28 
stands for the sum of products of every one of abc . . . p into 
the square of every other, and (X^Y) is the like function of 
ABC . . . Q. 
Thirteen more equations all independent, are, (X^)=Ci, 
(X^Y)=a, (X'A^^)=C3, (X^YZ)zz:C„ (XV«)=C3, (r^)=Ce, 
(r^5)=C;, (rV)=Cs, {rht)^C„ (rOT)=C,o, (XYrs)=C„, 
(XV-)=;Cio , (XYZV)=Ci 3 ; M'here (XV“) is jo'Q' products : 
but we cannot add {rstu^^Cui for a reason just given, nor 
(XV)=B', which is merely 2e(X^)=2eBi. The quantities 
Ai . . . Bj . . . Cj . . . are linear functions of the symmetric 
sums of variables M„, which appear in them multiplied only 
by simple numbers and multiples of 2e. Of these 25 symme- 
tric sums we can eliminate 21, and obtain a final equation 
containing any four, say, t^=S(X,.YfZ,), x='2;{y {L^^, 
y=2(X,X,Y,Z„), ^=(V,X,Y.Z,) ; viz., 
Jy-|-Kz=:L ; 
where HIJKL are made up of symmetric functions of (ABC 
. . . Q abc . . . p). 
If no solution exists of this equation in whole positive values 
of wxyz, no polyedron is described in equation (A), and to 
every distinct polyedron therein described must correspond a 
distinct solution. And I conceive that every such solution 
will give a distinct polyedron ; but 1 have not yet examined 
this point closely. At any rate, the limits of wxyz will be 
easily determined by obvious considerations. For autopolar 
polyedra {Q!=p', A=a, &c.), y=z, and the 25 variables 2, 
are reduced to 16.* 
Thus we have at last before us, in terms perfectly general, 
a solution of this difficult and celebrated question of the poly- 
edra; and it is highly probable that the method here opened 
will supply the key to a multitude of tactical problems which 
have hitherto defied our analysis. I cannot help thinking, 
that the symmetric functions of double discontinuous variables 
here handled, will reveal the long sought secret of the alge- 
braic expression of tactical conditions, and thus enable us to 
lay the foundation of a purely Tactic Calcuhis. 
* Vide my Memoir, “ Autopolar Polyedra,” in the last Volume (1857) of 
the Philosophical Transactions. 
