REFERENCE TO THE ENUMERATION OF POLYHEDRA. 251 



way only, the second process in two ways, the third in 

 five ways ; these being in fact the numbers of ways of 

 partitioning the basic polygon. 



We may in like manner, by the converse process of the 

 addition of a summit, convert an w-acron into an {n+ i)- 

 acron; viz., it is only necessary to take on the «-acron a 

 polygon of any number of sides, and make this the basic 

 polygon of the new summit of the {n+ i)-acron, and for 

 this purpose to remove the faces within the polygon and 

 substitute for them a set of triangular faces standing on 

 the sides of the polygon and meeting in the new summit : 

 the same figures exhibit the process for the cases of a tri- 

 pleural, tetrapleural and pentipleural summit respectively, 

 which (as for the subtractions) are the only cases which 

 need be considered. It may be noticed that for the same 

 basic polygon the process is in each case a unique one; 

 the process is said to be the first, second, or third process, 

 according as the new summit is tripleural, tetrapleural, or 

 pentipleural. • 



Now, reverting to the before-mentioned division of the 

 polyacrons into three classes, an (?*-t- i)-acron of the first 

 class may by the first process of subtraction be reduced 

 to an 7«-acron, and conversely it can be by the first process 

 of addition derived from an %-acron. An {n+ i)-acron of 

 the second class, as having a tetrapleural summit, may by 

 the second process of subtraction be reduced to an ?2-acron, 

 and conversely it can be by the second process of addition 

 derived from an w-acron. And in like manner, an {n+ i)- 

 acron of the third class, as having a pentipleural summit, 

 may be by the third process of subtraction reduced to an 

 w-acron, and conversely it may be by the third process of 

 addition derived from an w-acron. 



Hence all the {n+ i)-acrons can be by the first, second 

 and third processes of addition respectively derived from 

 the w-acrons. It is to be observed that all the (n+i)- 



