( 330 ) 

 spherical triangle, from which further the rule of Neper immediately 



follows. 



The train of thoughts followed here will be found back entirely 

 in the following. 



2. A hyperspherical tetrahedron I shall call doublerectangular, if 



two opposite edges stand each normal to one of the faces. 



Let us suppose the letters A v A^ A z and A A at the vertices of 

 the telahedron in such a manner that A t A t is perpendicular to the 

 face A i A s A 4 and A t A 4 perpendicular to the face A l A i A i . 



To make the tetrahedron doublerectangular it is necessary and 

 sufficient for the angles of position on the edges A x A % , A t A 4 and 

 A, A t to be right : J ) 



from which then ensues: 



A 3S = A ui — A Zi = A ix = 2 n , 



^81 = "m > 



A t4 = « ls . 



If we do not count the rectangular elements and if Ave count 

 those which arc equal only once the doublerectangular hyperspherical 

 tetrahedron has 15 elements, namely a xa , « l3 , a 14 , a,„ a s4 , a 84 , a ls , 



f< 38' ft 34' ""IS' ^18' "^14' ^41' ""4 8' "^48' 



3. We now form, starting from a doublerectangular hyperspherical 

 tetrahedron A l A t A M A 4 of which we think the edges all <C±n, a second 



hyperspherical tetrahedron (Fig. 2) by prolonging the edges meeting 

 in A A = A\ through this vertex, namely the edge A v A 4 with a seg- 

 ment A\ A' a , the edge A % A 4 with il', A' s and the edge 4, .1 4 with 

 A\ A' 4 , so that A x A\ — A % A\ — A t A\ = \ x. 



By very simple geometrical considerations we find that the tetra- 

 hedron A\ A' 2 A' s A' 4 is again doublerectangular, that namely A\ A\ 

 is perpendicular to A'. 2 A\A\ and A\ A' 4 perpendicular to A\A % A\ ; 

 furthermore it is evident that the following relations exist between 



2 ) The signs used here I have derived from Prof. Dr. P. H. Schoute, Mehr- 

 dimensionale Geometrie, 1st vol., page 267, Sammlung Schubert XXXV, Leipzig, 

 G. J. GÖSCHEN, 1902. 



So I understand 



by « 12 the edge A\ A. 2 ; 



by « 12 the angle of position formed by the faces lying opposite the vertices 

 Ai and A-2, i. e. the angle of position on the edge A A A t ; 



by A 12 the facial angle having A x as vertex and lying in the face opposite A 2 , 

 i. e. the angle A s A x A±. 



