DERIVED FROM THE REGUIAR POLYTOPES. 27 



PR 



the truncation fraction r h is the ratio — — and now we have the 



PQ 



theorem : 



"If a line cuts any three parallel spaces /S (l) n _ i , S 2) n _ 1 , /S {3) n _ l 



of S n represented by the three equations b ± x\ -\~ b 2 x 2 -\~ 



-\- b q x q = c t , {t = 1, 2, 3) in the points P, Q, R, we have 



PB c x — c 3 

 PQ q — c 2 



For this is obviously true for the edge A x ^ q + i of the simplex 

 of coordinates, the values of x [ for the three points of intersection 

 with this line being determined by the relations b x x x — c L , (t = I , 2, 3); 

 therefore it is true for any transversal, according to a well known 

 theorem, already used implicitly in the preceding article, the ratio 

 in question being the same for all possible transversals. 



Now in the case under consideration the spaces JS (i \_ x , S <:2 \_ lf 

 # (3) n _ x may be represented by the equations x x -j- x 2 -[-.•• ~\~ œ ic + i — c n 

 (7=1,2,3), where c x and c 2 are the maximum and minimum 

 values of the left hand member with respect to the vertices of the 



extended simplex (w, 00 . . . .0), while c 3 is the maximum value of 

 the same expression with respect to the vertices of (m i9 w 2 , w 3 , ... ,0). 

 So we have 



c x = in, 0.2=0, c s = m -\- m x -\- , . . + »?* 



giving tor the truncation traction the result 



m 



and therefore T k = m — (m -\- m x -(-... -j- m k ). 



So we find in the case of the (P) ;) of art. 12 represented by 

 (5443322210) m= 26, r = 21, 71=17, r 2 = 13, r 3 = 10, 

 T 4 = 7, t 5 = 5, t 6 = 3, r 7 = 1. 



For n= 3,4, 5 the extension number and the truncation numbers 

 are indicated in Table I, the seventh column containing the exten- 

 sion number e, the eighth column giving what may be called the 

 "truncation symbol". So in the case of e x e 2 <? 4 S(G) the extension 

 number is 11, the truncation symbol is 7,4,2,1 where these 

 numbers represent successively the values of r , T l ,T 2 ,T 3 ; so we 

 find mentioned here a truncation y 7 T at the vertices, T 4 r at the edges, 

 y 2 y at the faces and A^ at the limiting bodies. 



