Ex. 1. To determine the multiplicity of the point x = 0, y = 0,

Hence, for the determination of the tangents at the multiple point, which we see is a triple point, we have, substituting for the partial differential coefficients in the equation

which is equivalent to x' = 0, and y' = 0:

the former of which equations shews that the axis of y touches one branch of the curve, and the latter, that the tangents to two branches coincide with the axis of x. The form of the curve is exhibited in the diagram:

Ex. 2. To determine the multiple points of the curve

(y2 − 1)2 = x2 (2x + 3).

In this case u = (y2 − 1)2 – x2 (2x + 3) = 0,

and, as conditions for a multiple point,

These three equations are satisfied by each of the following

becomes, for the first system of values of x and y,

Thus we see that there are three double points.

is subjoined.

The figure

Multiplicity of a Multiple Point at the Origin.

Let

119. The existence and multiplicity of a multiple point at the origin may be ascertained more simply by inspection. the equation to a curve, arranged in groups of terms of different dimensions, be

12

u, denoting generally a series of terms of r dimensions, and u denoting those of lowest degree in the equation. Then, in the immediate neighbourhood of the origin, we may neglect terms of higher compared with those of lower orders, so that the equation will become

the dimensions of this equation determining the degree of the multiplicity, and the simple factors, into which it may be decomposed, defining, when equated each of them to zero, the directions of the branches. This method of finding the multiplicity of a multiple point, may be readily deduced from the general equation given in the preceding Article.

Ex. 1. Taking the first example of the preceding Article, we have, retaining only the term of the third dimension,

xy2 = 0,

which shews that the axis of y touches at the origin one branch, and that of x two branches, of the curve.

Ex. 2. Take the curve

x* - 2ax3y - 2x2y2 + ay3 + y1 = 0.

Then, retaining terms of the lowest dimensions, we have

− 2ax3y + ay3 = 0,

whence, for the equations to the tangents at the origin,

120. A point of osculation is a multiple point in which the several branches of the curve have a common tangent. Thus cusps are a species of points of osculation.

Suppose that there are only two branches at the point, then the roots of the equation

must be equal: hence, as a necessary condition for a point of osculation,

If this condition be not satisfied for any point of a curve corresponding to the three equations

in which case there will be a double point with two distinct tangents; or

d2u dx dy

2

<

d2u d2u
dx2 · dy2'

when the equation (1) will give impossible relations between x' and y', and (x, y) will be the coordinates of a conjugate point.

Remark on the Theory of Multiple Points.

121. If there be a multiple point in a curve, its position and its multiplicity may be ascertained by investigating the pairs of values of x and y which satisfy the three equations

and by determining the order of the lowest partial differential coefficients of u which do not all vanish at the point. The directions of the tangents will be ascertained by the formula of Art. (118). If the relations between x' and y', expressed by this formula, be impossible, the existence of a conjugate point is at once indicated. We cannot however be sure that the point is not a conjugate instead of a multiple point or cusp, even when all the relations between ' and y' are possible. Additional considerations are necessary in order to ascertain this to a certainty: an examination of the general nature of the curve in the neighbour

hood of the point, by an algebraical discussion of its equation,

a being supposed to be less than b. Then, putting

u = (y - cx3)2 - (x − a)o. (x − b)5 = 0,

6cx2 (y - cx3) - 6 (x − a)3 (x − b)3 – 5 (x − a)° (x − b)* = 0,

2 (y - cx3) = 0,

y = a3c. d'u

Then also

=

18a*c2,

dx2

=

2,

y' - 6a2cx'y' + 9aac2x22 = 0,

which might seem to indicate a point of osculation, when the equation to the common tangent of the two-branches would be

y' = 3a2cx'.

It is easily seen, however, that the point is really a conjugate point. For

y = cx3 ± (x − a)3 (x − b)a,

which shews that y is impossible when x differs very slightly from a.

For further information on the subject of this Article, the reader is referred to a paper on the General Theory of Multiple Points in the Cambridge Mathematical Journal for November,

1840.