THE IN-AND-CIECUMSCEIBED POLYGON. 
231 
IL 
Application to the several Polygons up to the Enneagon. 
If the equation of the inscribed conic is U= 0, and that of the circumscribed conic 
is V=0, and if the discriminant (U + |V) is iii tiie first instance represented by 
then the square root of the discriminant is 
l + 2i3H-2(y-/3^)r+2(^-2^y+2/3^)r+&c., 
so that the condition for the triangle is 
and that for the quadrangle is 
S-2^7+2/3®=0. 
It is obviously convenient to introduce into the formulae, in the place of 7 and 5, the 
quantities 
c = 7-/3^ 
(7=S-2/37+2/3*; 
and writing also, for symmetry of notation, h in the place of /3, we have 
3 = ^, 
y = C+l)\ 
I =fZ+ 2 Jc, 
so that the discriminant will be 
= 1 + 4^1 + 4(c+ + 4((Z+ 2^c)f , 
which is 
=(i+2^i+2cr)^+4(fzr— cT)- 
But for homogeneity I introduce the quantity a=l, and put the discriminant 
={l-\-‘2.hl-\-2acef+ 
The square root, divided by is 
=i(l+ 2 J|+ 2 «cr)v/l + 4a^(6Zr-cT)^(l + 2 ^|+ 2 «cr/J 
or developing, this is 
l,(l+2^|-f2ac|) 
+ 2 (f7f-cr*r) Hl-\-2hl+2acl^) 
- 2a\dl^-(fU^{\-\-2hlA-‘^aclJ 
+ 4«^( fZf - (1 + ^ + 2«cf )* 
- 10 ««((Zr - ^ (1 + 2^1 + 2 «cr/ 
4 - &c.; 
