250 MR. A. CAYLEY ON THE A FACED POLYACRONS IN 



6n- i2 = '^c + ^d + ^e + 6f+'jg + Sh + ^c., 



and therefore 



i2 = ^c + 2d+e + of- g-2h- ^c, 

 or 



3c + 2c? + e= 12 +^ + 2A + <SfC. j 

 whence if c=o and d=o, then e= 12 at least. It appears^ 

 moreover (since n cannot be less than e), that any poly- 

 acron with less than 12 summits cannot belong to the 

 third class^ and must therefore belong to the first or the 

 second class. 



An {n + i)-acron;, by a process which I call the subtraction 

 of a summit^ may be reduced to an w-acron ; viz., the faces 

 about any summit of the {n-\- i)-acron stand upon a poly- 

 gon (not in general a plane figure) which may be called 

 the basic polygon, and when the summit with the faces 

 and edges belonging to 

 it is removed, the basic 

 polygon, if a triangle, will 

 be a face of the w-acron ; 

 if not a triangle, it can 

 be partitioned into tri- 

 angles which will be faces 

 of the /i-acron. The an- 

 nexed figures exhibit the 

 process for the cases of 

 a tripleural, tetrapleural 

 and pentipleural summit 

 respectively, which are 

 the only cases which need 

 be considered ; these may 

 be called the first, se- 

 cond and third process 



respectively. It is proper to remark that for the same 

 removed summit the first process can be performed in one 



