one and the same line; that is, the radius of curvature is normal to the involute, and tangent to the evolute.

The evolute may therefore be constructed, by drawing a curve tangent to the normals at the different points of the involute.

From what precedes, it is plain that the evolute may be regarded as formed by the intersections of the consecutive normals to the involute, and that the point of intersection of any two consecutive normals may be taken as the centre of the osculatory circle, which passes through the two consecutive points of the involute at which the normals are drawn.

108. Equation (6) of the preceding article, combined with (2), gives

Substituting this value in (1), we have, after reduction,

Substituting the same value in (4), reducing and squaring both members, we obtain

Dividing this by (7), member by member, and taking the root,

But if z represent the arc of the evolute, we have

109.

If any two radii of curvature be drawn, as one at M and the other at M'; the first being denoted by R, the second by R', and the corresponding arcs BC and BC' by z and z', we have

M

that is, the difference between any two radii of curvature is equal to the arc of the evolute intercepted between them.

If in the equation Rz+c, we make z = o, and denote by r, the corresponding value of R, we shall have

r = 0 + c = c;

that is, the constant c is always equal to the radius of curvature which passes through the point of the evolute, from which its arc

is estimated.

If we estimate the evolute of the ellipse from the point C, we have

If the evolute and one point of the involute be given, and a thread be wrapped around the evolute and drawn tight, passing through the given point M, fig. (a); when the thread is unwrapped or evolved, the point of a pencil first placed at M, will describe the involute; for, by the nature of the operation, CC' is always equal to M'C' - MC.

110. The equation of the evolute of any curve may be found thus: Differentiate the equation of the involute twice; deduce the values of dy and dy, and substitute in the equations

combine the results, which will contain the four variables a, ß, x and y, with the equation of the involute, and eliminate x and y; the final equation will contain only a, B, and constants, and will therefore be the required equation.

As an example; let it be required to find the equation of the evolute of the common parabola.

Substituting these values in (1) and (2), and reducing, we

and putting for y, in (3) and (4), its value √ 2px = (2p)‡x3, w

we

M

If we make ẞ = 0,

(α - p)3,

we have

ACP,

απ P, and laying off C will be the point at which the evolute meets the axis of X. If we transfer the origin of co-ordinates to this point, we have

Since every value of a' gives two values of B', equal with contrary signs, the curve is symmetrical with the axis of X. If a' be negative, B' is imaginary, and the curve does not extend to the left of C. The branch CC' belongs to AM, and CC" to AM'.

TRANSCENDENTAL CURVES.

111. The most general division of curves is into the classes, Algebraic and Transcendental.

When the relation between the ordinate and abscissa of a curve can be expressed entirely in algebraic terms [see Art. (5)], it belongs to the first class; and when such relation can not be expressed without the aid of transcendental quantities, it belongs to the second class.

112. One of the most important of the latter class is

THE LOGARITHMIC CURVE,

so named, because one co-ordinate is the logarithm of the other. Its equation is usually written

y = log x,

or, if a be the base of the system of logarithms,

x = a.