DERIVED FROM THE REGULAR POLYTOPES. 19 



But the extension number of the first poly tope is 1 -)- 5V/2, that 

 of the second is 5. 



Remark. In the application of the method of measuring the 

 amount of truncation introduced for the simplex to the measure 

 polytope we experience that the truncation fraction may become an 

 improper fraction. This means that the point of intersection R of 

 the truncating space /S n _ i with the edge PQ lies on PQ produced 

 at the side of Q. 



If we wish to avoid this inconvenience we can determine the 

 amount of truncation in the following new way. If O is once more 

 the centre of the polytope and M the centre of the limit M L {2e) of 

 the extended measure polytope M^ 2£) at which the truncation is to 

 take place, whilst the truncating space S n _ i normal at OM cuts 



PM 



OM in P. we may consider — - — as measure for the amount of 



J OM 



truncation. Then we find 



n — Jc — 1 



PM ^ — ® m '° ~ ? m ' { 



OM (n — k) m\ 







from which it ensues that the new truncation fraction is deduced from 



2 

 the old one by multiplication by — -. 



But instead of altering our method of measuring the amount 

 of truncation we prefer to put up with the inconvenience indicated. 

 So in Table IV the truncation numbers are indicated, after the 

 extension number where q =\ -j- q \/2 and q" = q \/2, according 

 to the original system in column seven. 



D. Expansion and contraction symbols. 



58. We now prove the theorem: 



Theorem XXXII. "The expansion e,,, {k = 1 , 2, 3, . . . , n — 1), 

 applied to the measure polytope M^ 2) of JS n changes the symbol of 



coordinates [1 , 1 , . . . , 1] of that polytope into an other symbol 



which can be obtained by adding \/2 to the first n — h digits. 



Proof. The operation of expansion e k is performed by imparting 



to all the limits M k (2) of M n {2) a translational motion, to equal 



distances away from the centre O of M n (2) , each M L (2) moving in 



the direction of the line OM joining O to its centre M , these M k 2 



remaining equipollent to their original position, the motion being 



2* 



