[128 
128] 
IN-AND-CIRCUMSCRIBED POLYGON. 
141 
: 2 + y 2 - P 2 = 0 
г-gons, is for 
respectively, 
lius being of 
and inscribed 
= 0, 
m of 
^onis symme- 
etrop. t. xiii. 
id to). Fuss 
which, he remarks, is satisfied by r = —p and r ~jU+q’ an< ^ conse q uen Hy the 
rationalized equation will divide by p + r and pq — r (p + q) ; and he finds, after the 
division, 
p3g3 q_.p2g,2 (p + q) r —pq(p + q) 2 r 2 — (p + q) (p — q) 2 r 3 = 0, 
which, restoring for p, q their values R + a, R-a, is the very equation above found. 
The form given by Steiner (Grelle, t. n. p. 289) is 
r (R — a)=(R + a)*J(R — r + a) (R — r — a)+(R + a)^(R — r — a) 2R, 
which, putting p, q instead of R + a, R — a, is 
qr=p\/(p- r) (q -r)+p V(gr - r) (q + p) ; 
and Jacobi has shown in his memoir, “ Anwendung der elliptischen Transcendenten 
U. s. w.,” Crelie, t. ill. [1828] p. 376, that the rationalized equation divides (like that 
of Fuss) by the factor pq-{p + q)r, and becomes by that means identical with the 
rational equation given by Fuss. 
In the case of two concentric circles a = 0, and putting for greater simplicity 
jR 2 
= M, we have 
r 2 
A+BÇ + Op + D(? + Ej? + bc. = (l + f) Vl + MÇ. 
This is, in fact, the very formula which corresponds to the general case of two 
conics having double contact. For suppose that the polygon is inscribed in the conic 
U= 0, and circumscribed about the conic U + P 2 = 0, we have then to find the 
discriminant of %U + U + P 2 , i.e. of (1 + %) U + P 2 . Let K be the discriminant of U, 
and let F be what the polar reciprocal of U becomes when the variables are replaced 
by the coefficients of P, or, what is the same thing, let — F be the determinant 
obtained by bordering K (considered as a matrix) with the coefficients of P. The 
discriminant of (1 + £) U + P 2 is (1 + f) 3 K + (1 + £) 2 .F, i.e. it is 
(1 + £> 2 [K{ 1 + f) + F), =(K + F)(1+1) 2 (1 + iff), 
where M = + ; or, what is the same thing, M is the discriminant of U divided 
by the discriminant of U -f P 2 . And M having this meaning, the condition of there 
being inscribed in the conic U = 0 an infinity of n-gons circumscribed about the conic 
U+P 2 — 0, is found by means of the series 
A + + C£ 2 + Dip + Fp + &c. = (1 + £) Vl + M£.