259] THE PROBLEM OF POLYEDRA. 185 
Then, forming the substitution symbol 
A'B'C'D'E'... k'l'm'... abode ... IILM..., 
if, operating with this upon the symbol system of the first polyhedron, we obtain the 
symbol system of the second polyhedron, the second polyhedron is said to be polar - 
syntypic with the first; and, as in the case of syntypicism, this may happen in 
several different ways. 
Lastly, if there be a polyhedron having the same number of vertices and faces, 
and if the vertices and faces in any order be represented by 
abed... KLMN..., 
and the faces and vertices in a certain order be represented by 
A BCD ... klmn...; 
then, forming the substitution symbol 
ABGD ... klmn ... abed ... KLMN..., 
if, operating with this upon the symbol system of the polyhedron, we reproduce such 
symbol system, i.e. in fact, if the polyhedron be polar-syntypic with itself, the polyhedron 
is said to be autopolar; and in accordance with a preceding remark, this may happen 
in several different ways. It is clear that the substitution symbol, operating on the 
symbol system of the vertices, must give the symbol system of the faces, and 
conversely; but operating on the symbol system of the edges, it must reproduce such 
symbol system of the edges: and this last condition will by itself suffice to make 
the polyhedron autopolar, i.e. the polyhedron will be autopolar if the substitution symbol, 
operating on the symbol system of the edges, reproduces such symbol system.