by xa, yb, but if when x is increased or decreased by a quantity, however small, y is affected with, then at such a point the curve, which lies in some other plane, pierces the plane of reference; and the point is a conjugate or isolated point; and of course one or two or more branches of a curve may pass through such a point: as for instance if the equation to a straight line is y-b = (-) (-a), the equation is satisfied by xa, y = b, which indicates a point in the plane of reference; but every other point of the line is in the plane passing through the line BD, see fig. 71, OA = a, AB = b, and perpendicular to the plane of the paper.

When two branches of the curve simultaneously pierce the plane of the paper, the two roots of (126), Art. 248, are impossible, as is the case in Ex. 8, which is traced below in Art. 260. And a curve may have any number of such conjugate points, by continually passing through the plane of the paper, such as in the subjoined example:

y = ax2 + (bx) sinx.

......

The curve is traced in fig. 72, the dotted line indicating the branches in a plane perpendicular to that of the paper. y = ax2, which represents the diametral curve, is a parabola, в'o B, drawn in the figure; and the ordinate to the curve is periodically reduced to its ordinate when x = 0, or = π, or = 2π, or any multiple of ; but when x is negative, the part of the ordinate to be added to or subtracted from the ordinate of the parabola is affected with (−), except at the points where x = some multiple of π, at which the branches of the curve pierce the plane of reference: and thus it continues ad infinitum, the curve itself being continuous, but there being a series of discontinuous points, if we consider only those points which the plane of the paper contains.

Curves such as the last have been called "Courbes Pointillés *;" which name however has been given by writers who discard the mode of interpretation of the symbol of impossibility which we have employed in this Treatise, and are therefore obliged to allow that algebraical expressions admit of discontinuous geometrical interpretation: a result surely at

* See page 382 of a "Treatise on the Differential Calculus," by Augustus De Morgan, M. A. Baldwin and Cradock, London, 1842.

variance with the algebraical nature of such functions, which admit of differentiation, and thereby indicate that they fulfil the law of continuity.

4) On the method of determining the simultaneous increase and decrease of x and y nothing more need be said; but we dy must be careful to investigate the points at which = 0,

dx

and = ∞, and to observe whether or not there is a change of sign, as such is the criterion of maxima and minima. With

dy
dx

this object we shall equate to 0 and ∞, and examine the course the curve takes at these critical points.

dy

dx

changes sign or

5) In regard to asymptotes, and the course of the curve with respect to them, we must examine the finite values of x for which y is infinite; and the values of y for which a is infinite, x as such will be asymptotes parallel to the axes of y and x respectively; and by investigating whether not for these asymptotic values, we shall determine whether the infinite ordinate is a maximum or a minimum; that is, whether it returns, or whether it continues round the circle of infinite radius, such as we described in the last chapter; and which of the forms delineated in figs. 24, 25, 26, 51 the curve takes. We must also be careful to determine whether there are rectilineal asymptotes inclined at oblique angles to the axes of coordinates, general rules for the discovery of which have been given in Art. 234; and whether the curve is above them or below them. It may happen that two distinct branches of a curve will approach the same asymptote. Sometimes also a curve sin x will cut its asymptote; as e. g. if y = a the axis of x is an asymptote, and the curve cuts it whenever x = an integral multiple of π.

x

6) The general character of a curve with regard to the curvature of it in a particular direction and its points of inflexion has been sufficiently discussed in Section 3 of the present

d2y

dx2

Chapter. Practically however is of little use in enabling us to trace a curve, unless it assumes a simple and explicit form; and should also at any point of the curve

day dx2

= 0, and

not change its sign, we may conclude that more elements of the curve than two, and therefore more points of the curve than three, are in one and the same straight line.

7) There is nothing more to be added on the theory of multiple points and their varieties.

8) And generally it is of little use to examine the values of x and y, except at such critical points as we have above described; and except when x = ∞, and y = ∞, in order that we may determine the course the curve is taking at such distances from the axes.

259.] Thus, in the discussion of any particular equation representing a plane curve, the method indicated by the following rules is the most convenient to adopt :

I. Reduce the equation if possible to the explicit form, and simplify it, as far as may be, by means of a change of origin, or by a transformation into polar coordinates.

II. Discover, arrange, and tabulate with their proper signs, all the critical values of y and x, both in and out of the plane of reference.

III. Discuss and tabulate the critical values of

dy dx

as c. g.

determine the angles at which the curve cuts the axes, the maximum and minimum ordinates, &c.

IV. Find the equations to the asymptotes, and determine whether the curve is above or below them.

V. Find, if it is possible in a convenient form,

determine the direction of curvature, and the points of inflexion.

VI. If at any point

dy 0
dx 0

evaluate the quantity, and

determine the several double, triple points, &c.

260.] Examples illustrative of the preceding theory. Ex. 1. Trace the curve whose equation is y = a2x3.

From (152) it is plain that the curve is symmetrical with respect to the axis of a; and since the curve passes through the origin, the former value of dy nate form, the value of which, as before shewn in Ex.5, Art. 249, is such as to give a cusp, both branches of which touch the axis of x, and which are in the plane of the paper on the positive side of the axis of y, and out of it on the negative side. The

at that point assumes an indetermidx

same thing is also apparent from the second value of

= 0 at the origin, and is affected with ±√

d2y

dy

dx'

which

when x is nega

tive, and with when x is positive; also being affected

with, shews that one branch of the curve is convex, and the other concave, downwards.

From 1 it appears that the curve passes through the origin, with two branches, both touching the axis of x, one of which is convex, and the other is concave, downwards, and which are out of the plane of the paper on the negative side of the axis of x, and are in it on the positive side. From 2 and 3 it appears, that as a increases, whether positively or negatively, y increases dy dx

also, and since approximates too, that the curve approaches

to parallelism with the axis of y; the only critical value is x = 0; the curve is drawn in fig. 64.

Ex. 2. Discuss the curve

y = x3-2x2-5x+6= (x+2) (x-1) (x-3);

As the equation does not admit of expansion in descending powers of x of a form such that the highest positive power of x may be unity, it follows that the curve has no rectilineal asymptote. The table of the critical values is as follows.

x=

On examination of which table it appears that, when ∞, the curve is at an infinite distance below the axis of , approaching to parallelism with the axis of y, and being concave downwards; whence it cuts the axis of x, when x = = −2, as shewn by 1, at a large acute angle, and the ordinate attains a maximum at the value of x given by 6; whence the ordinate decreases, cutting the axis of y at a distance + 6 from the origin, and being concave downwards until x = at which

2
3'

point, as shewn by 7, there is a point of inflexion; and the curve being convex downwards cuts the axis of x, when a = 1, and decreases until x is equal to the value given in 5, at which point there is a minimum ordinate: after which the ordinate again increases, cuts the axis of a when x = 3, and goes off to