Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/211



Substituting the last two results in (F), we get

Since the slope of one tangent is the negative reciprocal of the slope of the other, they are perpendicular. But a line perpendicular to the tangent at P is a normal to the curve. Hence

A normal to the given curve is a tangent to its evolute.

Again, squaring equations (D) and (E) and adding, we get

But if $$s' =$$ length of arc of the evolute, the left-hand member of (G) is precisely the square of $$\tfrac{ds'}{ds}$$ (from (34), §94, where $$t = s, s = s', x = \alpha, y = \beta$$). Hence (G) asserts that

That is, the radius of curvature of the given curve increases or decreases as fast as the arc of the evolute increases. In our figure this means that

The length of an arc of the evolute is equal to the difference between the radii of curvature of the given curve which are tangent to this arc at its extremities.

Thus in, §118, we observe that if we fold $$Q^v P^v (= 4 a)$$ over to the left on the evolute, $$P^v$$ will reach to O', and we have:

The length of one arc of the cycloid (as OO'Qv) is eight times the length of the radius of the generating circle.

Let a flexible ruler be bent in the form of the curve $$C_1 C_9$$ the evolute of the curve. $$P_1 P_9$$, and suppose a string of length $$R_9$$, with one end fastened at $$C_9$$, to be