‘27 
can assume always that the first A of A^A^Ac • • • are units, 
and that the first a of AaB^Ca . . . are units. We can then 
eliminate all the variables, except BaBjB,, . . . , having no 
capital but B. If there is but one B-gon under considera- 
tion, the inspection of this result of elimination will, if I mis- 
take not, inform us at once how many and what systems of 
summits the B-gon may have ; and thence we can determine 
how many and what systems of summits the C-gon may have, 
&c. ; i.e., we can proceed with certainty to construct as well 
as to enumerate the polyedra described in equation (A). 
What precedes amounts already to a solution of our problem 
in a theoretical point of view. But fortunately we are not 
under the necessity of dealing in detail with a whole table-full 
of variables. We can expel them all under the sign of sum- 
mation, except four symmetrical groups. 
The addition of either set of equations (S) gives the con- 
dition 
in which sum X and Y stand for every pair of ABC . . . Q 
in turn, and rs for every pair in turn of ahc . . . p. 
There are three symmetric functions of variables of the 
second order, six of the third, and fifteen of the fourth, besides 
the sixteenth just w’ritten. These are, 
(x,x,), (XA.) (X,.A); (XAA.), (x,x»Y,), (X,XA0, (XAA,-)> 
(XAA.), (XA.Z0; (X,XAA„). (X,X.XA„), (XAA.Y,), 
(XAAAO, (XAAAO, (x,.xA.z„), (X,xA»z,), (xaA<z,), 
(v,xa;.z.), (V,x.yAs), (xaaa.), (y,.xa:,z,), (xaa,.z.), 
(V,XAA.), (WXA.Z0. 
Here (X,.X^YfZu) denotes the sum of all products of four 
symbols, of which two have, the same capital, and all have 
different subindices. So (V,.X,.Y«Zi), putting small for large. 
Among these 3-f6+l6=25 variables under the sign ( ) of 
summation, we have 22 equations, including the last written, 
viz., 
(X'^j^Ai, (XY)=A2, (?•-)= A3, all independent; 
(X^)=Bi, (XA)=B2, (XYZ)=B3, (r»)=B„ 
all independent; but W'e cannot add (/•5^)=Be, because this is 
connected with the other five by our fundamental equation, 
{(A+B+ . . . -irQ)—(n-\-b+c+ . . . -f-p)}"=:0. Here {r^s) 
