SOLUTION OF THE PROBLEM OF THE POLYEDRA. 97 
We have p’Q’ unknown quantities M,, discontinuous 
variables, which can have any of them the value zero or 
unity, and no other value. The columns headed by A BC 
- ++, the faces in (A), give the equations, 
(M,) A=A,+A,+A,+++++A,=AZ+A5+--++A3, 
B=B,+B,+B,+---+B,=Bi+ Bj+---+By, 
Q=Q,+0,+Q,4+++-+Q,=Qi+Qj+---+Q), 
Q’ equations for any whole value >0 of 7, because M7, = 
M,,, whether this be unity or zero, and the rows give the 
p equations for any value integer and positive of 7. 
(M,) a=A,+B,+C,+---+Q,=A7+Bi+---+Qi, 
6=A,+B,+C,4+---+Q,=A}+B+---+Q}, 
p=A,+B,+C,+---+Q,=A%+Bi4+---+Q5. 
The (p'—1+Q'—1)-edges have next to be expressed. 
If one of these is the intersection of the faces I J, and 
connects the summits m and n, we shall have 
LlndnIn=1, and 
Pike K=O = 3 Sede, 
whatever be K and p; because the three faces IJ K, 
having no common line, cannot all contain the same two 
summits m and n, and the three summits mn p, not being 
in the same line, cannot all be in the same two planes I 
and J. 
As the I-gonal face I has I edges, we have the following 
equations, by what has just preceded, 
(E) (A, ABnBrn+AmAnOnC,+AmAnDnDrat ++: 
+A,A,Q,Q,)=A 
3(B,,B,AnAn+ BrBrCmCn+BrB,DnDnt +++ 
+B,,B,Q,,Q,)=B 
(Cp CpAmAn+ CnOnBmBn+ CmCnDmD, + + >> 
+C,,C,,Q,,Q,,) =C 
3 (QQ AmAn a Q,,Q,B,B, “fs Q,,Q,D,,D,, a 
VOL. XV, i) 
