100 REV. T. P. KIRKMAN ON THE 
We next eliminate all capitals but A and B, obtaining an 
equation containing only A, A,---B, B,---. Putting in 
this for A,, &c., in turn the systems of A values not proved 
before impossible, we can read a certain number of sets 
of B values (B,,=1 B,=1, &c.) of which we can say that 
none besides are possible; that is, we can construct a 
certain number of sets of two columns under A and B, of” 
which we can say that every possible system is written 
down among them. Next, forming the equation contain- 
ing only A, A,---B, B,---C, C,-++, we can construct a 
limited number of sets of three columns, of which we can 
say with certainty that every possible polyedron is thus far 
constructed. And thus we can proceed till we have before 
us a certain number of sets of Q’ columns, of which we 
can pronounce that every possible Q’-edron is once, and 
only once, constructed; that is, we can, by a strictly de- 
monstrative process, both construct and enumerate the p’- 
acral Q’-edra described in equation (A). And this is a 
complete and rigorous solution, from the theoretical point 
of view, of the problem of the polyedra; for they are all 
comprised in descriptions of the form of (A). 
The result U=0, above mentioned, will show at once 
whether any polyedron is described in equation (A); for, 
if none exists, U=O will not be satisfied by the values of 
AB---Q ab---p under consideration. 
It appears highly probable that the same equation U=0 
will be sufficient for the enumeration of the different poly- 
edra under (A), for the reasons following. 
Let }X,X,Y,Y, be the number of pairs of lines, edges or 
diagonals, drawn or drawable from the a-ace a to any two 
other summits 7 and s, a7 being in the X-gon and as in 
the Y-gon, which have the common summit s. No two 
different polyedra can have this sum alike at every summit 
a; for if they have, they will have the same faces in the 
