DERIVED FROM THE REGULAR POLYTOPES. 71 



on #0 (n -\- 1) (1) and to investigate under what circumstances the 

 zero symbol of the one is the inversion of that of the other. If 

 each of the two products 



^a ^b @c • • • &r ^s &t » ^a' ^b' "c' • • • ^r' ^s' ^t' > 



where we have 



a<b<c<+ . .<r<s<t , a < b' < c < . . . < r' < s' <t' 

 bears k factors, the two results are represented by 



«-fi b — a c—b s — r t — s n — t 



(k,k,. .k } k—\,k—\ y . Jc — l,/c—2,/c—2,. Jc—2,. . .22. .2,11. .1,00. . 0) 



and the same expression in which the a, b, c, . . r, s, t are dashed. 

 So the conditions are 



a -J- 1 = n — t' , b — a = t' — s , c — b = $' — r', . . 

 . . , s — r = c — b' , t — s = b' — a, n — t = a -\~ 1 , 



giving immediately 



a-\-t' ' = b-\-s ' = c-\-r '= . . . =r-\-c =s~\-b' = t-\-a =n — 1. 



So the first part is proved and the second is deduced from this 

 by suppression of the dashes. In this second part the unpaired 

 middle expansion e n _ ± occurs, if and only if both n and h are odd. 



It is an easy task to return to the e and c symbols referring 



n 



to the simplex (1 00 . . 0); to that end we have to omit the e 

 symbol and to add c to any expansion form, where e is lacking. 

 In doing so we arrive for n = 3, 4, 5 by means of the first part 

 of the theorem to all the cases, as e 2 e 3 S(b) = — e 1 e^8{^)), of 

 equal and concentric polytopes of opposite orientation mentioned in 

 the table, and by means of the second part to all the cases, as 

 ce 2 8(6), of central symmetry. 



39. In the joint paper of M rs Stott and myself quoted in the 

 preceding article , the notion of reciprocal polytopes has been exten- 

 ded to that of reciprocal nets by considering a net of 8 n as a 

 polytope with an infinite number of limits (l) n in 8 n+1 . In this 

 case the centre of the circumscribed spherical space of the polytope 

 lies at infinity in the direction of the normal to the space S n bea- 

 ring the space filling, from which it ensues that the poles of the 

 limits (l) n coincide with the centres of these polytopes. So one 

 obtains a net reciprocal to a given one by considering the centres 



