These three equations are satisfied by each of the following systems of values,

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

3x22 – 2y12 = 0,

and, for each of the two second systems,

3x2 + 4y = 0.

Thus we see that there are three double points.

The figure

Multiplicity of a Multiple Point at the Origin.

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

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 a two branches, of the curve.

Ex. 2. Take the curve

x1 - 2ax3y - 2x2y2 + ay3 + y* = 0.

Then, retaining terms of the lowest dimensions, we have

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

so that there is a triple point at the origin.

Point of Osculation.

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 d2u <

dx dy dx2 dy'

when the equation (1) will give impossible relations between ' 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 x' 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, is sufficient for the purpose.

Ex. Take the curve

(y − cx3)3 = (x − a)o. (x − b)3,

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

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

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

2 (y - cx3) = 0,

du

dx

du

dy

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)*,

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.

Points d'Arrêt or Points de Rupture.

122. An algebraic curve never stops abruptly in its course, that is, it never possesses singular points of the kind called by French writers points d'arrêt or de rupture. Such points are however of frequent occurrence in transcendental curves. instance, in the curve belonging to the equation

For

The impossibility of the existence of such points in curves represented by algebraical equations depends upon the fact that impossible roots enter algebraical equations, involving one unknown letter, by pairs. Suppose in fact that, when x is equal to ah, y has an impossible value for each small value of h however small, and that y has a possible value when x is equal to a + h. Then, when x passes from a h to a +h, one value of y, and therefore, by the nature of algebraical equations, two values of y, the two values being of the forms a± √(- ẞ), must change into possible ones, which will evidently, in consequence of the correspondency of their values, be equal to each other when his indefinitely diminished. The existence of two equal values of y, corresponding to the value a of x, shews that there is no abrupt termination of the curve at the point of which the abscissa is a.

Points Saillants.

123. A point saillant is a point of a curve where two branches of the curve stop abruptly and have tangents inclined to each