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Every plane curve manifestly has an evolute, and has only one; but it has an infinite number of involutes, because in the unwrapping of the string which has been wound round the curve, every point of the stretched string describes a curve which is the involute corresponding to that point.

294.] Again, multiplying the former of (38) by dx, and the latter by dy, and adding, we have

dxd£+dydr) = 0; (41)

dr\ dx
d£~ dy'

And since ^| and ^ are respectively the tangents of the angles

made with the axis of x by the tangents to the evolute, and to the involute, it follows that the tangent to the evolute is perpendicular to the tangent to the involute, or that the normal to the involute is a tangent to the evolute. This result might have been anticipated from what is said in Art. 290.

The radius of curvature at any point of the evolute may be expressed in terms of the radius of curvature of the corresponding point of the involute by the following process.

Let p and p be the radii of curvature at corresponding points of the involute and the evolute: the angle contained between two consecutive normals to the evolute is equal to that contained between two consecutive normals, or between two consecutive tangents to the involute; that is, is equal to + dr or to + d\J/; see Art. 218: and if da is the length-element of the evolute, d<r dp; also ds = + pdr; therefore

, dap=±dr

= ±dfsP. (42)

Hence p'= 0, when dp = 0; that is, if the radius of curvature of a curve is constant, or attains to a maximum or minimum value, that of the evolute at the corresponding point vanishes, and its curvature is infinite.

Other properties of the involute are discussed in Vol. II of our Treatise.

295.] Let us however investigate these properties of curvature from a geometrical point of view.

The equations (35), (36), (37), connecting x, y, p, f and 7;, shew that the centre of curvature is the centre of a circle passing through three consecutive points in the curve. These three points are necessary to render the circle definite. If it passes through only two points, its centre may be any where on the normal which is perpendicular to the tangent passing through the two points, and thus there may be an infinite number of circles satisfying the condition: but if the circle is to pass through three points, its centre must be on the normal perpendicular to the tangent passing through the second and third points, as well as on the normal corresponding to the first and second points; and as these two normals will intersect in one point, this point must be the centre of the circle, and the circle becomes definite; in other words, the two consecutive elements of the curve which are delineated in fig. 108, viz. Pq and Qr, will form two sides of a triangle, and by joining Pb the triangle will be completed, and the circle described about this triangle will be a definite circle, and pass through the points p, Q, R, which are three consecutive points on the curve. We may on this conception determine the radius of curvature as follows:

The angle of contingence, or dr, is the angle between the two consecutive elements Pq and Qr, whence by Art. 284,

dr = ± . (43) P

Complete the parallelogram, of which Pq, Qr are two adjoining sides, and draw the diagonal Sq, and the other lines as in the fig. 108; then

Pq = ds, Qr = ds + d2s;

Pk = dx, Qt = dx + d2x = Pe, ■." sp is parallel to Rq;

.-. Ek = — d%x;

Qk = dy, Rt dy + d2y = Se, •.' Sp is parallel to no.; .-. Sy = d2y.

Therefore d2x and d2y are the projections of Sq on the coordinate axes.

Now the area of the parallelogram Pqrs = Pq x QR x sin PQK, = ds (ds + d2s) sin rfr, ds3

= ± • (44)


omitting d2s because it is added to ds, and replacing sin rfr by dr in terms of equation (43).

Again, the area of the parallelogram Pqrs = the area of Pquv,

= PK X UQ = PK X RW = PK (RT — WT),

= dx | dy + d*y (dx + d*x) ~ j ,

= dx d2y dy d2x. (45) Hence equating (44) and (45), we have

1 d2xdy-d2ydx

p ~ ± d? {m

Hence if three points of a curve are in one and the same straight line, Pqrs = 0, and the radius of the circle of curvature is infinite. Also if of three elements the first and third are equally inclined to the second, the circle of curvature passes through four consecutive points.

296.] And now let s be equicrescent; so that in fig. 109 Pq = Qr = ds, Om = x, MP = y, Mn = dx, xi = dx + d2x, NQ = y + dy, Lr = y + 2dy + d2y; let us complete as before the parallelogram Pqrs, which is in this case equilateral; and therefore the radius of curvature lies along the line nsQ, which is coincident in direction with the diagonal Sq. Join Pr, which is, as is plain from the geometry, perpendicular to Sq; then, if p is the radius of curvature, by the property of the circle,

2p : RQ I I RQ : VQ;

Rq* ds2

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Hence also we have a geometrical proof of the values of cos r and cos yff determined in Art. 285, equations (23) and (24). As Sq is coincident in direction with the radius of curvature,

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297.] Ere we conclude this part of our subject, we must investigate some general properties of evolutes; and I will assume the equation to the base-curve or involute to be, as heretofore, an algebraical equation of the rath degree, of the form (45), Art. 207: although the geometrical properties which follow are generally true of all curves.

Let the equation to the curve be given in the implicit form

F(x,ij) = c. (48)

Now if we express f and »j, the current coordinates of the evolute, in terms of the partial-derived functions of this equation we have,

each of which expressions is evidently of 3(n—1) dimensions iu terms of x and y, and does not admit of further reduction. And as the equation to the evolute is found by the elimination of x and y between these equations and (48), which is of n dimensions, the evolute itself is of 3n(n 1) dimensions.

Also the class of a curve, see Art. 223, is the same as that of the number of tangents which can be drawn to it from any given point. Now the number of normals which can be drawn to the involute from a given point is n2; and as every normal to the involute is a tangent to the evolute, so the evolute is of the »2 class, so far as tangents can be drawn in the plane of reference; yet this does not affect the relation which has heretofore been shewn to exist between the order and class of a given curve. Thus the evolute to a conic is of the sixth degree, see Ex. 3, Art. 289, and of the fourth class: the evolute to a general curve of the third degree is of the 18th degree, and of the ninth class.

As a normal drawn at any point of an involute is a tangent to the evolute, so if there is any peculiarity at a poiut of the involute a corresponding peculiarity will exist in the tangent of the evolute: the two curves will therefore have reciprocal properties. Some of these I proceed to indicate.

To a point of inflexion on the involute, an infinite branch of

the evolute corresponds; because at a point of inflexion -~ 0,

and the radius of curvature is infinite; and the normal to the involute is the asymptote to this infinite branch of the evolute; and as the direction, in which the radius of curvature is drawn, changes at a point of inflexion, so does the infinite branch of the evolute return on the side opposite to that in which it went off; that is, the evolute travels round the sphere of infinite radius; see Art. 186, and fig. 24; and has at infinity a point of inflexion. Hence it appears that the evolute has as many rectilinear asymptotes as the involute has points of inflexion; and thus as the involute is of the wth degree, it may have 3«(n—2) points of inflexion within a finite distance, and therefore the evolute may have that number of different rectilinear asymptotes within a finite distance. The evolute of a conic therefore has no rectilinear asymptotes. The evolute of a curve of the third degree may have nine rectilinear asymptotes.

To a point on the involute at which the curvature is a maximum or a minimum, a cusp corresponds on the evolute; the cusp being at a distance from the involute equal to the maximum or minimum radius of curvature.

If two branches of the involute pass through the same point,

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