A cusp is a point where two branches of a curve stop abruptly and have a common tangent. Thus, the curve belonging to the equation

y2 = x23 (a2 + x2)

has a cusp at the origin, the common tangent of the two branches coinciding with the axis of x. The following is the form of the curve:

As another example we may take the equation

x1 – ax3y – axy2 + } a3y2 = 0,

which belongs to a curve of the form

which has a cusp at the origin of coordinates.

There are two species of cusps: the ceratoid, so called from its likeness to the horns of cattle, the curvature of the two branches lying in opposite directions, and the ramphoid, so called from its likeness to the beak of a bird, the curvature of the two branches lying in the same direction. The former figure affords an instance of a ceratoid, the latter of a ramphoid.

Analytical Property of Multiple Points in Algebraical Curves. 115. If u = f (x, y) = 0 represent the equation to an algebraical curve cleared of radical and negative indices, the values of x and y, at a multiple point, will satisfy the equations

Let a, b, be the coordinates of a multiple point. It is clear that, since two or more branches pass through the point (a, b), there must be two or more values of y corresponding to the value a ± h of x, where h is indefinitely small, one value for each branch; and that when h= 0, that is, at the point (a, b), all these different values must become equal values; hence it appears that the equation f(a, y) = 0 must contain two or more values of y, each equal to b, and that therefore the

d

equation dyf(a, y) = 0, its derivative, must (by the theory of equations) contain one or more roots, each equal to b. Similarly, the equation f(x, b) = 0 must contain two or more roots, each equal to a, and its derivative f(x, b)

d

dx

= 0 must

contain one or more roots, each equal to a. Hence the values of x and y, which correspond to a multiple point, must satisfy the equations

du

du

0,

dx

dy

du

du

where

and are the partial differential coefficients of u dx dy

with regard to x and y.

COR. Differentiating the equation u = 0, we have

Analytical Property of Cusps in Algebraical Curves.

116. The analytical property which we have established in the case of a multiple point holds good also in relation to cusps.

First, let us suppose that the tangent at the cusp is not parallel to either of the coordinate axes. Then it is evident that the very same reasoning is applicable to cusps as to multiple points, in consequence of the common feature which they possess, viz. that the value a + h or a - h of x corresponds to more than one value of y, and the value b + k or b − k of y to more than one value of x.

Secondly, let the tangent at the cusp be parallel to the axis of y. Then, when y = b + k or b−k, x will have more than one value, and therefore, when k is equated to zero, and the values of x are thereby made equal to a, the equation ƒ(x, b) = 0 will have more than one value a of x. It follows, therefore, that at such

a cusp

du

dx

= 0. Again, differentiating the equation to the curve, we have

du

du dy

+

= 0:

dx dy dx

but = ∞, since the tangent is parallel to the axis of y; and

dy dx

Hence, at a cusp as well as a multiple point,

Analytical Property of Conjugate Points in Algebraical Curves.

117. The same property may be proved to hold good also for conjugate points.

Let (a, b) be the coordinates of a conjugate point. Then, when x = a+h, or y=b+k, h and k being very small quantities, the values of y and x respectively must, by the nature of a conjugate point, be impossible. But, as we know by the theory of

equations, impossible roots enter rational equations by pairs, and must therefore, on the alteration of the values of the coefficients, by pairs degenerate into possible ones. Hence, when we put x = a, the equation f(a, y) = 0 must have at least two equal values b for y; and therefore, by the theory of

d

equations, the equation dyfa, y) = 0 must have one of these

d

dx

=

roots. Similarly, the equations ƒ(x, b) = 0 and 1 f(x, b) = 0, must have at least one root a in common. Hence, for the existence of a conjugate point it is necessary that, as in the case of multiple points or cusps,

Determination of the Multiplicity and of the Directions of the Tangents at a Multiple Point.

be the equation to a curve free from radicals and negative indices, x, y, being the coordinates of any point whatever in the curve. Let (x, y) be a multiple point: then at this point we know that

Let x+h, y+k, be another point of the curve near to the multiple point: then, from (1), putting x + h, y + k, for x1, y1, respectively, and expanding f(x+h, y + k) by Taylor's theorem, we have

1

Suppose that at the point in question the lowest partial differential coefficients of u, of which at any rate all do not vanish, are of the nth order; then the equation (3) is reduced to

Put hλx', k = λy': then, dividing out by λ" and multiply

=

Now in the limit when hk :: dx: dy, the quantity λ will become less than any assignable quantity, and therefore the equation will ultimately become

This equation, which is homogeneous in x' and y', is equivalent to n linear equations in x' and y', which will represent the tangents to the several branches of the curve, n in number, at the point (x, y), the point (x, y) being considered the origin of coordinates. Thus the degree of plurality of a multiple point is defined by the order of the lowest partial differential coefficients of u which do not vanish.