50 REV. T. P. KIRKMAN ON THE REPitESENTATION 



Cor. 3. The greatest number of edges in a ^-edron cannot 

 exceed 3p — 6. 



The greatest number of edges is an s-peak cannot exceed 

 3s— 6. 



This follows from Cor. 2 and Theo. I. 



Cor. 4. The number of faces in an s-peak is never less 

 than i(s+4), nor greater than 2s — 4. 



The number of summits in a p-edron is never less than 

 J(p-|-4), 7ior more than (2p — 4). 



This is deduced from Theo. I. and the two preceding corol- 

 laries. 



3. Here we may add conveniently the two following theo- 

 rems. 



Theo. III. No p-edral q-peak has a face of so many as 

 either p or q angles : 



No ^-edral q-peak has a summit of so many as either p 

 or q edges. 



First : No ^'-peak has a face of q angles, for all its -5' sum- 

 mits cannot be in one face. 



Secondly : No j9-edral g'-peak has a face of p angles ; for 

 if it has, it has at least 2p edges terminating at those angles, 

 viz., the p sides of the face, and p others through their inter- 

 sections. The remaining q — p summits of the solid, not in 

 that face, are connected by not less than q — p — 1 edges, 

 which number, added to the other 2p edges, cannot exceed 

 q-{-p—2, by Theo. I. : i. c, 



which is absurd. 



The second part of the theorem is the polar property to the 

 first. 



Theo. IV. No ^h-edral q-peak has a face of more than 

 2q — p — I angles: 



