AND ENUMERATION OP POLYEDRA. 57 



11. Returning to our 15-edron P, whose triangular faces 

 are cr, c, f^ and w, we may choose at pleasure two ridge- 

 summits from the series 



6 c? e ^ A O' A / n, 

 (in which the zeros denote the triangles) under the restrictions 

 just laid down. Let bdj^ knb be the triplets. We have to 

 complete the system 

 kab Abe kcd Ade Aef Afg Agh Ahi Aij Ajk Akl Aim 



bed de efg gh hi ij jk kl 



Amn Ana b'dj k'wb 



Imn nab dj jb kn bk eg In-, where the points in b'd'j 

 and k*n% shew duads of non-consecutive letters. 



We dispose first of dj and /cw, between either of which 

 lies a single triangle. The eAge dj leads from the point bdj 

 to the triangle f^ along d and e on one hand, and jih and g 

 on the other. We must have either dej or d'ij for a summit 

 lying between bdj and /"; for dj cannot lead from bdj to any 

 others constructive: thus we must have one of the two 

 systems 



dejyjve, hve, gh'e; d'ij, d-hi, d-gh, de*g; 

 either of which conducts along eg to ejg, by triplets of the 

 proper form, which repeat every duad that they introduce, and 

 dispose of de, ij, hi, gh, and ge. The summits dej, ji'e, 

 &c., known by the single point, I call wall summits. 



The edge kn leads along n to the triangle m, and the only 

 triplet possible is kl'n, which disposes both of kn and In as 

 well as of kl. 



The edge jb can enter no triplet but bjk, for no duad con- 

 taining J is disposable besides ^A: this uses bk; and all our 

 duads are now twice employed. We have then these two 

 systems, each of 26 triplets ; 



Aab Abe Aed Ade Aef Afg Agh Ahi Aij Ajk 

 nab bed j'de efg e'gh e'hi e'ij bjk 



Akl Aim Amn Ana b'hn 

 n*kl Imn b'dj ; 



I 



