98 REV. T. P. KIRKMAN ON THE 
+Q,,Q,PP,)=Q, 
or more briefly, 
Dy) LE Cap, Eh 
where X is every letter in turn but I out of ABC---Q, 
and m n is every pair in turn of a bc-+ +p. 
In like manner, since the m-edral summit m has m 
edges, we must have 
(E) ¥(V.XaVeXe+ VaXaVeXe+ VaXaVadat*** 
+V,XiV,X») =a 
3(V,X,VaXat VeXzVeX%e+ VeXeVaXat °° 
+V,X,VpXp»)=5 
3 (VpXpVaXat VpXpVeXo+ VpXpVeXet+**: 
+V,X,V.X.) =P, 
or more briefly, 
SV Ska \ Mem, 
where a is every one in turn but m of a dc---p, and V X is 
every pair in turn of ABC.---Q. 
We have thus completely expressed both the tactical 
and arithmetical conditions of the problem. 
I fear that there is no shorter mode of arriving at the 
solution than by the way of elimination of this table-full 
of p'Q’ variables. This is possible; but the enormity of 
the operations implied is such as to give a very lively idea 
of the difficulty and vastness of this problem of the poly- 
edra. 
The p'+Q’ equations (M,,)(r=1) give the means of eli- 
minating as many variables from the p’ + Q’ equations (EF). 
The values thus found for those p’+Q’ are then to be 
substituted in the equations (M,,),(r=2), 
A=A?2-+ A? &., B=B2+B} &c., 
a=A?2+B? &., b=A}+ Bj &c., 
which will introduce certain duad products; after which 
all exponents may be erased from the variables, and prQ 
