AND ENUMEEATION OF POLTEDRA. 49 



the summits, we have, since every edge is thus twice 

 enumerated, 



2c=3*, 

 and since c-f 2=»-f «, by Theo. I., 



.'. g-|-6=3w, 

 and «4-4=2w. 



This gives us the following : 



Theo. II. In any polyedron whose summits are all 

 triedrah twice the number of the edges is thrice that of the 

 summits ; the number of edges plus six is thrice that of the 

 faces ; and the number of summits plus four is twice that of 

 the faces. 



From this follows the truth of what may be called its polar 

 theorem ; 



In any polyedron whose faces are all triangular, twice 

 the number of the edges is thrice that of the faces ; the 

 number of edges plus six is thrice that of the summits ; and 

 the number of faces plus four is twice that of the summits. 



Cor. 1. No polyedron having only triedral summits can 

 have an odd number of them. 



No polyedron having only triangular faces can have an 

 odd number of them. 



Cor. 2. The number of edges in a p-edron is never less 

 than 4p. 



The number of edges in a ^edron is never less than 4 of 

 the number of its summits. 



For the /?-edron has the fewest edges when the faces have 

 the fewest sides, i, c, when they are all triangles ; and of all 

 the polyedra which have q summits, that will have the fewest 

 edges whose summits have the fewest edges, that is, whose 

 summits are all triedral. 



Definition. An s-peak is a polyedron having s summits. 

 This being premised and permitted, the second part of Cor. 2 

 is better stated thus : 



The number of edges in an ^peak is never less than 4s. 

 II 



