( 497 ) 



centres of the bounding simplexes Sf_., Sn~p> S',., S'^—p with these 

 points as vertices. Then the five points B, C, B' , C' , P lie in sucb 

 a w^ay upon the same right line, that B and C' lie on one side of 

 P and B' and C on the other side, and we have 



p BP—{n-p)PC ) AP _BP _CP 

 (n-p) CP= p . PB' y PA' " PB' " PC' ' 



We can now assert that the bounding simplex S/j of the vertices j^ 

 of Sn lies entirely or partly inside S'n when B is between C' and P, 

 whilst S^j lies entirely outside S'n when C' lies between B and P. 

 In other words: as AP increases, the bounding simplex Sp of Sn 

 comes entirely outside Sn when B coincides with C' and the spaces 

 Spij—\ and Sp,.—p—\ of Sp and S'n—p, crossing each other in general 

 entirely perpendicularly, become incident because they get the point 

 B = C', then common centre of gravity, as point of intersection. 

 Under the condition BP ^=^ C'P follows from the equations 



BP _ 71— p PC _ AP 



PC'~~^ ' CP^PA' 



the relation 



{n-p)AP = p .PA', 



which shows that P must coincide with the p'^* dividing point Ap 

 of AA'. 



10. If P coincides with Ap the spaces Spp—\ and Spn—p—\ of 

 Sp and S'n-p have, as we saw above, the common centre of Sp and 

 S'„-p in common. As this point of intersection of Sp and S'n—p 



P 

 becomes vertex of the section, — if we call this again ~ {Mn) in 



n 



connection with preceding investigations — the theorem holds : 

 "The centres of the [ J bounding simplexes ^/, of a regular simplex 



Sn{py"^) form the vertices of a polytope congruent to - {Mn) for 



n 



p = l,2,..., n-1." 



For even n ^^Inl we have specially : 



f2n\ 

 "The centres of the I ^ I bounding simplexes Sn' of a regular 



simplex Sin' {n']/2) form the vertices of a Z)2>/— i •" 



11. If P lies between Ap and Ap^i the vertices of the section 

 of the two simplexes Sn and S'n are furnished by the points of 

 intersection of each bounding simplex Sp-^i of Sn with the p -\-l 



