969 



consist of arcs measuring i tt, eacli of which is composed of an 

 edge 03 and an edge 23. On the former part lies the projection of 

 a vertex of e, C,^^, on the other that of a vertex of ce^ e, C,,^. 

 Hence, viewed in this manner, the vertices of these two polytopes 

 correspond one-for-one to eacli other. 



Since the whole C^^^ considered possesses the symmetry of tlie 

 pentagonal (= parallel or pyritohedral) hemihedrism of the regular 

 crystal system, we know that also the said arcs measuring ^.t can 

 be classed into groups possessing that symmetry, each set of corre- 

 sponding points of which may be represented by one symbol. Hence 

 it is clear that also the coordinate symbols correspond one-for-one 

 to each other, and that the corresponding coordinate-symjiols represent 

 the same number of vertices. 



They likewise agree in the number of zeros. For besides the 

 vertices the said arcs have no point in common with a coordinate- 

 space unless when lying entirely in it. 



Two corresponding symbols do not always agree in the number 

 of equal coordinates. It is found indeed, that some groups of the 

 said arcs are intersected by spaces bisecting the angles of the co- 

 ordinate-spaces exactly in the projections of the ejCg„(,-vertices. In 

 this case the (?j,6^eoo'Symbol has two equal coordinates and the indica- 

 tion of the hemihedrism can be omitted in it, while this is not the 

 case with the corresponding ce^e^C^f^^-symbo\9,. 



7. We shall next consider the dashed lines in the figure. 



These contain in succession the vertices 2 1312 0, etc. They 

 are found to consist of arcs measuring \ jr, each of which contains 

 an edge 02, an edge 12, and an edge 13. 



We conclude from this, thai a relation similar to that which 

 connects e^C^^^ and ce^e^C^^^, likewise exists between the three 

 polytopes d,Ceoo, ce.e^C,,, and ce.e^C,,,'). 



The full lines contain no other edges than OJ. Hence the poiytope 

 ^iCgo, stands alone. 



8. We shall now proceed to consider the division of the sphere 

 into spherical triangles. 



The dashed lines divide the sphere into rectangular spherical 



1) The set of points whose coordinate-symbols are mentioned on p. 25 (Table B) 

 in Dr. Elte's dissertation, are projected in the foot-points of the perpendicular 

 arcs drawn in each triangle 012 from the vertex 1 to the opposite side 02. It is 

 evident that these symbols must show a similar resemblance to those of the 

 polytopes egCgoo» ce^eaCGoo and ce^ggCgjo as exists between these three polytopes. 



