(sin 6 X . sin fa . sin fa) 4- (sin # 2 . sin <f> 2 . sin fa)
is constant.
[Use the fact that in Ex.' 4 the product of the parameters of three
collinear points is — 1.]

Ex. 6. Show that no real tangent can be drawn to a crunodal cubic
from a point on the loop ; and that from any other point of the curve
two real tangents can be drawn, one point of contact being on the loop
and the other not.
[If <f>, ty are the parameters of a point and its tangential,
2^ + ^ = (mod. ttj).]

Ex. 7. Considering an acnodal cubic (supposed continuous at the two
ends of an asymptote) as divided into three parts by the three real
inflexions, prove that from any point on one part two real tangents can
be drawn to the curve, one point of contact lying on each of the other
two parts.
Ex. 8. The conic of closest contact at P with a crunodal cubic meets
the line of inflexions at unreal or real points according as P does or does
not lie on the loop.
[See Ex. 4.and Ch. X, § 2, Ex. 16 (ii).]

Ex. 9. The line of inflexions of

(a^x* + 2a i xy + a 2 y 2 ) z + A^x 3 + %A l x 2 y + ZA^xy 1 + A s y s =
is 4 (n a 2 - o^ 2 ) z + 3 (a A 2 + « 2 A - 2 a x A ,) x + 3 (a A 3 + a % A x - 2 a, A t ) y = 0.
