102 REV. T. P. KIRKMAN ON THE 
broken, and which are satisfied by the values of A B, &c., 
under review, will be the number of p’-acral Q'’-edra de- 
scribed under equation (A). 
There is reason to believe that the result of this investi- 
gation may be made to take the shape 
H=Iv+Jy+Kz+--- 
where HI J K---are symmetric functions of A B---Q 
ab..-p, and x y z---are symmetric functions of the vari- 
ables M,,; and such a shape that the number of solutions 
of this equation in whole values of wy z---is the exact 
number of polyedra which it is our object to discover. 
This form, if it be attainable, would certainly be the most 
elegant issue that we could desire to our investigation. 
As the number of these symmetric functions is limited, 
and as they appear in the expression of symmetric func- 
tions of the numbers ABC---adc---, it is plain that all 
but certain ones can be eliminated; but I have not been 
able to obtain a final equation having no solutions in xv y z 
-- «foreign to the question. The equation above found, 
13 (K-X)(V2-Y) -e=wty 
is true for more values of w and y than those of which we 
are in quest. But as I have not formed the equation ob- 
tained by eliminating from all the equations of this kind 
that can be written, I cannot say whether a result, show- 
ing symmetric functions of higher degrees, can be obtained 
of the form desired. 
I have a demonstration that the number of p’-acral 
Q’-edra can be easily found, whatever be the numbers of 
AB.---Q ab---p, if that of the p’-acral Q’-edra be known 
for another set of numbers A, B,---Q, a ,:-+p,, of which 
either A, =B,=C,=.-- -=Q;=8, or else a,=),;=¢j/=*- - 
=p,=8. And thus the problem is reduced to the deter- 
mination of the number of Q-edra having only triedral 
summits, or, which is analytically the same thing, of p- 
