115] NOTE ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED POLYGON. 89
and substituting these values, the determinant divides by
1, Jc , Jc 2 ,
1, Jc', Jc' 2
1, Jc", Jc" 2
the quotient being composed of the constant term G, and terms multiplied by Jc, Jc', Jc" ;
writing, therefore, Jc = Jc' = Jc" = 0, we have 0=0 for the condition that there may be
inscribed in the conic U = 0 an infinity of triangles circumscribed about the conic
V = 0 ; G is of course the coefficient of f 2 in A]f, i.e. in the square root of the
discriminant of %U + V; and since precisely the same reasoning applies to a polygon
of any number of sides,—
Theorem. The condition that there may be inscribed in the conic U = 0 an
infinity of w-gons circumscribed about the conic V = 0, is that the coefficient of | n_1 in
the development in ascending powers of £ of the square root of the discriminant of
f U + V vanishes. [This and the theorem p. 90 are erroneous, see post, 116].
It is perhaps worth noticing that n = 2, i. e. the case where the polygon degene
rates into two coincident chords, is a case of exception. This is easily explained.
In particular, the condition that there may be in the conic 1
ax 2 4- by 2 + cz 2 = 0
an infinity of n-gons circumscribed about the conic
x 2 + y 2 + z 2 = 0,
is that the coefficient of f 1 ' 1 in the development in ascending powers of £ of
vanishes ;
V(l + af)(l +&f)(l + cf)
or, developing each factor, the coefficient of £ w_1 in
(1 + 1 a ç _ a 2 ^ 2 + T V a 3 f - & a 4 f 4 + &c.) (1 + £ &! - &c.) (1 + * c£ - &c.)
vanishes.
Thus, for a triangle this condition is
a 2 + b 2 + c 2 - Zbc - 2oa - 2ab = 0 ;
for a quadrangle it is
a 3 jp c 3 _ _ ]f c — ca 2 _ c 2 a — ah 2 — a 2 b + 2abc = 0,
which may also be written
(b + c - a) (c + a — b) (a + b — c) = 0;
and similarly for a pentagon, &c.
1 I have in order to present this result in the simplest form, purposely used a notation different from
that of the note above referred to, the quantities a*+ *■ + «* and , 2 + 2/ 2 + * 2 being, m fact, interchanged.
C. II.
