54 ANALYTICAL TKEATMENT OF THE POLYTOPES REGULARLY 



We now pass to the determination of the centres G ai G b , G L of 

 (P) a , {P) b ,\l) c ! i -\ which are collinear, as G t G u and G l G b are normal 

 in the centre G t to the space S n _ ± bearing (/)£i_i, in order to 

 prove that G { lies between O a and G b , i. e. that (P) n and (P) b lie on 

 different sides of that space <8' nr _ 1 . If we consider the a? n+1 of these 

 three points we find for 



G a . . . ra n+i + ^-— - r/> r -f- (r— 1) A-i + • • • + An (r — / + m) 



-)-. . . + /?o(r— O + ZV-ifr— ^— *) + • • - + A+i » 

 G & . . . ra n+1 -\ r— r — A_i — 2/?i_ 2 — . . . — putf—m) 



n 



+ 1 



1 I 

 (An— /V0( r_ H-»0 + ••• + A<>— O+A-i^— ^— !) + ••• +A+i • 



^ . . . ra n+i -j- 



Now in comparing the three vaines œ (a \ af b) , af a,b) of œ n + i we 

 can omit the common part ra n + i . But then if we write y n + 1 for 

 a? w + 1 — ra n + i it is evident that we have if b) < y (a ' b) << y (a) . For y fl ' b \ 

 yisO f y<P) are ar ithmetic means, y (ai ^ of a series S of positive integers 

 1,2, . . . , r — l-\~m , each of them taken a certain number of 

 times, y rt) of an other series of integers consisting of S and of 

 positive numbers r — l-\- m, r — l-\-m-\-l, . . . . , r — 1 , r equal 

 to or larger than the largest of 8, y <b) of a third series of integers 

 consisting of 8 and negative numbers. So G a and G b lie on different 

 sides of the space 8 n _ i bearing {1)^—1, i- e. the system of polytopes 

 contained in the list admits no holes, every limit (/) H _i of an 

 arbitrarily chosen polytope P a being covered by an other polytope P b . 



We have still to show that no two polytopes of the net can 

 overlap. We do so by simply remarking that the vertices of 

 the polytopes of any group of constituents form together the total 

 system of vertices of the symbol derived from the partition cycle *) 

 (see above at the beginning of the treatment of the case r > 1 

 under consideration), as each of the groups of constituents of the 

 list has been deduced from that symbol according to the processes 

 of art. 27. For — while overlapping of polytopes of the same group 



*) This fact can also be put on duty in the proof about the position of two polytopes 

 with common (On — l on different sides of that limit. 



