48 \N\LYTICAL TREATMENT OP THE POLYTOPES REGULARLY 



n — 1 n — 4 



[1 +J-V%J-V/2 ) ... ) 1V / 2]V2=[1 +V2, 1, L,..'.,l] 



in accordance with the statement of the theorem. 



By the way we find : 



Theorem XL VII. "In the expansion e,, the limits 8(jfc -\- 1) of 

 CV 2) are moved away from the centre to a distance equal to k 

 times the original distance for k <] n — 1 and to a distance equal 

 to 1 -\-\nV% times the original distance for k = n — 1". 



This comes true, for this extension corresponds in both cases to 

 that deduced from the sum of the digits of the symbol of coordi- 

 nates of the new poly tope. 



As the distance O M was 1/ — - — , 



for /•<<>« — 1 and -^—p- * ov k = n - 



V n 



76. Theorem XLVIII. "The influence of any number of expan- 



n - 1 



sions e ki e h e ini . . of C 2 n(2) on its symbol [100. .0] V2 is found 

 by adding the influences of each of the expansions taken separately". 



Proof. Here likewise, in the succession of two expansions the 

 subject of the second is to be what its original subject has become 

 under the influence of the first. So in the case of e 2 <?i O of the 

 octahedron (fig. 17 a ) the original subject of e 2 (the triangle) is 

 transformed by e. { into a hexagon (fig. 17 b ) and now the hexagon 

 is moved out, in the case e 1 e L O the linear subject of e x (the edge) 

 is transformed by e 2 into a square (fig. 17°) and now this square 

 is moved out; in both cases the result (fig. 17 d ) is the same, a 

 tCO. In general, for /• > /, in the case e k e L C 2n { ' 2) the subject 

 S(k-\- 1) of e,, is transformed by e L into the form e t S(& -\- 1) of 

 the same number of dimensions, while in the case e l e k (7 2?l (2) the 

 subject /S y (/-j-l) of e t is transformed by e k into an h — 1 -di- 

 mensional limit g { of import /. Here also the geometrical condi- 

 tion : "that the two new positions of any vertex shall be separated 

 by the length of an edge" leads to the ordinary composition of 

 the motions of the centre accord. ng to the rule of the parallelogram 

 in the case of two expansions, etc. 



By the way we find: 



Theorem XL1X. "The operation e k can still be applied to any 

 polytope deduced from CV 2) in the symbol of coordinates of which 

 the k -f I s ' and the k -f- 2 m/ digit are equal." 



We indicate by means of this theorem the expansion symbol of 



