AND ENUMERATION OF POLYEDBA. 63 



hah Abe Acd kde kef Afa 

 Cab Cbc Ccd de ef fa Cda da\ 

 which determines either def and dfa to be the required trip- 

 lets, or else efa and ead^ giving 

 either k C ab c d ef *) 



6 4344634 >C3^ 



or k C a b c d ef ) 



64544549* ^ 



Of these two, one is merely the reflected image of the 

 other, the faces about the base being like and in like order. 



If C has two non-consecutive duads, they may occur at 

 adjoining or non-adjoining summits in C. This gives two 

 systems: — 



kab kbc kcd kde kef kfa 



Cab Cbc cd de ef fa Cce Cea ce ea, 



which demands cde and efa ; and 



kab kbc kcd kde kef kfa 

 Cab be cd Cde ef fa Cbd Cea bd ea, 

 which requires bed and efa ; and the faces stand thus, 

 kCabcdef (4) 



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kCabcdef (5) 



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C cannot be combined with three duads of non-consecutive 

 letters, for in that case there would remain to be repeated 

 those three besides five of those employed with A ; but to 

 complete the 12 there are only two triplets required, which 

 cannot dispose of eight duads. 



16. Next let C be triangular: it may be combined with 

 one or more duads of non-consecutive letters. If with one 

 only, the system is, requiring three triplets more, 



kab kbc kcd kde kef kfa 



Cab Cbc cd de ef fa Cea ca. 



