AND ENUMERATION OF POLTEDRA. 59 



in which the broken or branching ridge-line, which connects 

 the ridge-summits and vertices of the triangles, proceeds along 

 the intervening faces. The species considered in the pre- 

 ceding article has two varieties. It may be worth while to 

 work out another example. 



Let there be again 15 faces, one H-gonal face A, and 

 only three triangles, 6, f, and m ; and let d'j-n be the ridge- 

 summit. We have to complete the system 



hah Abe kcd Ade Aef Afg Agh Ahi Aij A/A 

 ahc cd de efg gh hi ij jk 



Akl Aim Amn Ana dj-n 

 kl Imn na dj,jn, nd, ac, eg. In. 



If we look at the series 



aOcdeOghijklOnf 

 we see that dj leads from the point djn to the triangle^ along 

 the faces jihg on one hand and de on the other. We have 

 the choice of the two triplets de'j and d'ij for the disposal of 

 dj : take dej. This make ei'j inevitable, for ej must be 

 repeated ; after which e'hi, e'gh are determined, and all our 

 duads thus far are repeated. Taking d'ij, we must have of 

 necessity d'hi, d'gh, and de'g, which closes our circle of 

 duads. Thus we have either 



dej, e'ij, e'hi, e'gh ; or else 

 d'ij, d'hi, d'gh, de'g ; 

 either of which disposes of de, eg, gh, hi, ij, jd. 



Next, we see that jn leads from djn to m only along n ; and 

 we can make no other triplets than 



jk'n, kl'n. 

 The edge nd leads to h along d and c. We can write either 

 na'd or cd'n ; that is, we may have either 

 na'd, a'cd; or cd'n, na'c. 

 Hence the species of 15-edron — having triedral summits, 

 a 14-gonal face, three triangular faces, between which inter- 



