DERIVED FROM THE REGULAR POLYTOPES. 



81 



where the small subscripts at the right indicate the number of limits 

 concurring in any vertex x ). So we find through any vertex 



five edges, 



two p z , two p ki six p 6 , 



one P 3 , five tl\ four tO, 



one (55311) = ce ± e 2 J6(6), two (5531— 1) '= e ± e 2 5(5), 

 two 1 [5311] = ee, e 2 C 16 , 

 and this gives in a transparent way in toto 



5.480 



2 

 2.480 

 ~~3 

 480 



i.e. 1200 (/),, 



6 

 480 



= S20p 3 

 = 80 P, 



2.480 

 ~4 

 5.480 

 T2~ 



= 240jö 4 

 = 200 tT 



6.480 



"~ "6 

 4.480 



"24" 



= 480j» 6 . . . 

 = 80 /O. . . 



?» 



(0i 



fa. 



(/) 3 



w* 



(/)-, 



ir wn 2 - 480 if «« 2 ' 480 in ^ 



== lOc^^/Xo ),———- = 16e,.e 2 3(5),- — — -= 10 c^^ C/ 16 „ 



So the result is 



(480, 1200, 1040, 360, 42) 

 in accordance with the law of Buler. 

 Case f [755311]. — 

 In the same way we find here the table: 



(75) 2f (5S) a ,(81) a> *i[ll] 1 



(755),, (75)(53) 2 , (75)(31) 4 , (553),, (531) 4 , (311) 1; 



(75)i[ll] 2 , (53)|[11] : 



(7553),, (755)(31) 2 , (75)(531) 4 , (75)(311) 2 , (5531) 2 , 



(5311) 2 ,(755)i[ll] 1 ,(75)(53)i[ll] 2 ,(553)i[ll] 1 , 



i[3H]i 



(75531) 2 , (755)(311) 1 , (75)(5311) 2 , (55311),, 



(7553}i[ll] 1 , (75)i[311] 2 , J[5311] 2 



(755311),, (755)-i[311] 1 , (75)|[5311] 23 £[558111 



So we find through any vertex 

 seven edges, 

 three p 3t ten /? 4 , six p 6) 



one CO, five tT, six P 3 , eight P 6 , two C, four tO, 

 two (32110), one (22100), two (32100), four P tT , four P t0 , 



one P co , one (3 ; 3), two (3 ; 0), two £[5311], 

 one (322100), one (p 3 ;tT), two P 4[531l] , one £[55311], 



two (432110); * 



*) So (531) is to bear the subscript 4, as the 5 may be related either to x<i or to ;r« 

 and the 1 either to x± or to Xt>\ so (31 — 1) is to admit the subscript 2, as the three 

 digits may apply either to 4~ a-g, +#4, — ^5 or to +#3, — #4, + ^5, etc. 



Verh. Kon. Akad. v. Wetensch. Ie Sectie Dl. XI N°. 5. E 6 



1040 {l\, 

 360 (/) 3) 



42 {l\. 



(31-1), 

 (75)(31-1) 4 ,(531-1) 4 



(755)(31-1) 2 ,(75)(Ö31-1) 4 , 



(5531—1), 

 (75531-1), 



