DERIVED FROM THE REGULAR POLYTOPES. 5 



a n , must be unity, which gives a n = j>, unless P and Q coincide 

 which happens for a n = 0. So in the case p = n we liave a n = -A-. 

 In the case p <C n we consider the points 



JT . . . 00^ ==z dn) 00 2 ■ ^ 5 00 <^ ~~ @\y 00 1^ ~" = ~ &2* ' ' ' 

 VX> • • • cZ?| === U , tl?2 ~~~' ^*/;J ^?3 ^ == ^1? ^4 "F" of > J ... 



passing into each other by interchanging x v and x ± . The distance 

 a p \/2 between these points is unity for a p = tV^2. 



Symbol + -J- [#i, « 2 » • • ■ » a n}> Here a n differs from zero; for the 

 supposition « n = is incompatible with the division of the vertices 

 represented by the symbol [a i} a 2 , . . . , a n ~\ into the two groups 

 + -j- \a A , 0%, . . . #,J, the inversion of the sign of zero having no 

 effect whatever. 



Here the point 



JT ... iJL^ u ? c ? /i 1' 3 jî 9? ... 



must be considered in combination with the points 



hi • • • 00 ^ Ct n \, 00 2 CL n > 1^3 @n—2> ' • ' 



Jaj . . . X\ ■ ~ ct,, , c2/.> " ~ Ct ,j jj 663 iZ., 9, ... 



corresponding with it as to the coordinates oo- è , oo k , . . . oo n + i , as these 

 points Q and R are the nearest ones to P obtainable either by 

 interchanging two digits or by inverting the signs of two digits. 

 Now we have under these circumstances 



PQ 2 = 2 [a n — 0„„i) 2 , PR 2 = 4 {a\ -f a\_ ± ), 



from which ensues PQ << Pi£. So we must have PQ == 0, PR = 1, 

 giving « n = #„_! = -| V/2. 



48. In the case of the first symbol [a i} a 2 , . . . , # n ] we are confronted 

 with two possibilities, as we have to choose between a H = A and 

 a n = 0, i. e. between a group containing the measure polytope 

 jr, |-, ... -£] and an other group containing the cross polytope 

 ^ V/2, 0, ... 0]. Do the two regions lying on different sides of 

 the limiting demarcation line cover the same ground as the group 

 of the measure polytope on one side and the group of the cross 

 polytope on the other? The answer to this question depends on 

 the manner of deduction of these two groups. If we follow closely 

 the geometrical manner of deduction developed by M rs . Stott the 

 contraction forms derived from the measure polytope do possess 

 coordinate symbols winding up in zero, whilst on the other hand 

 the form derived from the cross polytope by means of a set of 

 expansions under which e n _ i occurs are represented by coordinate 

 symbols containing no zero. These two exceptional facts which 



