40 
ON THE A FACED POLYACRONS IN 
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w-acron a polygon of any number of sides, and make this the basic polygon of the 
new summit of the (n + l)-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 tripleural, 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 (w-fl)-acron of the first class may by the first process of subtraction be reduced 
to an w-acron, and conversely it can be by the first process of addition derived from 
an w-acron. An (n + l)-acron of the second class, as having a tetrapleural summit, may 
by the second process of subtraction be reduced to an ?i-acron, and conversely it can 
be by the second process of addition derived from an n-acron. And in like manner, 
an (?2+l)-acron of the third class, as having a pentipleural summit, may be by the 
third process of subtraction reduced to an ?i-acron, and conversely it may be by the 
third process of addition derived from an w-acron. 
Hence all the (n +1 )-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 + l)-acrons of the first class are obtained by the first process; the second process 
is only required for finding the (n + l)-acrons of the second class; and these being 
all obtained by means of it, the third process is only required for finding the 
(w + l)-acrons of the third class. Hence the second process need only be made use 
of when the w-acron has no tripleural summit, or when it has only one tripleural 
summit, or when, having two tripleural summits, they are the opposite summits of two