1 2 ANALYTICAL TREATMENT OF THE POLYTOPES EEGULAELY 



two regular hexagons in parallel planes, or we have to combine the 

 system a? 4 ,# 5 = (10) with each of the six possibilities 321, 312, 

 231, 213, 132, 123 for x it x 2 ,x 3 contained in x ± , x 2 , x 3 = (321) 

 giving six parallel edges of the same length. So in order to prove 

 that the result is a hexagonal prism P 6 we have only to show yet 

 that the planes of the two hexagons are normal to the six edges. 

 But this follows from the theorem in art. 5, as the planes of the 

 hexagons are parallel to the plane x k = 0, x 5 = 0, i. e. to the face 

 A ± A 2 A 3 of the simplex of coordinates, while the six edges are 

 parallel to the line x ± = 0, x 2 = 0, x 3 = 0, i. e. to the opposite 

 edge A k A- of that simplex. 



The hexagonal prism jP 6 obtained in this manner occurs ten times, 

 as for the subscripts 4 and 5 in the equation x k -j- x 5 = 1 we can 

 take any combination of the five numbers 1, 2, 3, 4, 5 by two. 



c). For x i -\- x 2 = 5 we have either x ± = 3, x 2 = 2 or x\ = 2, 

 x 2 = 3 and in both cases x 3 , x hi x 5 = (HO). So we find here ten 

 prisms P 3 . 



d). Finally x. v = 3 gives x 2% x 3 , # 4 , x 5 = (2110); so we find here 

 five CO. 



All in all we have got the limiting polyhedra 



5 tl\ 10 P 6 , 10 jP 3 , 5 CO; 



so their number is 30. 



The number of the faces is easily found. As the numbers of 

 faces of tT, P Q , P s , CO are respectively 8, 8, 5, 14 we get 



|(5 X § + 10 X 8 + 10 X 5 + 5 X 14) = 120. 



So the final result (e, k, ƒ, r) x ) is (60, 150, 120, 30), in accor- 

 dance with the law of Euler. 



Remark. In the case of the simplex (1000) of S± we would have 

 to consider the equations 



a) . . . x ± -f- x ± -j- x z -j- x /k . == 1 or x b = , 



b) . . . x i -\- x 2 -f- x 3 = 1 or # 4 -\- x 3 = , 

 <?) . . . ^ -(-- 1^. 2 =1 or c^ 3 -j- a? 4 -\- x :) = , 

 d) . ._. x l '= I or ^2 ~f~ ^3 ~h ^4 ~\r a 'b = , 



containing — as we remarked already in art. 6 — respectively a 

 limiting tetrahedron, a face, an edge, a vertex of the simplex £(5). 

 Therefore in the expressive language of M rs . Stott the limiting 

 polyhedra of (32110) are distinguished, as to their orientation, as 



^•This is the general symbol I used for S„ in my textbook; here e, /c, ƒ", r stand 

 for "Ecke, Kante, Flâche, Ka.um," i.e. for vertex, edge, face, limiting body. 



