5*2" REV. T. P. KIRKMAN ON THE REPRESENTATION 



which contain each two consecutive duads of the above written 

 n — 1, in other words, which contain each three consecutive 

 letters out of ahc , , , Im, 



Let these two triplets be actually abc and jkl. The edges 

 Kb, ah, and be, are in the face b ; and the two edges ob and 

 he which pass through the summits hah and hbc, meet in the 

 summit abc. That is, ^ is a triangular face ; and in the same 

 way it appears that A is a triangular face. We have thus proved 

 that our w-edron P must have at least two triangular faces. 

 If, now, the restriction that all the summits shall be triedral 

 be removed, the base being still (n — l)-gonal, it would not 

 be difficult to show algebraically that the same necessity 

 remains for at least two triangular faces; but I shall here 

 content myself with a geometrical proof of the theorem fol- 

 lowing, which is under no restriction. 



5. Theo. Y, If a p-edron has a f p — 1 J-gonal face, two 

 at least of its faces are triangular. 



If a q-peak has a (q — \)-gonal summit, two at least of 

 its summits are triedral. 



For, since every side of the {p — l)~gon is the intersection 

 of two faces of the jo-edron, there can be no face which 

 does not contain a side of the (/? — l)-gon. The summits 

 of the solid not in the plane of the {p — l)-gon must there- 

 fore be connected by a broken or branching line, which 

 encloses no space, and which will have at least two extremities 

 H and H'. The summit H, having only one edge (part of 

 the broken line), which passes through a second summit out 

 of the plane of the (/? — l)-gon, must have at least two edges 

 which pass through angles m and n thereof: i, e,, Yimn is 

 a triangular face. In the same way it is demonstrated that 

 Wm'n is a triangular face of the j^-edron. 



The second part of the theorem is the corresponding polar 

 property. 



