SOLUTION OF THE PROBLEM OF THE POLYEDRA. 101 
same order about every summit, i.e. they will not be dif- 
ferent polyedra. It is easy to form equations containing 
such sums. Thus, if we multiply the square of A by that 
of B, we have 
A?= J A?+235A,A,=3A,4+25A,A,=A+4254,A, 
B?= 3 B?+25B,B,=>B,+2>B,B,=B + 2>B,B,, 
4(A?—A)(B?-B) =3A,A,>B,B,, 
where 7s is every pair in turn of a0---p, in each of the 
sums of this product. Hence, keeping A invariable, and 
giving to X 7's every possible value, 
31 (A? A)(X?-X =F A,A,X,X,+ FA,AX,X,+ FA,A,X,X,, 
=A+3A,A,X,X,+ SA,A,X;X,, 5 
which, by adding up Q’ such equations, one for every face 
' in turn of A B---Q, becomes 
33(X?-X)(Y?-Y) =2e + 25X,X,Y,Y,+22X,X,Y,Y,, 
where X and Y are every pair in turn of AB---Q, and 
rstu are every set in turn of four out of ad---p. In 
like manner it is proved that 
43\(7?—1r)(#—s)=e+ 2X,Y,Y,Z,+ 5X,Y,V,Z, 
where 7's are every pair in turn of @b-+-p, and XYVZ 
are every four in turn out of A B---Q. 
So far as I have examined cases, I find no two different 
polyedra having the same value of the sum 3X,X,Y,Y, 
taken over all the summits s. 
Now there will be for every polyedron a certain equation 
S=0, containing only symmetric functions of the variables 
of different degrees in them, and true for no other poly- 
edron. This will be translateable into an equation V=O, 
containing only symmetric functions of the numbers A B 
-+-Q ab---p, This condition V=O will therefore, in all 
probability, be a factor of our result of elimination, U=O. 
If so, the number of factors of distinct forms (such that 
the evanescence of one does not for all values of A B, &c., 
follow from that of the other), into which U=O can be 
