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and for a value a-h impossible, or vice versa, the curve stops short at the point in question, and is doubled back on itself, forming what is called a cusp. The cusp is said to be of the first species or a ceratoid* when the branches touch the common tangent on opposite sides; and of the second species or a ramphoid when they touch on the same side. These may be distinguished by the consideration day

that in the first the values of are of opposite, and in the

dx2

second of the same signs. It is to be observed that at a cusp the two branches of the curve never make with each other an angle the trigonometrical tangent of which is of finite magnitde: we cannot properly say that the angle itself is infinitely small, as in fact it is equal to two right angles, the inclination of the one branch of the curve being measured in a direction opposite to that of the other. Although the condition of when of the form having

dy

dx

0

impossible values always indicates a conjugate point, yet it may happen that

dy
dx

and any number of the differential coefficients are possible at a conjugate point. In such cases the impossible branch of the curve does not pierce the plane of the axes, but touches it at the conjugate point, the order of contact being that of the highest differential coefficient which is possible. To determine with certainty whether a point be or be not a conjugate point or a cusp, it is always necessary to try whether the equation to the curve gives possible values for both variables on each side of the point in question.

If some of the values of

dy
dx

be possible and some im

possible for the given value of a, there is a conjugate point situate on the curve; that is, a branch in the impossible plane pierces the plane of reference in a point through which there passes a possible branch of the curve.

Képas, a horn.

† Ράμφος, a beak.

For a fuller development of the relation between the dy 0

various kinds of points indicated by the condition dx 0 the reader is referred to a paper by Mr Walton in the Cambridge Mathematical Journal, Vol. II. p. 155.

If the equation to the curve be put under the more symmetrical form

u = f (x, y) = 0,

we easily obtain analytical conditions for distinguishing between the three classes of double points indicated by the

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Proceeding to the differential of the second order, we find in consequence of the preceding condition

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Now for a true double point we must have two possible

values for ; for a point of osculation we must have the

dy dx

two values equal; and for a conjugate point we must have

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If the point be more than double, it is necessary to proceed to higher differentiations, but the formulæ become too complicated to be of much use.

The second of the preceding conditions furnishes an easy demonstration of the following general property of curves of the third order. "The three asymptotes of a curve of the third order being given, the locus of the points of osculation is the maximum ellipse which can be inscribed in the triangle formed by the asymptotes: the locus of the conjugate points is within, and of the double points without this ellipse."

If we refer a curve of the third order to two of its asymptotes as axes, their intersection being the origin, its equation must evidently be of the form,

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(ax + b + cy)2 — acxy = 0,

or a2x2 + acxy + c2y2 + 2 ab x + 2bcy + b2 = 0,

which is the equation to an ellipse.

That this ellipse is the maximum ellipse inscribed in the triangle formed by the asymptotes is easily shown. The equations to the three asymptotes are

x = 0, y = 0, and ax + cy + 2b = 0.

From the last it appears that the interceps of the axes

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the points of contact bisect these two sides of the triangle.

dy dx

If in the value of derived from the equation to the

b

ellipse we substitute the values and

we find

dy dx

=

a

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c

a

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which is the same as that derived from

the equation to the third asymptote, and as these values of x and y satisfy both the equation to the ellipse and that to the asymptote, it appears that the ellipse touches all the three sides of the triangle in their middle points, which by Chap. VII. Ex. 19, is the property of the maximum ellipse. The latter part of the theorem is too obvious to need demonstration. This proposition is due to Plucker, Journal de Mathématiques, (Liouville) Vol. 1. p. 11.

Points of Contrary Flexure or of Inflexion.

Ex. (1) The equation to the Witch of Agnesi is

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d2y dy
and
da

both become infinite when x = 0 and when

x=

dx

x = 2a, but neither of those values gives a point of inflexion, since y is impossible when a is negative or greater than 2a.

(2) The curve whose equation is

x3-3bx2 + a2y = 0

has a point of inflexion the co-ordinates of which are

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(3) Let the equation to the curve be

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There are two points of inflexion, the co-ordinates of the one being

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(4) Let the equation to the curve be

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(5) Let the equation to the curve be

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y = b + (x − a)",

where m and n are both odd.

m

If >1, x = a gives a point of inflexion, the tangent

n

being parallel to the axis of x.

to

m

If <1, x = a gives a point of inflexion corresponding

d2y

dx2

n

∞, the tangent being perpendicular to the axis of a.

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