( ^^-^ ) 



4. We now pass to the systematic treatment of tlie difïereiit par- 

 ticular cases, piittini»- together under the ditferent headings, containing- 

 the expansion and conti-action symbol, the symbol with the numbers 

 of vertices, edges, faces, limiting bodies, and the symbol with the 

 limiting bodies in the order of body, face, edge, vertex import, several 

 remarks pertaining to facilitate the interpretation of the drawings. 



e, S{d) — (20, 40, 30, 10) — (5^7', — , — , 57'). 



The resnlt is given in tig. 3. By the operation e^ of the moving 

 ont of the edges the T of fig. 2 becomes a tJ' (fig. 4) with four 

 hexagons of face import and four triangles of vertex import. As each 

 vertex of T assumes three different positions if it moves out with 

 each edge passing through it, tlie vertices of this tT must bear two 

 digits, the first indicating the original vertex of 7', the first in com- 

 bination^with the second the edge of T moved ont. By retracing in 

 fig. 3 the same pairs of digits one easily finds again the ^ J' deduced 

 from (2345), though for the reason stated above no dotted lines have 

 been admitted. If we rotate this tT around the centre of fig. 3 to 

 an amonnt of 72° in the indicated sense it is brought into the posi- 

 tion with (54, 45) as bottom-edge and (13, 31) as top-edge in coincidence 

 with a second tT, having in common with the first — in its original 

 position — the hexagon (54, 53, 35, 34, 43, 45), deduced by the co- 

 operation from the triangle (345) common to the tetrahedra (2345), 

 (3451) of fig. 1". Or rather: the centre of fig. 3 is found by drawing 

 the tT of fig. 4 twice and by putting these two tT" in such a way 

 upon each other as to get a limiting hexagon in common; then this 

 centre is the point of intersection of the lines bisecting orthogonally 

 the two edges (43, 34) and (54, 45). Or still otherwise : the limiting 

 polygon of the projection is a semiregular decagon with sides alter- 

 nately equal to z and d and from this fact the circumcentre can be 

 deduced ^). 



It goes without saying that the vertices of each following /7M)ear 

 pairs of digits deduced from those at the corresponding vertices of 

 the preceding tT by adding unity to each digit, in which process 

 the 5 becomes 0. 



The four different positions 12,13,14,15 of the original vertex 1 

 foi-m the vertices of a T of vertex import. 



It is easily verified that the ten limiting bodies 5tT,5T, now 



') From tig. "2 upwaitl \vc use in oil the diagfams for s and fZ the same measures 

 in order to show by tlie projection the swelling of the polylopes corresponding to 

 the operations of expansion, 



5 



Proceedings Royal Acad. Amsterdam. Vol. XIV. 



