An elementary treatise on cubic and quartic curves, by A. B. Basset.

18

THEORY OF CURVES.

excluding OP, OQ not more than n (n- 1)- 2 or (n + 1) (n - 2)
tangents can be drawn from 0; but if 0 moves up to coincidence
with P, the two tangents OP, OQ coincide; hence excluding the
tangent at P, not more than (n + 1) (n- 2) tangents can be drawn
from P.
26. From a point of inflexion, not more than n(n-I)-3
tangents can be drawn to a curve.
At a point of inflexion P the curve cuts its tangent, and the
latter has a contact of the second order with the curve. From a
point 0 near P, draw three tangents OQ, OQx, OQ2, touching the
curve at points near 0. Then two of the points of contact will lie
on the same side of the tangent at P that 0 does, whilst the third
one will lie on the opposite side. But when 0 moves up to
coincidence with P all three tangents will coincide with the
tangent at P; hence the number of remaining tangents that can
be drawn from P to the curve is n (n - 1) - 3.
27. From a node, not more than n (n- 1) - 4 tangents can be
drawn to a curve.
Let 0 be a point on the curve near the node; then we have
shown in ~ 25 that (n + 1) (n - 2) tangents can be drawn to the
curve from 0. But two of these tangents will touch the branch
which does not pass through 0 at two points P and Q which are
near the node. Hence when 0 coincides with the node, these two
tangents will coincide with the other nodal tangent, and therefore
not more than (n + 1) (n - 2)- 2 = n (n- 1)- 4 tangents can be
drawn from the node.
28. From a cusp, not more than n (n- 1)- 3 tangents can be
drawn to a curve.
Let 0 be a point on the curve near a cusp; then only one
tangent can be drawn from 0 to touch the other branch in the
neighbourhood of the cusp, and when 0 coincides with the cusp,
this tangent coincides with the cuspidal tangent. Hence the
number of tangents which can be drawn from a cusp is
(n + 1) (n - 2) - 1 = n (n - 1)- 3.
The last four propositions may be stated in a somewhat
different form. If m be the class of a curve, the number of
tangents which can be drawn from any point 0 which is not on