AND ENUMERATION OF POLYEDRi. 51 



No p-edral q-peak has a summit of more than 2p — q — I 

 angles. 



For, let a face of a /7-edral ^'-peak have 2^--p+r sides or 

 angles. The remaining p — 1 faces cannot have less than 3 

 sides each, nor can the sum of all the sides of the faces of the 

 solid exceed twice the number of the edges, because no edge 

 can be counted in more or less than two faces : t, e., 



2q—p+r-\-3(p^\)'2^P+g—^). or 



2qi-2p+r^3'2^P'{-2q—4, 

 which is absurd. 



4. Let now P be any w-edron with only triedral summits. 

 It cannot have a face of more than n — 1 angles, by Theo. III. 

 Let it have an (n — l)-gonal face A, which is bounded by 



the n — 1 faces, abc klm, in that order. The 



n — 1 summits of P about A are, in order, represented by the 

 11 — 1 triplets 



Aaby Abc, Acd Aim, Ama; 



and since all the n — 1 edges ab, be, cd , , . . Im, ma, which 

 meet A, must be found each at some second summit of P, these 

 n — I duads will appear in the n — 3 triplets which represent 

 the remaining summits of P; the whole number of those sum- 

 mits being 2w — 4, by Theo. IL This is impossible, unless 

 some triplet contains two or more of these n — 1 duads: now no 

 triplet can exhibit three of them, for it would then contain 

 more than three letters; nor can any two non-consecutive 

 duads, as ab and cd, form a triplet. Let then abc be one 

 triplet, in which ab and be are exhibited ; there remain n — 3 

 duads to be disposed of in n — 4 triplets, which cannot be, un- 

 less some one of these contains two of the duads. 



There must then of necessity be, among the n — 3 triplets to 

 be added to those containing A, two of the form, abcy jkl. 



