DERIVED FROM THE REGULAR POLYTOPES. 21 



of the given polytope (P) ni if we fix either the values of n — d 

 coordinates and allow the remaining d -\- 1 to interchange their 

 values, or the values of n — d — 1 coordinates and split up the 

 remaining d -\~ 2 into two groups of interchangeable ones, or the 

 values of u — d — 2 coordinates and split up the remaining d-\~ 3 

 into three groups of interchangeable ones, etc., this process winding 

 up for n < 2d in a symbol with n — d -\- 1 and for n > 2 (d — 1) 

 in a symbol with d groups." 



"This (P) d will be limiting polytope of ÇP) n , if: 

 1°. each syllable of the unextended symbol with middle digits 

 exhausts these digits of the symbol of (P) n , 



2°. no two syllables without middle digits have the same end digits." 

 Proof. The first part of the new theorem is a consequence of 

 this that in the different cases communicated the corresponding 

 extended symbol is always consisting of n — d -\~ 1 syllables, i.e. 

 of k syllables with more than one digit and n — d — h -\- 1 syl- 

 lables with only one digit for k= 1,2, . . .,d; so it is equivalent 

 to part a) of the proof of theorem III. The second part of the 

 new theorem is equivalent to part b) of the proof of theorem III; 

 for the only cases in which it is impossible to put the syllables 

 of the extended symbol behind one another so as to obtain the 

 order of succession of the pattern vertex are the two excluded by 

 the two items 1° and 2°, i. e. 1° that a syllable with middle 

 digits does not exhausts these digits and 2° that two syllables 

 without middle digits do have the same end digits. Finally the 

 part c) of the proof of theorem III can be repeated here. 



By means of theorem III' we find e. g. in the case of the (P) } 

 represented by (5443322210) the following 58 different kinds of 

 limiting (P) 6 : 



(5443322), — (544332) (21), (544332) (10), — (54433) (221), 

 (54433) (210), —(5443) (3222), (5443) (322) (21), (5443) (322) 

 (10), (5443) (32) (221), (5443) (32) (210), (5443) (2221), (5443) 

 (2210), — (544) (33222), (544) (3322) (21), (544) (3322)(10), (544) 

 (332) (221), (544) (332) (210), (544) (32221), (544) (3222) (10), 

 (544) (322) (2 10), (544) (32) (2210), (544) (22210), — (54) (433222), 

 (54) (43322) (21), (54) (43322) (10), (54) (4332) (221), (54) (4332) 

 (210), (54) (433) (2221), (54) (433) (2210), (54) (43) (32221), (54) 

 (43) (3222) (10), (54) (43) (322) (210), (54) (43) (32) (22 10), (54) (43) 

 (22210), (54) (332221), (54) (33222) (10), (54) (3322) (210), (54) 

 (332) (2210), (54) (322210) — (4433222), — (443322) (21), 

 (443322) (10), — (44332) (221), (44332) (210), — (4433) (2221), 

 (4433) (2210), — (443) (32221), (443) (3222) (10), (443) (322) 



