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It has also a series of asymptotes parallel to the axis of y, and at the general distance (22 + 1) a from the origin, à being any integer positive or negative, but not zero.

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Ex. 11. To trace the Cardioid from its polar equation

p = a (1 – cos 6).

It is evident that, as 0 increases from 0 to , p increases from 0 to 2a; and that the expression for r remains unaltered when is substituted for + 0. Also

log r = log a + log (1 - cos 6),

dr sin e rdo 1

cos O

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Hence the form of the curve must be such as that exhibited in the diagram, there being a ceratoid cusp at the origin of polar coordinates.

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Ex. 12. To trace the spiral of Conon or Archimedes from its equation

αθ. The dotted line in the figure indicates that portion of the curve which is due to the negative values of 0.

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Ex. 13. To trace the Logarithmic Spiral of Descartes from its equation

p = CE".

This curve is called also the Equiangular Spiral, because it cuts all its radii at the same angle: in fact

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Ex. 14. To trace the Lituus from its equation

a? 72


This curve touches Ox asymptotically, and approaches O by an infinite number of circumvolutions.

For methods of constructing geometrically the curves which have been above considered, and deducing their equations from their geometrical properties, which is the converse of the course which we have adopted, as well as for historical information respecting them, the student is referred to Peacock's Gregory's Examples.


Ex. 15. To trace the curve represented by the equation

X* + y = aʼzy.

It is evident that x and y must have the same sign : hence the curve can lie only in two quadrants. In the neighbourhood of the origin, neglecting small quantities of higher orders than the second, we have

XY = 0,

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which shews that the axes of x and y are both touched by a branch passing through the origin. If – x and - y be written for x and y, the equation is not altered, and therefore the curve

, is the same in both quadrants. It is impossible for either x or y to be infinite, since ** + ywould then be a positive quantity of an infinitely higher order of magnitude than aʻxy. The curve must therefore be of the form

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Ex. 16. To trace the curve represented by the equation

x* + y* = 2axy. At the origin there is

Xyo = 0, which shews that the axis of x is touched by two branches and the axis of y by one branch of the curve. It is evidently impossible for x to have a negative value. The curve is symmetrical with respect to the axis of x, because its equation remains the same when

y is substituted for + y. Neither x nor y can be infinite, since 2* + ya would then be infinite compared with 2axy?

Hence the curve must be of the form

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Homogeneous Curves. 163. Curves represented by equations of the form

U = C,

where u is a homogeneous function of x and y, and where c is a constant quantity, may be traced very conveniently by assuming y = tx, and obtaining x and y in terms of t: a series of values must be assigned to t, and the corresponding values of x and y must be tabulated. It is desirable, however, first to ascertain whether there be any asymptotes by the method of Art. (110), Chap. II. Ex. 1. To trace the curye

x®y xy' = ao. λ

M Putting X = y

we get


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λμ – λμ = αθη

da and

M (524 – M) -1 (54* 14)

dr From (1) we see that

(a) (u) {(2)* - ()"} = 0, and, from (2),

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(xv) {5 (2)*— (x)} ( ) - (a) {5 (w)* – (1)"}


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we obtain four asymptotes to the curve, represented by the equations

x = 0, y = 0, y - x = 0, y' + i = 0.

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Again, putting y = tx, we see that


t(1 – t)'

t' and therefore, observing that the ratio of y to x must be always of the same sign as t, we have the following table of corresponding values :

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The form of the curve will therefore be the following.

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