22 ANALYTICAL TREATMENT OF THE POLYTOPES KEGULARLY 



(210), (443) (32) (2210), (443) (22210), — (4332221), — (433222) 

 (10), — (43322) (210), — (4332) (2210), — (433) (22210), — (43) 

 (322210), — (3322210). 



13. We will insert a few remarks about the character of the 

 limiting (JP) 6 obtained. 



In the case (5443322) of one syllable we find a non specialized form 

 (3221 100) which will prove to be an e 2 e k 8(7) in M rs . Stott's language. 



In the cases (544332) (21) and (544332) (10) we find right 

 prisms on (3221 10) = e 2 e k 8(6) as base. 



In the case (54433) (221) we find a prismotope the constituents 

 of which are a (21100) = e 2 8(5) and a (110)=j» 3 . So this (P) 6 

 can be generated in the following way. Consider a space 8^ and a 

 plane 8. 2 perfectly normal to each other. Take in 8 /t an e 2 8(b), 

 in 8 2 a p 3 , and let P be a definite vertex of the former, Q a 

 definite vertex of the latter. Now move either c 2 8 (b) parallel to 

 itself in such a way that P coincides successively with all the points 

 inside p s , or p 6 parallel to itself in such a way that Q coincides 

 successively with all the points inside e 2 8 (5). Then the (P) can 

 be considered as the locus either of the e 2 8 (Q) in the first case 

 or of the P 3 in the second ; its vertices are given in the first case 

 by the three positions of e 2 8 (&) in which P coincides with one 

 of the vertices of p s , in the second by the thirty positions of p s 

 in which Q coincides with one of the vertices of e 2 8(b). We 

 represent it by the symbol \e 2 8(5); 3). 



In the case (54433) (210) we find an \e 2 8(b); 6). 



In the three cases (5443) (3222), (5443) (2221), (5443) (2210) 

 we find successively (CO ; T) , (CO ; T) , (CO ; tT). 



In the cases (5443)(322)(21),(5443)(322)(10),(5443)(32)(221) 

 we have to deal with right prisms on a (CO; 3) as base, whilst 

 (5443) (32) (210) is a right prism on a (CO- 6) as base. These 

 prisms may also be represented by the symbols (CO; 3; 2) and 

 (6'0; 6; 2) as prismotopes of the second rank. But a prismotope 

 proper of the second rank is the (P) 6 represented by (544) (322) (210), 

 which may be represented as such by the symbol (3 ; 3 ; 6). To 

 generate it we have to start from three planes a i9 ci 2 , & 3 two by 

 two perfectly normal to one another, and to place in a x and ct 2 

 equilateral triangles and in ci 3 a regular hexagon; then the (P) 6 is 

 obtained by the parallel motion of the hexagon in such a way that 

 a definite vertex of that hexagon coincides successively with all the 

 points inside the fourdimensional prismotope (3 ; 3) determined by 

 the two triangles. 



