DERIVED FROM THE REGULAR POLYTOPES. 



65 



In the following table for S 5 we give only the indices of the 

 ^-symbols which are to be applied to the constituent of the prin- 

 cipal net in order to obtain all the other ones. 



Constituents 



I 

 II 



III 

 IV 



V 



VF 



VF 

 VII 



Ver- 

 tex 







1 







2 

 1 



3 



2 



4 



3 



gap 

 4 



1 



2 



3 



4 



— 







2 



3 



4 



— 







1 



4 



— 







1 



2 



3 



3 



4 



— 







1 



2 



1 



02 



13 



24 



3 



04 



4 







01 



12 



23 



34 



12 



23 



34 



4 







01 



2 



03 



14 



2 



03 



14 



13 



24 



3 



04 



1 



02 



34 



4 







01 



12 



23 



3 



04 



1 



02 



13 



24 



14 



2 



03 



14 



2 



03 



23 



34 



4 







01 



12 



24 



3 



04 



1 



02 



13 



VIII 



IX 



X 



XI 



XII 





Constituents 









1 



2 



3 4 



12 



023 



134 



24 



03 



14 



02 



013 



124 



23 



34 



04 



01 



012 



123 



123 



234 



34 



04 



01 



13 



024 



13 



024 



13 



24 



03 



014 



12 



023 



124 



23 



034 



14 



02 



234 



34 



04 



01 



012 



23 



34 



14 



02 



013 



134 



24 



03 



014 



12 



123 



0234 



134 



024 



013 



124 



023 



0134 



124 



023 



134 



024 



013 



0124 



123 



234 



034 



014 



012 



0123 



1234 



234 



034 



014 



012 



1234 



0234 



0134 



0124 



0123 



Ver- 

 tex 



gap 



014 



034 



234 



012 



024 



134 



013 



123 



124 



023 



0124 



0134 



0234 



1234 



0123 



We only remark here that the number of ways in which the 

 principal net can be transformed into any other one is equal to 

 the number of different cyclical permutations of the partition symbol 

 of the latter, if we make allowance for the fact that two of these 

 ways may be essentially the same as they pass into each other by 

 interchanging the different positions of the constituents without 

 central symmetry, etc. 



34. In the outset of this paragraph (art. 22) we have excluded 

 prismatic nets, restricting ourselves to uniform ones; moreover we 

 have disregarded 1°. all cases in which not all the constituents 

 are of simplex extraction (the hybridous nets of art. 24, b) and 

 2°. the nets with two systems of vertices (art. 25, c). Now that 

 our general considerations about simplex nets are come to a close we 

 wish to add a few words about these two exceptional groups of nets. 



Hybridous nets. In order not to become too circumstantial we 

 only mention the decomposing symbols of the three plane hybridous 

 nets indicated in art. 24. They are (in the notation of art. 24): 



Verhand. Kon. Akad. v. Wetensch. (l Bte Sectie) Dl. XI. 



C 5 



