967 



The whole hypersphere is thus divided into 14400 double rectan- 

 gular ^) hyperspherical tetrahedra, which I shall call fundamental 

 tetrahedra, the vertices of which are : 



the vertices of the Cgoo (points 0);- 



and the projections of: 



the middle points of the edges (points J); 



the centres of the faces (points 2) ; 



the centres of the limiting bodies (points 3). 



The vertices of each fundamental tetrahedron are a point 0, a 

 point 1, a point 2 and a point 3. 



The elements of the fundamental tetrahedron may be easily cal- 

 culated from the 6 dihedral angles. We know indeed, that the 

 dihedral angle at the edge 01 is = i rr, each of those at the edges 

 23 and 03 = -|^ -t, each of the others = i -t. ") 



3. Deducing from the polytope C\oq, in the manner as described 

 in the memoir of Prof. Schoute already quoted firstly the polytopes 

 <^^i C'goo, ce, Cjoo, and ce^ C\oo^ and then by combination the other 

 polytopes of the family, we easily see: 



1. that the vertices of these primitive polytopes are projected on 

 the hypersphere respectively in the points 1, 2, and 3; 



2. that the vertices of the polytopes each derived from two ot 

 these primitive ones are projected in definite points of the corre- 

 sponding edges of the fundamental tetrahedra; 



3. those of the polytopes derived from three of the primitive ones 

 are projected in definite points of the corresponding faces of the 

 fundamental tetrahedra; 



while finally the vertices of the polytope e^e,e,Ctoo obtained by 

 combination of the four primitive ones are each projected in a 

 definite point within one of the fundamental tetrahedra. 



4. Taking one of the faces of a fundamental tetrahedron we 

 produce the spherical surface to which it belongs through the three 

 edges. 



Since at every edge an even number of dihedral angles come 

 together, we find, that in the produced part of one of the faces 

 there lie three faces of other fundamental tetrahedra. Hence we see, 



1) W. A. Wythoff. The rule of Neper in the four-dimensional space. These 

 Proceedings IX 1, p. 529-534; Verslagen, Vol. XV 2, p. 492—497. 



") This tetrahedron is treated as an example by P. H. Schoute, Mehrdimensionale 

 Geometrie I, § 9, N». 133, Aufgabe 802. 



The vertices 0, 1, 2, 3 are called here Ai^ A^, A^, Ai. 



