90 
NOTE ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED POLYGON. 
[115 
Suppose the conics reduce themselves to circles, or write 
U — x 2 + y 2 — R 2 = 0, 
V = {x — a) 2 + y 2 — r 2 = 0 ; 
R is of course the radius of the circumscribed circle, r the radius of the inscribed 
circle, and a the distance between the centres. Then 
%U+V =■(£■ + l,f+l,- %R 2 - r 2 + a 2 , 0, - a, 0) (x, y, l) 2 , 
and the discriminant is therefore 
- (f + l) 2 + r 2 - a 2 ) - (£ + 1) a 2 = - (1 + f) {r 2 + £(r 2 + R 2 - a 2 ) + fJR 2 }. 
Hence, 
Theorem. The condition that there may be inscribed in the circle x 2 + y 2 - R 2 = 0 
an infinity of n-gons circumscribed about the circle (x — a) 2 + y 2 — r 2 = 0, is that the 
coefficient of in the development in ascending powers of £ of 
\/(l + £) {r 2 + Ç(r 2 + R 2 -a 2 ) + ?Rr} 
may vanish. 
Now 
(A +BÇ + 6’p)‘ = VJ |l + \B I + (UC - iff) f s +...}, 
or the quantity to be considered is the coefficient of £ n_1 in 
(1+«-№•••) {l + iBI + (*40-^1,+ 
where, of course, 
H = r 2 , B = r 2 + R 2 —a 2 , C = R 2 . 
In particular, in the case of a triangle we have, equating to zero the coefficient 
of r, 
(A-B) 2 -4AC= 0; 
or substituting the values of A, B, C, 
(a 2 - R 2 ) 2 - 4>r 2 R 2 = 0, 
that is 
(ia 2 — R 2 + 2Rr) (a 2 — i2 2 — 2Ær) = 0 ; 
the factor which corresponds to the proper geometrical solution of the question is 
a 2 — R 2 + 2 Rr = 0, 
Euler’s well-known relation between the radii of the circles inscribed and circumscribed 
in and about a triangle, and the distance between the centres. I shall not now discuss 
the meaning of the other factor, or attempt to verify the formulae which have been 
given by Fuss, Steiner and Richelot, for the case of a polygon of 4, 5, 6, 7, 8, 9, 12, 
and 16 sides. See Steiner, Grelle, t. n. [1827] p. 289, Jacobi, t. III. [1828] p. 376; 
Richelot, t. v. [1830] p. 250; and t. xxxvm. [1849] p. 353. 
2 S'tone Buildings, July 9, 1853.