AND ENUMEKATION OP POLYEDItA. iSj 



19. When of two adnate crowns one is a triangle, there 

 may intervene in general between the faces joining them at 

 their common summits one or more faces, or none. 



If the crowns of our octaedron are a pentagon C and a 

 triangle ly, those faces at the common summits must be 

 contiguous on the base, otherwise the summits would be 

 angles of at least hexagons, containing each three angles on 

 the two crowns, one in the face intervening between those 

 non-contiguous faces, and two in the base. The common 

 summits must therefore be CD'a CD'6. 

 We have to complete 



kah Abe Acd Me kea CUa CD'6 

 D'ab he ed de ea Qa Cb. 

 As a and 6 cannot occur more than a fifth time, we must 

 write Cea and Cftc, whence of necessity Cee and cde. 

 This gives the faces 



kCWabcde. (10) 



5 5 305535 



20. Two quadrilaterals, or a triangle and a quadrilateral, 

 cannot be our two crowns, the base or most-angled face being 

 pentagonal ; for, as there must be seven summits out of the 

 base, it will happen in either case that a face will contain 

 three summits on the two crowns, two in the base, and one 

 neither in the base nor the crowns. Two triangles cannot be 

 adnate crowns, the summits being all triedral, because, of their 

 common summits, only one can present to the base an angle 

 less than two right angles ; and that which is not less than 

 two can be no angle in any face. 



The base or most-angled face of an octaedron having only 

 triedral summits cannot be less than pentagonal; for there 

 must be 3X12 angles counted in all the faces, but 4x8=32 

 only. 



I have thus found all the octaedra which have not a 

 heptagonal face, and which have all their summits triedral. 

 They may be collected briefly as follows : 



