SOLUTION OF THE PROBLEM OF THE POLYEDRA, 99 
more can be eliminated as before from equations (E). The 
values substituted for the expelled variables are next to be 
introduced in 
A=A3+ A? &., B=B3+B}3 &c., &c., 
a=A°+B &., b=A}+B} &c.; 
by which operation certain products of three, four, five 
* and six variables will be exhibited in the last written 
p+Q equations. We can then erase exponents and pro- 
ceed linearly to expel p'+Q’ more variables from equa- 
tions (E), and can thus operate till all are eliminated ; 
when an equation U=O will be the result, expressing a 
condition or conditions among the quantities A B---Q, ab 
-+ +p, which will be different from (A). 
The work will simplify itself as we proceed, by the 
erasure of all those products of variables which contain 
the capital J with more than J subindices, or the subindex 
2 under more than 2 capitals; for we know that not more 
than J of J, J, J,, &c., can have value, nor more than 7 
of A; B; C, &c. Or, if we wish to obtain a result in 
terms perfectly general, without any choice of value for J 
or i, we can content ourselves with erasing all exponents. 
Of course no product of more than Q’ different capitals, or 
more than p’ different subindices, can possibly remain. 
If we wish to construct and enumerate the polyedra of 
which (A) is the common description, we may do it thus. 
First eliminate all capitals but A. The mere inspection 
of the result in A, A, A,, &c., will show what systems of 
A values are possible, for A,,=0 A,,=1 are the only values 
that can arise, that is, no system of summits can be about 
the A-gon, which is not readable in that result; and if 
there be only one A-gon, it will often happen that only 
one set of summits is readable and admissible. We can 
pronounce with certainty that all are impossible but those 
(m n, &c.) readable in our result (A,,=1 A,,=1, &c.) 
