DERIVED FROM THE REGULAR POLYTOPES. 09 



e a e b . . . e s e t e n _\ M n = n _£-i 6 n _ s _i . . . c n _ b _^ <? n _ a _ 1 <? n _i C/ 2 w 



c e a e b . . . e s e t 6 n _ i M n = e n _ ( _ i e n _ s _ i . . . e n _ b _ i e n _ a _ i C 2 n 



G a e b ... e s e t l\l n = c e n _ t _i e ll _ s _ i . . . e n _ b _^ e n _ a _ i c n — 1 2 n 



c e a e b ... e s e t M n = c e n _ t _ { e n _ s _ i . . . e n _ b „ A e a _ a _ x C 2 n 



The proof of these general laws can be based on the remark 

 that each pair of polytopes forming the two members of any of 

 these four equations admits the same symbol of coordinates; if k is 

 the number of the symbols e ai e bi . . . , e s , e t these symbols of coor- 

 dinates are successively: 



n—t—\ t — s b — a a \ 



[1 4-(/-+l)V / 2,r+WI,I-f-(/-— l)Y/2,...,"f+V% T ] 



n — t — 1 t—s b—a . h 



[i + 1, k, .... ï, J^-i,. .,/— i, ..., Ï7TTT7T, Ö77»] 1 



n — t T—s b — a a 



[T+TV2, 1 +(/— l)V / 2,...,rFv 7 2, Ï ][ 



t — s b — a a \ 



\t , *_ i, . .,i—i,..., ÎTT.TT, öT^ö] ' 



By introducing the operation <? corresponding to the generation 

 of the regular polytopes starting from a point and representing 

 this point for M n by 7 J , for C\n by P ' we can unite thèse four 

 general laws in : 



Theorem LIII. "The two polytopes 



G a G b e c • • • e r &s ^t -1 Q ■> &u' G b' G c' ' • • e r' &s' e t' -M) 



are equal *) if and only if we have generally 



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



86. The influence of theorem LUI on the results laid down in 

 art. s 65 and 66 is evident. 



By polarizing an expansion or contraction form derived from the 

 cross polytope C 2 n of S n with respect to a concentric spherical space 

 (with oo n_1 points) as polarisator we get a new polytope admitting 

 one kind of limit (/)„_! and equal dispacial angles 2 ), to which corres- 

 ponds the inverted symbol of characteristic numbers of the original 

 polytope, etc. 



*) This theorem gives for M n and C 2 n what theorem XX11I contains about the two 

 differently orientated positions of the simplex; it holds not only for M n and C 2 n , n being 

 general, but also for the polytopes C120 an( ^ Qoo °f ^4 an ^ i u the same way there 

 exists a theorem analogous to theorem XXIII for the cell C04 of 84 in its two different 

 positions with respect to the system of coordinates. We shall have to come back to this 

 point in the fifth section of this memoir. 



2 ) Compare for S'i the foot note of art. 65. 



