and its rectilinear asymptote y = μx+ß for a point whose abscissa is x is

$”„(μ)[Þn_1(μ1)]2 – 2¢ ́„(μ)Þ'n−1(μ)Þn-1(μ) + 24n-2(μ)[$'n(μ) ]2 ̧ 2x [(μ)]3

assuming that no other asymptote is parallel to this one. Show from this result that the curve at opposite extremities is in general also on opposite sides of the asymptote.

50. Show that the curve

(y-2x) (y + x)+(y+3x) (y − x)+x=0.

has the parabolic asymptote 3y2 – 12xy + 12x2 + 5x = 0.

51. Show, by transforming to the point h, k, that the asymptotes of the general curve of the nth degree

and that the co-ordinates of that point are

[Professor Cayley uses the notation (α, α,

the general binary quantic of the nth degree:

ax, y)" for

CHAPTER IX.

SINGULAR POINTS.

237. Concavity. Convexity.

In the treatment of plane curves the terms concavity and convexity with regard to a point are applied with their ordinary signification. Thus, for example, any arc of a circle is said to be concave to all points within the circle; whilst to a point without the circle the portion lying between that point and the chord of contact of tangents drawn from the point is said to be convex and the remainder of the circumference concave.

238. In general the portion of a curve in the immediate neighbourhood of any specified point lies entirely on one side of the tangent at that point. This is clear from the definition of a tangent, which is considered as the limit

ing position of a chord. There is an ultimately coincident cross and recross at the point of contact, as shown at the ultimately coincident points P, Q in Fig. 31; so

that the immediately neighbouring portions AP, QB must in general lie on the same side of the tangent PT.

239. Point of Inflexion.

The kind of point discussed in Art. 238 is an ordinary point on a curve. It may however happen that for some point on the curve the tangent, after its cross and recross, crosses the curve again at a third ultimately coincident point. Such a point can be seen magnified in Fig. 32.

R

Fig. 32.

In this case it is clear that two successive tangents coincide in position: viz, the limiting positions of the chords PQ, QR. The tangent at such a point is therefore said to be "stationary," and the point is called a "point of contrary flexure" or a "point of inflexion" on the curve. The tangent on the whole crosses its curve at such a point, and the curve changes from being concave to points on one side of the tangent to being convex to the same set of points.

240. Point of Undulation.

Again, there may be a point on the curve for which the

Р Q R S

Fig. 33.

tangent crosses its curve in four ultimately coincident points, P, Q, R, S, as seen magnified in Fig. 33, and the

point is then called a "point of undulation" on the curve. There are now three contiguous tangents coincident, and the tangent on the whole does not cross its curve. And it is clear that singularities of a higher order but of similar character may arise.

241. Analytical Tests. Concavity and Convexity.

It is easy to apply analysis to the investigation of the form of a curve at any particular point.

Let us examine the point x, y on the curve y= p(x). Let P be the point to be considered, P1 an adjacent point on the curve. Let PN, P1N1 be the ordinates of P

1

1

1

and P1, and suppose P1N1 to cut the tangent at P in Q1. Then ON=x, NP=y=4(x).

h2

= p(x)+hp′(x)+21Þ”(x)+....., ...(1)

by Taylor's Theorem. Again, the equation of the tangent

Hence the ordinate of the curve exceeds the ordinate of the tangent by

Now, if h be taken sufficiently small, the sign of the right-hand side will be governed by that of its first term; and this term does not change sign with h because it contains an even power of h, viz., the square. Hence in general, on whichever side of P the point P1 be taken, N1P1-N1Q1 will have the same sign-positive if "(x) be positive, and negative if "(x) be negative; and therefore the element of the curve at P is convex or concave to the foot of the ordinate of P according as "(x) is positive or negative.

1

We have drawn our figure with the portion of the curve considered above the axis of x. If, however, it had been below, the signs of N1P1 and N1Q1 would both have been negative and we should have had the contrary result. But observing that p(x) is positive for points above the axis of x, and negative for points below, we may obviously state the unrestricted rule that the elementary portion of the curve y = p(x) in the neighbourhood of the point (x, y) is convex or concave to the d2y

dx2

foot of the ordinate according as p(x)p"(x) or y is positive or negative.

242. Points of Inflexion.

If "(x)=0 at the point under consideration, we have

and, as before, the sign of the right-hand side, when h is