DERIVED FROM THE REGULAR POLYTOPES. 99 



So from this table we deduce A 3 = D s by remarking that the 

 digits of the first syllable of D are the last three digits of the 

 unique syllable of A ; in order to facilitate comparison of A and D 

 we have reversed the order of A, B, C. 



So we find A t = B 3 (or rather A t = — B 3 ) as we get the same 

 form by placing the four digits of A between round brackets after 

 having taken the last unit with the negative sign and by placing 

 the digits of B, multiplied by V/2, between round brackets; etc. 



105. Before passing to the case n = 5 we will put the last two 

 small tables of the preceding article on duty as to the general 

 results they may suggest for n > 4. 



We begin by fixing our attention on the extreme case of the relation 

 between the two constituents A and C, being governed in the case 

 e 3 NE k by a vertex only. Here A , the limit of vertex import of 

 A y is still a vertex; so we have to accept for (7 the polytope deduced 

 from Cr k which admits as limit C 3 of body import a vertex and 

 this is the eightcell ce 3 Cr k . The same remark holds for e 2 NH 3 

 already, i.e. for the third of the four cases treated in art. 103. 



But the first of our two tables, i. e. the table of contacts, 

 suggests a remark of much wider scope. We deduce it from the 

 fact that each constituent with three kinds of limits (/) 3 is in contact 

 with the three others, whilst the only one with four different kinds 

 of limits (/) 3 is in contact with the three others and with itself. 



This fact suggests that in space S n we will want in all n different 

 constituents A, B, C, . . . , of which B only admits at most n different 

 limits (/)„_! and all the others at most n — 1. We have used this 

 suggestion as working hypothesis and found by its help the sixteen 

 limpd. nets of Sf 5 ; this was an easy task : as theorem LXVII gives the 

 three principal constituents A , B, C and the prism D can be deduced 

 from them, the table of contacts shows immediately which limits 

 (/) 4 remain uncovered and these limits reveal the character of the 

 fifth constituent. l ) 



'But there is an other method of deducing the new constituent, 

 much more capable of being extended to 8 n , viz the determination 

 of their coordinate symbols by transformation of the net symbol to 



l ) It may seem in accordance with this suggestion that in the cases e. z NH 3 and 

 e L e 2 NH 3 of S 3 we have found no fourth constituent i.e. no prism, though they require 

 the operation e 3 with respect to the two groups of HM 3 of different orientation , driving 

 these groups asunder. But this not appearing of the prism is rather due to the fact 

 that two adjacent HM 3 of difierent orientation are in contact by an edge only instead 

 of by a face , so that the separation intercalates a square instead of a prism. 



7* 



