47 



III. — On the Representation and Enumeration of 

 Polyedra, 



By the Rev. T^homas P. Kirkman, A.M., Rector of Croft- 

 with-Southworth. 



IRead December 18<A, 1863.] 



1 . The relation between the faces, summits, and edges of 

 a polyedron is expressed in the theorem following : 



Theo. I. "The number of faces and summits in any 

 polyedron taken together, exceeds by two the number of its 

 edges." 



For, let P be any polyedron having an (c— /)-gonal face 

 E, contiguous at its edges and angles to e other faces of P. 

 With /of these faces E will have no common edge; but if E 

 be removed by a section E' meeting all those e faces, P will 

 become P, a polyedron having an e-gonal face E', contiguous 

 to 6 other faces of P' ; P' having the same number of faces 

 with P, but having /more summits and /more edges than P. 

 If the e faces about E' be produced to meet any face F pro- 

 duced of P', P' is included in the prismoid solid P", whose 

 two opposite c-gonal faces E' and F are connected by e qua- 

 drilateral faces. Let now F be carved out of V by sections 

 along faces of P. 



The first frustum p cut off from P' will have k of the 

 divided, and / of the undivided and removed edges of P^. 



If /=ro, f) is a pyramid on a A-gonal base K ; and it is evi- 

 dent that P' diminished has lost a summit, in the vertex of 



