DERIVED FROM THE REGULAR POLYTOPES. 23 



take with either of the two signs the units printed in heavy type 

 only. Of the M 2 brought to the fore by this symbol itself the 

 centre is the point 2a iy 2a 2 , . . . , 2a n . So [2a i , 2a 2 , . . . , 2a n ] may 

 be called the "frame" of the net, and this symbol may be written 

 quite as well with round or even without brackets, as the faculty 

 of taking for the a { all possible integer values includes permutation 

 and changing of signs. 



62. If we consider the net N(M n (2) ) as a polytope 1 ) of S n + i 

 with an infinite number of limits {l) a which instead of bending round 

 in 8 n + ! fills JS n , we can apply to this polytope the expansions 

 e i ,e 2f . . . , e n and the contraction c, either separately or in possible 

 combination ; in this simple way the measure polytope nets e Y A(M n ), 

 e 2 N(M n ), etc. have been determined by M rs . Stott. We introduce 

 the corresponding analytical considerations by the following : 



Theorem XXXVII. "Let any expansion or expansion and con- 

 traction form (P) n of M n (2) be represented by the symbol of coor- 

 dinates [a lf a 2 , . . ., a n _i, «„]. Let M^ 2,,) be the measure polytope 

 with edge 2a concentric and coaxial to this (P) n and J\\M^ 2a) ) the 

 net of measure poly topes to which the M n (2,,) belongs. Let us suppose 

 in each of the oo H measure poly topes of this net a concentric polytope 

 equipollent to (P) n . Then the vertices of all the oo n polytopes 

 obtained in this manner cannot form together the vertices of a net, 

 if a differs from a A and from a i -f- 1." 



This theorem of a negative tendency can be proved thus. If we 

 call two (P)„ "adjacent" if the measure polytopes M n {2a) concentric 

 to them have this position with respect to each other, i. e. if these 

 M n (2,,) are in M n _ i {2a) contact, and we consider the limits (l) n _ x 

 of the highest import of any two adjacent (P) n deduced from the 

 common M n _ 1 (2ff) of the two M n {2 ° concentric with these (P) H , we 

 see at once that these limits g n _\ coincide for a = a lf whilst they 

 are at edge distance from each other and form therefore the end 

 polytopes of a prism for a = a ± -\- 1 . In all other cases two adjacent 

 (P) n are either too near to each other or too far apart. 



What we shall have to show farther is this that the vertices of 

 the cc H polytopes (P) n do form together the vertices of a net in 

 each of the cases a = d t and a = a ± -\- 1 . We prepare the general 

 proof of this assertion by indicating by the special case of the 

 threedimensional net of truncated cubes [1 -\- \/2, 1 -j- \/2, 1] 

 included in larger cubes M s (2a \ where a = 2 -j- \/2, how the other 

 constituents are to be found. This will give us occasion to introduce 



x ) Compare art. 39. 



