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proposition being likewise true. Hence, if dp be positive, the

dr curve is concave towards the pole, and convex if it be negative.

dp The expression must therefore change sign as we pass

dr through a point of inflection. To ascertain then a point of inflection we must assume



or = 0,

dr and ascertain whether any values of r, obtained from either of these assumptions, corresponds to a point in passing through which we observe a change of sign in

dp dr




General Principles and Examples. 162. From any proposed equation between x and y, the coordinates of the points which constitute a curvilinear locus, the form of the curve may be ascertained. The following is a general sketch of the ordinary method of effecting this object. First ascertain those values of x and y, if there be such, which render y and a respectively either zero or infinity: we shall thus determine where the curve cuts the axes of coordinates and the positions of asymptotes parallel to them. It will frequently be desirable next to ascertain the angles at which the curve cuts the axes. We must then determine the position of those asymptotes which are not parallel to either of the coordinate axes. The determination of the position of maximum or minimum ordinates and of points of inflection, as well as of multiple points and cusps, will give additional precision to the diagram. If our object be merely to ascertain the general form of the curve, it is not usually necessary to determine the actual position of points of inflection or maximum or minimum ordinates, the algebraical discussion of the equation being ordinarily sufficient to give a general notion of their position.

The method of tracing polar curves from their equations is similar, mutatis mutandis, to that which we have described in relation to curves referred to two axes of coordinates.

Ex. 1. To trace the curve represented by the equation

x’y + aby - aʻx = 0.


Obtaining y explicitly in terms of x, we have

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When y = 0, then x O or = 0. Hence the curve passes through the origin and touches the axis of x asymptotically.

the angle at which the curve cuts the axis of x at the origin : then, supposing x to be equal to zero,


xx2 ab 6


tan o

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which shews that o

a positive angle depending upon the ratio between a and b.

Also, y is positive or negative accordingly as x is positive or negative. It is evident also that so long as x is finite y is

y finite.

The form of the curve will therefore be such as in the subjoined diagram.

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0, and

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is zero when x = -

= + Co, it is plain that y must have two geometrical maxima. Their exact position may be found by differentiation, or thus: from the equation to the curve there is

4x*y* 4aʻxy + a* = a* 4aby,

(2xy - a?)? = a* – 4aby?. In order that x may be possible, the left-hand side of this equation must not be negative; hence the greatest value of y, without reference to sign, is given by the equation

at - 4aby = 0.

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determine two points P, P' of the curve, the ordinates of which


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X =

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+ Co.

are geometrical maxima. The value

is an analytical mini

20% mum of y, being its greatest negative value. It is easily seen from the general form of the curve that there must be a point of inflection at the origin of coordinates, and two others between 0, and x = 0, x =

We will however prove this analytically, and determine the actual position of these points. Differentiating twice, we get

dy ab - 22

a dx

(oz+ ab)??

x (2c2 – 3ab) dx2

(aca + ab). from this result it appears that there are three points of inflection, one at the origin, and two others Q, Q', of which the coordinates are



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This curve was called the Anguinea by Newton, in consequence of its form, and is one of the seventy-two species of curves of the third order which he has enumerated.

Ex. 2. To trace the curve

2 azy - bạy = 0.

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which shews that the curve touches the axis of x at the origin. When x has any positive value or any negative value greater 12

72 than


is positive, being negative from x to x = 0. When x = + OC, y

This curve is one of a class to which the name of Trident has been given by Newton : its form is indicated in the following diagram.

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= + 0.

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- z).

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'(x + a) (62 xo)

22 a being supposed to be less than 6. When x = + 0 or =

+00, which shews that the axis of y is an asymptote, When x=-a, y=0; when x=+b, y=0.

0 If x have a greater value than b, without reference to sign, y will be impossible. For each value of x, there will be two equal values of y with opposite signs, which shews that the curve is symmetrical with relation to the axis of x.

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