( 498 ) 



bounding simplexes S'n-p of >S"„ which have the property of counting 

 among their n — j^ vertices only one vertex corresponding to a vertex 

 of this Sp-\^i ; in each bounding simplex Spj^-x these /> + 1 points of 

 intersection form the vertices of a new regular simplex S^j^-i which 

 is concentric to the assumed one but oppositely orientated. We 

 determine the length of the edges of this new simplex, for the definite 

 case that F lies just in the middle between Ap and Ap-\-\, with the 

 aid of reflections in quite close connection with the preceding. 



If B, C, B', C' (fig. 3) are successively the centres of gravity of 



FiR-. 3. 



the bounding simplex >S^,_|_], of the bounding simplex /S„_;,_i of the 

 remaining vertices of Sn and of the bounding simplexes S'p^\, and 

 S'n—p-i of the groups of vertices of /S\ corresponding with the vertices 

 of Sp-^\ and S'n-p—], these points lie on a same right line through 

 P again, viz. -. B and C' on one side and C and B' on the other 

 side of P. If furthermore M and M' are corresponding vertices of 

 Sp-\-\ and S'p-{-\ these points lie in parallel normals erected in B 

 and B' on BB' and the line connecting M and M' passes through 

 P. The point of intersection J^ of B3f and CM' is the vertex of 

 ^^+1 corresponding to the vertex M of Sp^\. From CM and CM' 

 being parallel follows 



BN _ C'B _ C'P — BP 



MB~BC~BP-^FC' 



whilst the relations 



AP _BP _CP _ 2jo 4- 1 

 PA' ~ PB' ~ PC' ^ 2n — 2p — T 



and 



BP _B'P _7i—p—l 



PC~PC'~ 2^ + 1 

 enable us to express CP and BP in PC. Substitution gives the 

 result 



