( 66 ) 



accounted for, want as limiting faces exactly all the faces shown in 

 projection, each face counted twice over, a triangle always being 

 common to a tT and a T, a hexagon to two tT. 



The 20 vertices present themselves in two wreaths (10,10). 



e,S{h) — (30, 90, 80, 20) — (5C0, — , 10^,50). 



For the result fig. 5 may be consulted. By application of the e,- 

 operation the T of fig. 2 passes into the CO of fig. 6, each edge of 

 T being broadened out into a square, the sides of which are pai-allel 

 to that edge and to the opposite one. Here the particularity enters 

 that two of the six squares project themselves as line segments, which 

 is due to the fact that in pentagonal projection the edges 25, 34 of 

 the T of fig. 2 are parallel. Here we have to indicate the vertices 

 of the CO by three digits, the first indicating the original vertex of 

 T and the two others, in irrelevant order, in combination with the 

 first, the face which is moved out. This CO in indicated in fig. 5 

 by the same triplets of digits placed at the vertices. By reproducing 

 it four times by means of the rotations indicated above, fig. 5 is 

 completed ; here any two CO have to be placed upon each other in 

 such a way as to have a triangle in common. 



After having inscribed all the triplets of digits at the vertices 

 according to the rule given above about the augmentation with unity 

 for each rotation in the right sense to an amount of 72° we find 

 that the 1 is foremost in six triplets, corresponding (fig. 7) to the 

 vertices of an 0, i.e. we find 50 as limiting bodies of vertex import. 

 Farthermore the notation shows that the edge (34) of the Z" presents 

 itself in fig. 5 in three positions, the triplets of digits of the endpoints 

 of which are found by putting behind 34 and 43 successively one 

 of the three remaining digits 1, 2, 5, passing — if we rearrange the 

 second and the third figure according to their value — into 314, 

 324, 345 and 413, 423, 435. So we get the P, of fig. 8, occurring 

 in five different positions, and likewise the edge 25 leads to the 

 differently projected P^ of fig. 9, occurring also in five positions. So 

 the ten P^ of edge import are accounted for. 



Here the circumpolygon is a regular pentagon with sides 6' -j- c/; the 

 30 vertices appear in four wreaths (5, 10, 10, 5). 



e,S{f>) — (20, 60, 70, 30) — (ST, IOP3, IOP3, ST). 



The pentagonal projection (fig. 10) exhibits central symmetry as 

 does e,S{^) itself. Here (21, 31, 4L, 51) is the Tof fig. 2 moved out, 

 by which remark the 52" of body import are accounted for, whilst 



