Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/210



NOTE. If we eliminate $$t$$ between equations (D), there results the rectangular equation of the evolute $$OO'Q^v$$ referred to the axes $$O'\alpha$$ and $$O'\beta$$. The coördinates of O with respect to these axes are $$(-\pi a, -2a)$$. Let us transform equations (D) to the new set of axes OX and OY. Then

Substituting in (D) and reducing, the equations of the evolute become

Since (E) and (C) are identical in form, we have:

The evolute of a cycloid is itself a cycloid whose generating circle equals that of the given cycloid.

From (A), §117,

Let us choose as independent variable the lengths of the arc on the given curve; then $$x, y, R, T, \alpha, \beta$$ are functions of s. Differentiating (A) with respect to $$s$$ gives

But $$\tfrac{dx}{ds} = \cos \tau, \tfrac{dy}{ds} = \sin \tau$$, from (26), §90; and $$\tfrac{d\tau}{ds} = \tfrac{1}{R}$$, from (38) in &sect; 100 and (39) in &sect;101.

Substituting in (B) and (C), we obtain

Dividing (E) by (D) gives