26 
p'-\- Q' equations, for any positive whole value of r ; for the 
only value of M„ is 1 or 0, so that 
Consider now the edge of the polyedron between the faces 
I and J, which is also between the summits vi and n. We 
shall have 
for the four factors are all units. 
!'■ T’"!'" T*" 
n 5 
And we have 
Jr Trlr Tr — a — Tr JrT^r XT 
r 
n 5 
because the three summits mnp, not being in a line, cannot 
be each in both the planes IJ, and the three faces UK, not 
having a common line, cannot be each in both the summits 
m and n. Also 
(S) 2IrIjXrX,=:I, for each of the Q,' faces, 
because the I-gon has I edges ; and in like manner, 
(S) 2X,„Y,„X.Y.=m, for each of the p' summits, 
because the m-ace has m edges. The former sum includes 
every pair rs of the p' numbers abc . . . , and every value of 
X out of AB . . . Q, except I, which is constant in the equa- 
tion; and the latter sum includes every pair XY of the Q' 
numbers AB . . . Q, and every value of z out of abc .../?, 
except m, which remains constant in the equation. 
If now we add to these equations (S) as many of 
(Act) as are sufficient for our purpose, taking different values 
of the exponent r, we can eliminate all the {p'Q') variables 
M„ from the equations (S) and obtain a relation U=0 among 
the numbers ABC . . . abc . . . , which will be different from 
that expressed in (A). This result will, I presume, break up 
into factors in various ways, thus, 
U=0=ViV2V3 . . . =WiWAV3 ... =0, 
each factor containing all the quantities ABC . . . abc .... 
The number of factors of distinct forms that are reduced to 
zero by the values of e and ABC . . . abc . . . that we choose 
to consider in equation (A), will, I conceive, be the number 
of distinct polyedra having that equation for their common 
description. But I do not presume here to express a con- 
fident opinion, on the subject of these factors. 
Let A be the greatest, or among the greatest, of the num- 
bers ABC . . . , and a be one of the greatest of abc . . . ; we 
