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c = the length of the string which is attached to the point, and whose extremity generates the involute; then

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which is the equation to a circle whose centre is at the point, and whose radius is equal to c.

As the involute of a point is a circle, so conversely the evolute of a circle is a point.

Ex. 4. To find the equation of the involute of the semicubical parabola whose equation is 27 an2 4§3, the length of пp being longer by 2 a than the arc A. Fig. 13.

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(4 §1+81 a2 n2)

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3 a

2

3(3a)

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{({+3a)*— (3a)*}

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;

Now

p = σ+2a, by the conditions of the problem;

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the equation to a parabola, situated as in the figure.

172.] On involutes of curves referred to polar coordinates. Let AP be the curve, fig. 14, whose involute is to be determined; and let its equation be

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Let PP' be the tangent at P, P' being the generating point of the involute. Draw from the pole s, sy perpendicular to PP', and sy

perpendicular to P'Y, which is the tangent to the involute at P'. Then sp = r, sy = p; sp′= r', sy'= p'; and our object is to find the relation between r' and p'. Let ds represent a lengthelement of the original curve; and, since PP' is the radius of curvature of the involute at P', let PP' p'; then, by (40), Art. 292, Vol. I, do' = ds.

and from the geometry,

= 8 + c ;

y'2 = p2+p'2,

p2 = No2+p22 −2p'p';

=

(106)

(107)

and after the elimination of r, p, s from these equations there will remain an expression in terms of r' and p', which is the equation to the involute.

Ex. 1. To find the equation to the involute of a circle. Let the centre of the circle be the pole: then, if a = the radius, its equation in terms of r and p is

r = p = a;
12 = a2;

whence (106) gives r2-p'

which is the equation to the involute of the circle.

This equation however may be found in the following way. In fig. 31, let qSA = 0, so that the arc AQ= a0; then, if s is the origin, and P is (x, y), PQ = a0, and

y = a sin 0-a◊ cos 0

x = a cos 0+ a0 sin 0

(108)

which simultaneous equations are those to the involute of the circle. From them we have

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which is the equation to the involute of the circle in terms of a

and y.

Also, since from (108) dy =a0 sin 0 de, dx = a0 cos 0 d0,

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Ex. 2. To find the equation to the involute of the logarithmic spiral.

Let a be the constant angle at which the curve cuts all the radii vectores; then its equation is

Therefore, see fig. from the pole to the

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15, if PP' is equal to the length of the curve point P, and if PP' = p', by (54), Art. 162,

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the equation to a logarithmic spiral, similar to the original one; that is, which cuts all its radii vectores at a constant angle, the same as that of the original spiral. From (110) it is evident that PSP' is a right-angle, and therefore SP'Y' SPY = a: the involute therefore is also the locus of the extremity of the polar subtangent.

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SECTION 6. Various problems of Plane Geometry solved by means of Single Definite Integration.

173.] Curves and classes of curves are generally defined by means of some salient geometrical property, and the equation of the curve is the expression of this property by means of mathematical symbols. Now, as many of these principal properties are only capable of expression in terms of differentials, it is evident

that the equation which defines a curve or a class of curves may be given in terms of these differentials in combination with, or independent of, finite coordinates, whether rectangular or polar. When the equation is given in this form, it is called the differential equation of the curve; and it is frequently necessary to obtain the finite equation from the differential equation. For this purpose integration is required.

The problem in its most general form is entirely beyond the present power of mathematical analysis. Many special forms of it however are capable of solution; and although in a subsequent part of this Treatise other methods will be investigated, yet it is expedient at once to apply to such problems the process of single definite integration so far as it is applicable, both for the sake of the geometrical problems, and for the illustration of the process of integration, which it so well exhibits. In some of the following examples, integration will be performed more than once; but only a series of successive integrations will be required, and these will be of the nature explained in Chapter V, Art. 149, 150. In all cases the limits will be given, and the integrations will be definite.

dx

174.] I will in the first place take that class of problems in which dy is given in terms of x and y in a simple form, capable of integration. This equation, geometrically interpreted, assigns the law of the ratio of the simultaneous increments of the coordinates at any point of a plane curve in terms of those coordinates; that is, gives the value of the geometrical tangent of the angle between the tangent to the curve at any point and either of the rectangular axes, in terms of the coordinates of the point.

dy

2 a

Ex. 1. Let = ; and let us assume that the curve passes dx У

through the origin, so that when x = 0, y = 0.

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dy

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(2a—~)*; and let us suppose that the origin

Ex. 3. (2)

dx

is on the curve.

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which is the equation to a cycloid, of which the vertex is the origin.

d2y dx2

175.] The process is similar when is given in terms of x

dy and y. In this case however, as the first integration will produce it is necessary that the value of this quantity at the limits, or at least at one limit, should be given. This condition requires that the direction of the axes should be given. Also, as the second integration will produce a finite equation in terms of x and y, of which the values at one limit at least must be given, this condition assigns the place of the origin.

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; and let us suppose

= 0, when x = ∞ ;

dx

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If the values at the inferior limits were such that when x = a,

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