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on expanding, and neglecting negative powers of x, we find

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as the equation to two asymptotes inclined at angles + 45° and - 45° to the axis of x.

On combining the equation of this asymptote with that of the curve, we find that there is a value of a corresponding to an intersection of the curve with the asymptote.

Differentiating the equation to the curve, we find

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This equated to zero gives a cubic equation, which must have one real root negative, since all the terms of the numerator are positive: this indicates a minimum ordinate. The course of the curve shows that the other two roots of the cubic must be impossible.

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The value of shows that the curve is always con

d'y da

cave to the axis of a when a is positive, and convex when it is negative.

The form of the curve is given in fig. 43, where OA = a, OB = b; ON is the abscissa corresponding to the intersection of the curve with the asymptote; and OM is the abscissa of the minimum ordinate.

(2) Let the equation to the curve be

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This curve, see fig. 44, has four infinite branches, and the equations to its asymptotes are

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The curve cuts the axis of x at right angles at the origin, and at distances + a and a from the origin: at the latter two points there are points of contrary flexure, while the origin is a cusp. There is a maximum value of y corresponding to a value of a between 0 and

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a.

Solving the equation with respect to y, we find

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which when a = 0 gives y = 0, and y +

2a2

= 0 or y

To determine the effect of increasing a positively, let us consider the two values of y separately. Taking the upper sign and expanding the radical in ascending powers of a, we

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Now when is small, the first term gives the sign to the series, and y is therefore positive; and as no value of a can make y = 0, this branch of the curve lies always in the first quadrant, and extends to infinity, since y, when a = ∞. Taking the lower sign and expanding the radical in descending powers of a, we have

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2x4

which when a is negative and infinite: expanding in ascending powers of a, we have

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which when x = 0 is negative and infinite; hence this branch lies wholly in the fourth quadrant.

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For the negative values of a it is sufficient to observe that as the original equation remains unchanged when - and - y are substituted for + and +y, it follows that the opposite quadrants are symmetrical, and we need therefore only investigate the form of the curve in the first and fourth quadrants.

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To determine the asymptotes: since y when x = 0, the axis of y is an asymptote to the branch in the fourth quadrant: also by expanding the value of y in descending powers of a we have, neglecting the terms involving negative powers of x,

y = ± x,

as the equations to two other asymptotes.

dy dx

Differentiating the value of y, we find that at the origin

=

0, and therefore that the curve then touches the axis of x. We also find a minimum value for y when x = ± 3} a. This minimum value of y belongs only to the branches in the second and fourth quadrants, and not to the branches in the first and third quadrants.

Without proceeding to find the value of

d2y

da

it is not

difficult to see that at the origin there is a point of contrary flexure, since the curve there both touches and cuts the axis The form of the curve is given in (fig. 45).

of x.

When the equation cannot be solved with respect to one or other of the variables, it is necessary to have recourse to particular artifices suited to the case under consideration.

(4) Let the equation to be discussed be

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When = 0, y = 0: the multiplicity of values of y shows that there is a multiple point at the origin. Differentiating, we have

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To find the true value of this fraction, differentiate its numerator and denominator; then

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therefore at the origin one branch touches the axis of a and the other that of y.

To find the points where the tangent is parallel to the

dy

axis of a make = 0, whence ay = x2; substituting this

dx

value in the equation to the curve, it becomes.

a-2 ax3 = 0;

whence = 0, x3 = 2 a3.

The former value gives the origin; the latter gives one possible value = 2a, to which corresponds y=23a. From x the symmetry of the equation it is easy to see that the curve is parallel to the axis of y when y = 2a and a = 23a. Hence it appears that in the first quadrant there is a closed curve forming a loop which at the origin touches the two

axes.

To find the asymptotes put y = x, then we have

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When 1 both x and y are infinite. The expression for the intercept of the tangent on the axis of x is

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Therefore a line inclined at an angle of 135° to the axis of a, and cutting it at a distance - a from the origin, is an asymptote to the curve. For the form of this curve, see

(fig. 51).

(5) The form of the curve whose equation is

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is given in (fig. 47), where OB = b, OA = a.

The reader will find a great variety of curves discussed in the work of Cramer, before referred to. For lines of the third order he may consult Newton's Enumeratio Linearum Tertii Ordinis, and Stirling's Commentary on that work

SECT. 2. Curves referred to Polar Co-ordinates.

When the equation to a curve is given by an equation
r = ƒ (0),

a fixed point is to be taken as origin, and a fixed line passing through it as the axis from which is to be measured. The values of which make ƒ (0) = 0 are then to be found; these give the angles at which the branches of the curve which pass through the origin cut the axis. By giving to the values 0 and n we find the values of r when the curve cuts the axis; and by giving to the value (2n + 1) π we find the values of when the radius is perpendicular to the axis.

By making

dr
dᎾ

=0 we find the values of 0, for which ris

a maximum or minimum. After determining these points in the curve, the asymptotes, both rectilinear and circular, are to be sought out; and when these are known there will generally be little difficulty in finding the form of the curve, except when singular points occur; and these are to be investigated by the usual process.

It is to be observed that in all cases we must substitute both positive and negative values of 0, and that when the result gives a negative value for r, it is to be measured along

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