REFERENCE TO THE ENUMERATION OF POLYHEDRA. 255 



therefore 3 octacrons. The heptacron has only two 

 tripleural summits^ and they are disposed in the 

 proper manner ; the second process gives therefore 

 I octacron. 



3. The heptacron 3344556 has five kinds of faces, viz. 



345, 346, 356, 456 and 455, and the first process 

 consequently gives 5 octacrons. The heptacron has 

 two tripleural summits, but they are not disposed in 

 such manner as to render the second process appli- 

 cable. 



4. The heptacron 3444555 has four kinds of faces, viz. 



355, 455, 445 and 444, and the first process gives 

 therefore 4 octacrons. The heptacron has one tri- 

 pleural summit, and the basic quadrangles 3545 

 which belong to it are of the same kind ; the second 

 process gives therefore i octacron. 



5. The heptacron 4444455 has only one kind of face, 



viz. 445, and the first process gives therefore i octa- 

 cron. There are two kinds of basic quadrangles, 

 viz. 4545 and 4445, and the second process gives 

 therefore 2 octacrons. 

 The number of octacrons would thus be 20, but by 

 passing back from the octacrons to the heptacrons, it is 

 found that there are in fact only 14 octacrons. Thus the 

 octacron 33336666 has only one kind of tripleural summit 

 666 (the summit is here indicated by the symbol of the 

 basic polygon) and the octacron is thus seen to be deriv- 

 able from a single heptacron only, viz. the heptacron 

 3335556 from which it was in fact derived. But the octa- 

 cron 33345567 has three kinds of tripleural summits, viz. 

 567, 557 and 467, and it is consequently derivable from 

 three heptacrons, viz. the heptacrons 3335556, 3344466 

 and 3344555, and so on. The passage to the heptacrons 

 from an octacron with one or more tripleural summits is 

 of course always by the first proeess, but for the last two 



