(6) In the curve of sines the equation to which is

there is a point of inflexion wherever the curve cuts the axis of x.

In polar curves points of inflexion are found by the

conditions that at such points

dp

=

0, and changes sign in

dr

side of the point corresponding to these values: the point is therefore one of inflexion.

Hence the origin is a point of inflexion for two branches

therefore when cose and y = a(1-e) there is a point of inflexion.

The preceding examples are taken chiefly from Cramer, Analyse des Lignes Courbes, Chap. XI.

Multiple Points.

Among these I include all those points for which we

dy 0

find =

dx 0

including points where several branches inter

sect, or nodes, points of osculation, cusps, and conjugate points.

Let u=0 be the equation to a curve free from radicals and negative indices, and assume

then if these three equations be satisfied simultaneously by x = a, y = b, (a, b) will be a multiple point. In order to determine its nature, suppose that the lowest partial differential coefficients of u, of which at any rate all do not vanish for these particular values of x and y, are of the nth order, then the multiple point will be one of n branches, the directions of their tangents being determined by the equation

By ascertaining every pair of values of x and y which will

in the same way, we may ascertain the positions and the plu

rality of all the multiple points of the curve.

Thus we see that there is a double point at the origin, its two tangents making with the axis of x angles the tangents

(2) Let the equation to the curve be

Hence there will be a triple point at the origin, the directions of its branches being defined by the equation

this equation is satisfied by dy = 0, which shews that one branch touches the axis of x, the two other branches being inclined to it at angles of which the tangents are and

-():

See fig. 29.

(3) The curve

x1 − 2 a x2y - 2x2 y2 + ay3 + y2 = 0

3 a

(2)

has at the origin a triple point, the values of

and 0. The form of the curve is given in fig. 30.

Both of these vanish when y = 0 and ∞ = ±a, and when 0. There are three double points corre

y = a and x =

sponding to these values of x and y.

we find

+

x2 + x2 y2 – бa x2y + a2y2 = 0

= 0 at the origin, or two branches there touch

each other as in fig. 32.

(6)

In the curve

x2 + bx1 − a3y2 = 0,

we find at the origin

u

2

=

(d) (d) - (ddy) - 0,

which indicates a point of osculation, and as

origin, the two branches touch the axis of x.

(7). The curve

(by − cx)2 = (x − a)3

(S)

dy

= 0 at the

dx

See fig. 33.

has a cusp of the first species when x = a; the common tangent is parallel to the axis of x.

(8) The curve

See fig. 34.

has at the origin a ramphoid cusp, the axis of a being the common tangent. See fig. 35.

has at the origin a ceratoid cusp touching the axis of x, and also a branch touching the axis of y. See fig. 36.

has a conjugate point at the origin, since x = 0, y = 0 satisfy the equation, but = ±h when h is small make y impossible.

(- -) *

which indicates that there are two impossible branches

passing through the plane of the axes at the origin.

(11) The curve whose equation is

(c3y - x3)2 = (x − b)3 (x − a), a<b,