EEV. T. P. KIEKMAJSr ON AUTOPOLAE POLYEDEA, 
185 
ah and AB meet at a summit h, they are nodal gamics in the nodal summit b and the 
nodal face B. From the first notions of poles and polars, it is evident that no edge can 
be its own gamic. 
Every polyedron is either autcypolar or heteropolar. A heteropolar p-edral g-'acron H 
is the sympolar of a g'-edral y)-acron H', whose summits are the poles of the faces of H, 
And let the vertices and faces of the first polyedron taken ia any order be represented by 
abode . . KLM . . , 
and the vertices and faces of the second polyedron taken in a certain order be represented by 
. K'L'M'. . 
Then, forming the substitution symbol 
a!Vdd}e\ . K'L'M'. . alcde . . KLM . . , 
which denotes that a! is to be written for a, 5' for 6 . . . K' for K, &c., if, operating with this upon the symbol 
system of the first polyedron, we obtain the symbol system of the second polyedron, the second polyedron 
wdl be syntypic with the first. It should be noticed, that there may be several modes of arrangement of the 
vertices and faces of the second polyedron, which will render it syntypic according to the foregoing definition 
with the first polyedron, i. e. the second polyedron may be syntypic in several different ways with the first 
polyedron. This is, in fact, the same as saying that a polyedron may be syntypic with itself in several 
different ways. Suppose, next, that the number of vertices of the second polyedron is -equal to the number 
of faces of the first polyedron, and the number of faces of the second polyedron is equal to the number of 
vertices of the first polyedron. And let the vertices and faces of the first polyedron in any order be repre- 
abode.. -KIM..., 
and the faces and vertices of the second polyedron in a certain order be represented by 
A'B'C'D'E'. . k'Vm'. . 
Then, forming the substitution symbol 
A'B'C'D'E'. . m'rn!. . abode . . KLM . . , 
if, operating with this upon the symbol system of the first polyedron, we obtain the symbol system of the 
second polyedron, the second polyedron 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 polyedron having the same number of vertices and faces, and if the vertices and faces 
in any order be represented by 
abed.. Kim.., 
and the faces and vertices in a certain order be represented by 
ABCD . . hlmm . . 
Then, forming the substitution symbol 
ABCD . . Ttlmn . . abed . . KLMN . . , 
if, operating with this upon the symbol system of the polyedron, we reproduce such symbol system, i. e. in 
fact, if the polyedron be polar-syntypic with itself, the polyedron is said to be autoj)olar ; and in accord- 
ance 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 polyedron autopolar, i. e. the polyedron will 
be autopolar if the substitution symbol, operating on the symbol system of the edges, reproduces such 
symbol system.” 
MDCCCLVII. 2 C 
