30 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



k n — A; 



i.e. the symbol (2 11. M 00 . . 0), what was to be proved. 



If by the operation e k the limits /S >(4) (k -j- 1 ) of S (i) (n -f- 1) are 

 moved away from the centre O to a distance equal to À times the 

 original distance, the extended simplex S^ x \n-\-\), the limits 

 jS (x \k-\-l) of which will contain these /S (i) (7c -j- 1) in their new 

 positions, will be a simplex £(*)(# ~\- 1), i. o. w. À = £-|-2 is the 

 extension number of the new polytope. This comes true, for accor- 

 ding to theorem V the sum h -j- 2 of the digits of the zero symbol 



n — A' 



(2 11..1 00... 0) is the extension number. So we find by 

 the way: 



Theorem VII. "In the expansion e,, the limits /S (i) (k-\- 1) are 

 moved away from the centre to a distance equal to k -j- 2 times 



the original distance." 



Remark. We may express the influence of the operation e k on 



n 



the symbol (1 00 . .0) of the simplex /S 0) (n -f- 1) presenting only one 

 unit interval between the first and the second digit by saying that 

 it creates a second unit interval between the h -\- I st and the /; -\- 2 nd 

 digit. This remark which holds also with respect to the symbol of 

 true coordinates will be of use in the following articles. 



19. Theorem VIII. "The influence of any number of expansions 



e k> e u e m> • • °f $ (1) (^ ~h 1) on ^ s zero symbol (1 00 . . .0) is found 

 by adding the influences of each of the expansions taken sepa- 

 rately". 



Indeed this gives (compare the small table n = 4 under art. 3) : 



e ± e 2 S(b) = (32100), e. k e z /S(5) = (32110), e 2 e a S 6 = (32210), 



e i e 2 e 3 JS(b) = (43210). 



Proof. We begin to prove the theorem for the case of two 

 operations of expansion only. 



It is stipulated expressly by M rs . Stott that in the succession of 

 two operations of expansion the subject of the second is to be what 

 its original subject has become under the influence of the first. So 

 in the case e.> e i T of the tetrahedron (fig. 4") the original triangular 

 subject of e 2 is transformed by e x into a hexagon (fig. 4'') and now 

 the hexagon is moved out, in the case e x e., T the linear subject 

 of e x is transformed by e 2 into a square (fig. 4 C ) and now the 



