968 



that tlie whole spherical siirlace is lilled by faces of fiindamejital 

 tetrahedra. 



5. It is not difficult to conceive in what manner such a spherical 

 surface is divided into faces of fundamental tetrahedra, all rectangular 

 spherical triangles. 



The figure given hei-e represents the eighth part of such a 

 spherical surface in stereographic projection. 



\ / . , - • • ' • \ 

 V :uri^ i-: iï V 



/ "u. ''''•■ / \ \ / \ 



'&¥- ' ' "^ ■■ 



k ■■■. vO/ ' V .• ' 



^ L ',/ 



The full circles in the figure divide the sphere into 12 regular 

 spherical pentagons and 20 equilateral spherical triangles arranged 

 like the faces of the polyhedron ce^l or ID. Of these the figure 

 represents three half pentagons, one whole and three half triangles. 

 The dashes connect the vertices in the said pentagons and triangles 

 with the middle points of the opposite sides. The dotted lines are 

 the diagonals of the pentagons. 



We see that the figure contains triangles of all four kinds, viz. 

 012, 013, 023 and 123. It follows therefore, since each triangle of 

 a certain kind is equivalent in C^^^, that all the spherical surfaces 

 met with in the division of the hypersphere here considered are 

 divided into triangles in the same manner. 



6. Let us now look first of all at the dotted circles in the figure. 

 These circles contain successively vertices 3 2 3 etc. They 



