DERIVED EROM THE REGULAR POLYTOPES. 49 



the example [5'4'4'3'3'2'2'2'ri] of art. 55, considered as a descen- 



dent of [1 . . 0]. Of the five intervals V2, indicated by (d if d 2 ) } 

 (d s , d^), (d 5 , d G ), (d H , d v ), (d di d l0 ) the first corresponds to the original 

 interval of the symbol of coordinates of C 2 S 2) whilst according to the 

 theorem the others result from the four operations e 2 , e 4 , e lt e s . 

 But as the symbol winds up in a unit instead of a zero we have 

 to add e 9 . So we find e 2 e k e n e H e 9 C 2 S 2 \ 



77. By means of the operations e k we can deduce from C^ 

 all the possible poly topes the square bracketed symbols of coordinates 

 of which are characterized by the fact that there is an interval V2 

 between the first and the second digits. If we wish to deduce from 

 C 2 ^ 2) also poly topes with square bracketed symbols the two digits 

 d ± , d. h of Avhich are equal we have to follow M rs . Stott by intro- 

 ducing the operation c of contraction, the subject of which is the 

 group of limits (l) n _i of vertex import. With respect to this opera- 

 tion we can prove the theorem : 



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

 form deduced from C 2 \'p the largest digit of the symbol of coordi- 

 nates of this form is diminished by V 2." 



Proof. Here we have to consider the two cases of the symbol 

 of coordinates, winding up either in 1 or in 0. 



Case [l+(a+l)\/2, l-|-aV2, 1 -f öV2, . . ., 1]. — If we 

 replace 1 -f- (a -\~ 1 ) V 2 by 1 -\-aV2 the limit y represented by 



œ i = 1 + (a-\-\)V2, x 2i a%, . . . , w n == (1 -\-a V/2, l-\-ö V 7 2, . . .,1) 



passes into 



œ i = l-\-aV2, cr 2 ,^3,. . .,a? w = (l-f-aV^2,l+*V / 2,.:.,l), 



i.e. that limit (/) rt _i moves parallel to the axis OX ± towards the 

 centre O over a distance V2. Evidently application of this process 

 to all the limits y corresponds to a substitution of 1 -j- aV2 for 

 the digit 1 ~\~ (a -j- l)\/2 within the square brackets. Evidently any 

 two adjacent limits represented originally by 



ar ± =l-\-(a-\-l)V2, a? 2 ,a? 8> . . ^ n =-{l + aV2, 1 + 0V2,.;., 1), 

 a?2= l_|_(^-f-l)V / 2, w ±9 œ 3 ,. • .w n = (l -i-«V / 2,l +bV2,...,l), 



which were separated by the right prism 



x u x 2 = (l + (a-{-\)V2,\-\-aV2), ar 8 ,.-. . ,x n = {\A r bV2 i . . .,1), 

 pass into the two limits 



Verh. Kon. Akad. v. Wetensch. I e Sectie Dl. XI No. 5. 



E4 



