5) is represented in projection 

 in fig. 16. According to the number of vertices ^) we hav^e to place 

 at each vertex four of the five digits, each of them with a meaning 

 as to the order of succession ; of these four digits the first indicates 

 the original vertex of >S(5), the second the new endpoint of the edge 

 moved out, the third the new vertex of the face moved out and the 

 fourth — according to what was stipulated under e^Sip) — the digit not 

 occurring at the vertices of the tetrahedron moved out: so 1234 

 denotes the position of the vertex 1, after this point has been moved 

 out with the edge 12, with the face 123, with the tetrahedron 1235. 



Likewise as in the case e^e.^, S{5) the 7' of fig. 2 passes here into 

 the tO of fig. 12, traced back easily in fig. 16 if one remarks that 

 the moving out of this tO under the influence of the t^j-operation 

 demands the digit 1 after the triplets of fig. 12. While now the 

 lower side (4351, 3451, 4321, 3421) of the projection of this tO 

 assumes the same length ,s -\- d as the upper side of the projection 

 of the F, of fig. 16, i. e. the side (1523, 1524, 1253, 1254), which 

 Pg corresponds in form and position with that of fig. 14, it is clear 

 that the circumpolygon is a regular decagon with side .v -|- d. So the 

 projection is once more central symmetric as is the polytope itself. 

 In connexion with this the limiting bodies of vertex import are like- 

 wise tO, w^hich is immediately verified by looking for the 24 vertices 

 in whose quadruples of digits the 1 is foremost; likewise, not only 

 the prisms of face import, but also those of edge import, are hexagonal. 



Evidently the centre of the figure can be deduced from the side 

 s -{- d of file regular decagon; moreover it is possible to use to that 

 end the property that two adjacent P^ of the ten of the form of 

 fig. 14 lying at the rim have in projection a S(piarc fiice j)rojccted 

 as a lozenge in common. 



1) It is easily verified that in each of the cases treated the notation corresponds 

 to the number of vertices, i. e. that the number of possible pairs, triplet, qua- 

 druples of figures is always equal to the number of the vertices. 



