DERIVED FROM THE REGULAR POLYTOPES. 33 



centre O of the expansion form, by which any of these limits gets 

 a vertex or some vertices in common with some of the other ones. 

 We illustrate this by the example of fig. 4. Here the results can 

 be tabulated as follows: 



c e ± T = O ^ c i e 1 T=2 1 1 c 2 e ± ^(impossible) 



c e,T = —T > , q<?,7 7 (impossible) -, c 2 e 2 T= T 

 c e, e 2 T = — e, T) c. { e, e 2 T=e ± T J c 2 e, e 2 T = e, T 



In this small table the negative sign indicates the inverse orien- 

 tation; the impossibility of c ± e 2 T and c 2 e i T is caused by the fact 

 that the polygons, in the first case of edge and in the ^second of 

 face import, forming the subject of contraction, have already a 

 vertex or an edge in common. 



But we can also account for the impossibility of c^ e 2 T and c. 2 e i T — 

 and for other similar results — by remarking that the contraction 

 c k undoes the expansion e k and that it can be applied, when and 

 only when the expansion form has been obtained by applying amongst 

 the different expansions the operation e k . So c is the only contrac- 

 tion operation which we have to introduce in order to be able to 

 deduce all the forms with a symbol satisfying the law of theorem I. 



As we will use henceforth exclusively the operation c Qi the sub- 

 script of the c can be omitted. 



2L We now prove the general theorem: 



Theorem X. "By applying the contraction c to any expansion 

 form the largest digit of the zero symbol of this form is diminished 

 by one". 



Proof. The groups of polytopes of vertex import of the expansion 



form represented by the zero symbol {a~\~\ 9 a, b, c, . . , 0), where 



a >"^ > c . . . , is found by putting so i = a-\~ 1 , leaving (a, b, c, . . , 0) 



for the other coordinates. By diminishing a -{- \ by one we get an 



other form with the zero symbol (a, a, b, c } . . , 0) possessing also 



polytopes of vertex import represented by (a, b, c, . . , 0). So the 



polytopes g Q of vertex import of the second form are congruent to 



and equally orientated with the corresponding polytopes of the first, 



but they lie in spaces x i = a nearer to the centre than œ-, = a-\- 1. For, 



if p is the extension number of the original form (a -j- 1 , a, b, c, . . , 0) 



of 8 n , and therefore p — 1 that of the new form (a, a, b, c, . . , 0), 



the true coordinate values of w i corresponding to the values a -f- 1 



p — 1 

 of the first and a of the second zero symbol are a -\~ \ — - 



Verhnnd. Kon. Aknd. v. Wetenscb. (l 9te Sectie) Dl. XT. C 3 



