64 REV. T. P. KIRKMAN ON THE REPRESENTATION 



6 3 745443-' 



The two latter are excluded, because one face is heptagonal : 

 the octaedron standing on that base has no crown, and ought 

 to be described among those that have a 7-gonal base. 



Let now C be combined with two duads of non-consecutive 

 letters. 



Here it will be useful to demonstate the following theorem. 



Theo. VII. Ifv of the summits in the crown C of an 

 n-edron on an (n — 2)-gonal base, having only triedral 

 summits, are bounded by pairs of faces which are not con- 

 tiguous faces about the base, that n-edron will have at least 

 T triangular faces about the base. 



For let C^6 be such a summit of C, 7 and e not being con, 

 tiguous faces about the base. The edge 7^ proceeding from 

 C^O must be a portion of a broken or branching ridge-line 

 which encloses no space, because there is no crown but C. 

 It will have at least one extremity H. This summit H, 

 having (mly one edge passing through a summit out of the 

 base, will have two edges passing through angles m and n 

 of the base; i. e., Yimn is a triangular face lying between 

 7 and Q. 



17. We must, by virtue of this theorem, have two triangular 

 faces about the base of our 8-edron. Let those triangles first 

 be a and c, separated by one face b only. 



