At a point of inflexion the Hessian = 0; and as

do not both simultaneously vanish, so at a point of inflexion p = ∞ ; that is, the curve degenerates into a straight line; and the radius of curvature, which corresponds to a point at which two consecutive elements are in one and the same straight line, is evidently infinite. The Hessian also = 0 at double points; dr but then it vanishes identically, because (7)

=

dr

dx dy'

= 0; and

therefore in this latter case p is indeterminate, and (31) must be evaluated. This is also otherwise apparent. At a double point the two branches may not have the same curvature, and as the same analytical expression gives the radius of curvature of both, it must necessarily take at that point an indeterminate form. If we multiply (31) by the denominator of the right-hand member, the expression becomes of 3n-3 dimensions in terms of x and y if then p is constant, the points of equal curvature on a curve of the nth degree are on a curve of the 3 (n − 1)th degree; and as the number of the points of intersection of these two curves is 3n(n−1), so on a curve of the nth degree there may be 3n(n-1) points at which the curvature is the same.

SECTION 2.-On evolutes of plane curves referred to rectangular coordinates.

288.] We have thus far determined the length of the radius of curvature, and the cosines of the angles at which its line is inclined to the coordinate axes; we shall now investigate the coordinates to the centre of curvature; and since it changes position as the point, at which the radius of curvature is drawn, moves continuously along the curve, it thereby describes a continuous curve, which we shall determine. This curve is, for a reason which will shortly appear, called the evolute of the original curve.

Let the equation to the curve be y = f(x), and let (x, y) be the point on it at which the radius of curvature is drawn; (§, n) the centre of curvature; so that, in fig. 102, we have

which formulæ determine the position of the centre of curvature corresponding to any point of the curve; and from eliminating x and y between these equations and the equation to the curve, viz. y = f(x), there will result an equation involving έ and 7, which will represent the locus of the centre of curva

ture.

The equations (32) assume various forms, according to the value given to p; i. e. whether we express p by one or other of its values (6), (7), (8), (16). Thus applying the value of the radius of curvature given in (9), when x is equicrescent, we have

If the equation to the curve is given in the implicit form, the equations (32) must be modified according to the equation (31) and those by means of which (31) has been determined; but as the expressions are long, not often employed, and easily found, it is not worth while to insert them at present.

289.] Examples on evolutes.

Ex. 1. To determine the evolute of the parabola.

substituting which in the equation to the parabola, we have

the equation to a semi-cubical parabola, whose cusp is on the axis of x at a distance 2a from the vertex of the parabola; see fig. 103.

Ex. 2. To find the equation to the evolute of a circle.

Ex. 3. To find the equation to the evolute of the ellipse.

the curve represented by which is delineated in fig. 104.

.

In reference to the properties of the evolute which are mentioned in Art. 229, I may observe that for all points within ECE'C' four normals may be drawn to the ellipse; from points on the curve three normals may be drawn; and for all space outside the curve only two can be drawn.

Ex. 4. To find the equation to the evolute of the cycloid.
Let the starting point be the origin; then

substituting which values in the equation to the cycloid, we have

$

§ − 2 ( − 2an — n2) ✯ — a versin-1 —— (−2 ay — n2)*,

-η

a

which is the equation to a cycloid, the highest point of which is the origin, equal to the original cycloid: § being parallel to the base, ʼn taken along the axis in a negative direction, as is maniη fest on a comparison of the last equation with (31) in Art. 201, and of figs. 50 and 105; which last represents the positions of the original cycloid and its evolute.

Since n = -y, Nп= MP; and therefore Рп = 2PG, or the radius of curvature is equal to twice the normal; see Ex. 3, Art. 283. Also the radius of curvature at B = BC= 2BA = 4a; also it is manifest that the radius of curvature at o = 0.

Ex. 5. To determine the equation to the evolute of the equitangential curve.

The equation to the curve is, see equation (27), Art. 200,

a + (a2—y2) * } − (a2 — y2) * ;

x = a log { a+ (a2 — y2) * }

y

...

a2y

'} —

1+

;

(a2 — y2)2

a2

(a2 — y 2)2

a2y