(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

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. x1.

find

Multiple Points.

Among these I include all those points for which we

dy

dx 0

including points where several branches intersect, 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 a 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.

(1) Let the equation to the curve be

3x2 - 2bx

=

x (3x + 2b)

= 0,

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

=

0,

du

dx dy

dy

=

0,

d2 u

dy

=

0,

d3 u

=

0,

=

dx dy2

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

a

(3)

The curve

x1 − 2 a x2 y − 2 x2 y2 + ay3 + y1 = 0

has at the origin a triple point, the values of

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

dy

being +2

dx

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

y =

a and a = 0.

= 0 at the origin, or two branches there touch

cach other as in fig. 32.

(6) In the curve

x + bx1 - a3y2 = 0,

dy

dx

which indicates a point of osculation, and as = 0 at the origin, the two branches touch the axis of x.

See fig. 33.

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

See fig. 34.

(8) The curve

x1 — a x2y — axy2 + a2y2 = 0

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 and also a branch touching the axis of y.

the axis of x, See fig. 36.

has a conjugate point at the origin, since x = 0, y = 0 satisfy the equation, but ah 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,