DERIVED EKOM THE KEGULAB, POLYTOPES. 5 3 



determined by making the sum x x -f- x 2 -\- ... -\- x v maximum , 

 the v digits which are to make that sum maximum consisting of p r 

 times r, p t _ x times r — 1, etc. and p (JL of the p m digits r — l-\-m. 

 Under these circumstances the poly tope (P) a is represented by the 

 symbol 



P r Pi—i Pm Pm-1 ?0 <V— 1 Pl+1 Pj—Pr 



( r , r — 1 , . . . , r — l-\-m , r — l-\-m — 1 , . . , r — / , r — I — 1 , . . , 1 , ), 



while this symbol passes into that of {l)^\ by the introduction of 

 intermediate brackets between the digits a v r -f- r — I -\- m and 



a 



v +1 



r _|_ r — / _|_ m> i. e. (/)^i b ! is represented by 



P r Pi-i Pp Pm-Pp Prn-i Po Pr—1 Pl+1 p l p r 



(r,r — l,..,r — l-\-m)(r — l~\-m,r — l-\~m — l,..,r — l,r — / — 1,. . ., 1, 0) 



• 



under the condition that of the two parts of this symbol the first refers 

 to the coordinates œ i} x. z , . . . ,œ v and the second to x v+i ,x v+2i . ..,x n+[ . 

 The determination of a second polytope ÇP) b of the list containing 

 {l)"n-\ as limit must be guided by the remark that in each of the 

 two parts of the symbol of (/)^'_i considered for itself we may 

 transfer the same amount from the immovable parts of the digits 

 to the permutable ones. But in order to obtain a symbol satisfying 

 the law of theorem I, when the intermediate brackets are omitted, 

 we have moreover to select these two amounts in such a way as 

 to obtain a set of permutable parts containing n -\- 1 integers 

 distributed over r different ones, succeeding one another with dif- 

 ferences unity. So we can either diminish all the digits r, r — ] , . . . , 

 r — I -\- m included between the first pair of brackets by r, coun- 

 terbalancing this by increasing a it a 2 , . . . , a v by unity, or — 

 which comes to the same — increase all the digits r — l-\-m, 

 r — I -\- m — 1 , . . . , included between the second pair of brackets 

 by r, counterbalancing this by diminishing a v+1 , a v+2 , . . . , a n+i by 

 unity; that these two results differ in form only can be shown 

 by remarking that the first passes into the second if we increase 

 all the digits 0, — 1, . ., — l-\-m, r — l-\-m, r — l-\-m — 1,. .,0 

 by r, counterbalancing it by diminishing a x -\- 1 , a 2 -\- 1 , . . ., 

 a v -\- 1 , a v+i , a v+2 , . . ., a it+i by unity. So we find one and always 

 one second polytope (P) 6 with (%_i as limit, represented by the 

 symbol : 



Pr_ ^_i_ p (M Pm-Pp ' Pm-1 Pp Py-\ Pl+1 PV P r 



(0, — 1/. -, — I -\-m,r — l-\~m,r — l-\~m — 1,. . , r — l,r — / — 1, . ., 1, ) 



(« + +l,a +1,...,^+!, a v + l,a y + 2 , ••• a n + \\ 



