DERIVED FROM THE REGULAR POLYTOPES. 103 



the last digits of £[53311], the symbol of A, and likewise why 

 the other syllables [l]V / 2 and [l'l]\/2 must correspond in the 

 same manner with either of the symbols [3'2'1'1'1] and [2 , 2 , 1 , 1'1] 

 of B and C. Also why B must be a prism and E a prismotope, 

 in connexion with the faculty of inverting the signs of \/2 in the 

 case of B, and of 2 -j- V / 2 and V2 in the case of E, these in- 

 version having no influence whatever on the distance of the vertices 

 obtained of the new origin which is to be the centre of the gap 

 filling polytope. 



Moreover the processes themselves indicate under which circum- 

 stances the prism B and the prismotope E present themselves. 

 If the symbol of B winds up in zero the second syllable of the 

 symbol of B is [0], i. e, the prism is lacking; but we know from 

 theorem XXXV that the last dmit of the svmbol of B is zero, if 

 the operation e k has not been applied to B. Likewise, if the last 

 two digits of the symbol of B are zero, the second syllable of 

 E is [0, 0J, i.e. there is no prismotope E, and the last two 

 digits of the symbol of B are zero, if neither e 3 nor è 4 has been 

 applied to B. 



Finally it is evident why we cannot add a fourth process to the 

 three considered ones and subtract 5-j-V / 2, 5-|-\/2, 0, 0, 0. 

 For then we would get 1 , — 1, [2 -f-\/2, 2-f-\/2, V"2], leading 

 to £ [11] [l'l'l]Y/2, i.e. — as £[11] is an edge instead of a 

 face — to a limiting body and not to a limit (/) 4 . 



107. Hmjjd. nets in 8 n . — It is easy to see how the processes 

 of the preceding article must be extended to S ni as the algorithm 

 always remains the same and the number of the subtractions has 

 to be augmented until only three digits of the subtrahend differ 

 from zero. So, if we indicate by A^ k) the constituent obtained by 



n — k k 



the subtraction of pp. .p 00.. we can formulate the general 

 result in the following theorem : 



Theorem LXV1II. — "In any net deduced from NH n we find, 

 besides the three principal constituents A, B , C always present, 

 under certain circumstances one or more prismotope» A^ for h = 1, 2, 

 . . . , n — 3, which may be called accidental constituents. The pris- 

 motope A (k) presents itself if — and only if — one or more of 

 the expansions e n _ k , e n _ k + i , e n _ i have contributed to the trans- 

 formation of Cr n into B; the two syllables of its symbol are the 

 last n — ■ h digits of A between square brackets preceded by £ and 

 the last k digits of B between square brackets/ 5 



