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61= $x (1

L ; (x + b)} on expanding, and neglecting negative powers of x, we find

y = + (x - 16) 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

dy 2003 + 3bx? + a? 2

dx (x3 - )(+ b)! 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.

When x = a, is infinite, or the curve cuts the axis at right angles.

The value of shows that the curve is always concave to the axis of x when ‘x is positive, and convex when it is negative.

The form of the curve is given in fig. 43, where 0 A = 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

X* a’ov?

2.0 a' 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 fexure, while the origin is a cusp. There is a maximum value of y corresponding to a value of a between 0 and - a. (3)

ry? + 2ay x3 = 0.
Solving the equation with respect to y, we find

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When x = 0, y = 0, and y = - . This will be readily seen by putting the original equation under the form

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

a l. 1 Q4 1.1 208

a 22.1.2 TS+ 1 m3 1.1 Remo

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Now when x is small, the first term gives the sign to the series, and y is therefore positive; and as no value of x can make y = 0, this branch of the curve lies always in the first quadrant, and extends to infinity, since y = 0, when x = c.

Taking the lower sign and expanding the radical in descending powers of w, we have

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which when x = oo is negative and infinite: expanding in ascending powers of X, we have

2a? 11.23 1.1 . y = -

(2 7 22.1.2 TS

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

For the negative values of w it is sufficient to observe that as the original equation remains unchanged when – w and - y are substituted for + Q 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.

To determine the asymptotes : since y = -00 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.

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

= 0, and therefore that the curve then touches the axis of x. dx 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 , 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 of x. The form of the curve is given in (fig. 45).

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

2013 – 3axy + y3 = 0. When x = 0, y = 0: the multiplicity of values of y shows that there is a multiple point at the origin. Differentiating, we have dy ay x20

-=- when x = 0, y = 0. dx - y - ax o

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

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which as y=0 when æ = 0, gives

dy dx

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

To find the points where the tangent is parallel to the axis of x make 0, whence ay = x?; substituting this value in the equation to the curve, it becomes

ao Cao = 0;

whence x = 0, x = 2a. The former value gives the origin; the latter gives one possible value x = 2a, to which corresponds y = 2ia. From the symmetry of the equation it is easy to see that the curve is parallel to the axis of y when y = 23a and x = 2:a. 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

3az

3az?

3+1' y = 3 : When x=-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 X, 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

(x +b) yo = (x + a) a*, b> a, is given in (fig. 47), where OB = b, 0A = 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:

p = f(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 0 which make f(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 no we find the values of when the curve cuts the axis; and by giving to the value (2n + 1)we find the values of go when the radius is perpendicular to the axis. By making 18 = 0 we find the values of 0, for which r is 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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