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Def. An edge of a polyedron is said to convanesce, when its two 

 summits run into one ; and it is said to evanesce, when its two faces 

 revolve into one. 



An edge (AB) is said to be convanescible, when neither of the 

 faces A and B is a triangle, and (AB) joins two summits which have 

 not two collateral faces, one in either summit, besides A and B. 



An edge (6) is said to be evanescible, when neither a nor b is a 

 triace, and the two faces about (ab) are not, one in either, in two 

 collateral summits, besides a and b. 



THEOREM II. Every polyedron, not a pyramid, has either a con- 

 vanescible or an evanescible edge. 



THEOREM III. Any p-edral q-acron, not a pyramid, can be 

 reduced by the vanishing of an edge, either to a (p l)-edral q-acron, 

 or to a p-edral (q \)-acron. 



By such a reduction of a ^-edral <?-acron P to P', of P' to P", &c., 

 P can be shown to be generable from a certain pyramid II ; by which 

 it is meant that n is the highest-ranked pyramid to which P can by 

 such reduction be reduced. 



Hereby it is evident that the problem of enumeration of the avedra, 

 is brought down to this: to determine how many (r + m)-edra are 

 generable from the r-edral pyramid. 



The autopolars so generable are first considered, as the heteropolars 

 are obtained by combination and selection of those operations with 

 which the theory of the autopolars makes us acquainted. 



Autopolarity is of three kinds, nodal, enodal, and utral. 



Every even-based pyramid is nodally autopolar ; i. e. it cannot 

 but have two nodal summits. For example, the 5 edral and 

 7-edral pyramids have the signatures of their faces and summits 

 thus arranged, 



1234 123456 



4321 65432 T 



the upper line showing the triangles, and the lower the triaces about 

 the base, which as well as its pole the vertex, is signed zero. The two 

 triaces in the triangle 5 are 3 and 2 ; the two triangles in the triace 

 1 are 6 and 1 in the 7-edron, and 4 and 1 in the 5-edron. The nodal 

 summits and faces are 3 and 1 in the 5-edron, and 4 and 1 in the 

 7-edron. No other mode of autopolar signature is possible in these. 



