THE IN-AJsTD-CIECIJMSCEIBED POLYGON. 
235 
The similarity of form for the relations corresponding to the pentagon and the hexagon, 
and for those corresponding to the heptagon and the octagon, is, I am inclined to think, 
accidental ; the functions are homogeneous as regards degree and weight ; and the 
degrees and weights of the two consecutive functions being identical, the literal parts 
must be similarly constituted. 
III. 
M. Mention’s Formulae for the Case of two Circles. 
In the case of two cmcles, if, as usual, the radii of the inscribed and circumscribed 
circles are put equal to r and R respectively, and the distance of their centres to a, then 
the equations of the inscribed and circumscribed circles respectively may be taken to be 
— r'=0, 
{x — = 0 ; 
and if, in the notation of M. Mextiox, we put 
■ — = 
^4 (r^+ R^+a^ - 2r"R^ - 2 r\e — =-v, 
then the quadratic radical is 
s/(l+45){[l+(2+2i)5]=+4.r> ; 
and comparing this with 
V^l + 4(c+ + 4(f?4- 2^-c)r, 
we have 
b =i + 2 , 
<^^-2k=4(^^+2^•+v + l), 
and thence 
h= ^^- 2 , 
c= 2^■+v + l, 
d=—2h—2i ; 
and by means of these values, or by effecting the development in a different manner, we 
find 
C= j i.2 
l+'+i, 
D=2 {i(-»-l)}, 
E = r 4 . 
< +i .4(. + l) 
I -('+in 
2 K 2 
