Ex. 11. Find the Pliicker's numbers of the locus of the point of
contact of tangents from a given point to a family of curves with
characteristic (p, I).
W = p + 1, m' = 2p + M, k = in general, where M is the class of the
envelope of the family. is a p-ple point of the locus. Consider the
intersections with the locus of any line through 0, and the tangents
from to the locus. Verify by considering conies through four points
or touching four lines.]

* Ex. 5 to 13 should be omitted on a first reading.
f2



68 SINGULARITIES OF CURVE IV 8

Ex. 12. Find the degree and class of the locus of the intersection of
tangents from fixed points A and B to a family of curves of degree n
and class m with characteristic (p, I).
[n' — l(2m — l), m'. — m(M+p)+2l{m — l), where M is the class of
the envelope. A and B are l(m — l)-ple points of the locus and the
tangents from A and B to the envelope are »»-ple tangents of the locus.]

Ex. 13. Find the degree and class of the locus of the foci of a family
of curves of degree n and class m with characteristic (p, I). The foci of
the envelope are w-ple foci of the locus.
[In Ex. 12 take A and B at the circular points.]



CHAPTER V

FOCI

§ 1. Definition of Foci.
If co and a> are the circular points, and the lines Sco, Sa>'
touch a given curve (not at a> and co'), S is called a focus of the
curve.
