Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/207

 of the radii of curvature at different points on the curve and then drawing them in and drawing the locus of the centers of curvature.

Formula (E), §116, gives the coordinates of any point $$(\alpha, \beta)$$ on the evolute expressed in terms of the cöordinates of the corresponding point $$(x, y)$$ of the given curve. But $$y$$ is a function of $$x$$; therefore


 * $$\alpha = x - \frac{ \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right] \frac{dy}{dx} }{ \frac{d^2 y}{dx^2} },\ \beta = y + \frac{ 1 + \left( \frac{dy}{dx} \right)^2 }{ \frac{d^2 y}{dx^2} }$$

give us at once the parametric equations of the evolute in terms of the parameter $$x$$.

To find the ordinary rectangular equation of the evolute we eliminate $$x$$ between the two expressions. No general process of elimination can be given that will apply in all cases, the method to be adopted depending on the form of the given equation. In a large number of cases, however, the student can find the rectangular equation of the evolute by taking the following steps:

General directions for finding the equation of the evolute in rectangular coördinates.

Find $$\alpha$$ and $$\beta$$ from (50), §117.

Solve the two resulting equations for $$x$$ and $$y$$ in terms of $$\alpha$$ and $$\beta$$.

''Substitute these values of $$x$$ and $$y$$ in the given equation. This gives a relation between the variables $$\alpha$$ and $$\beta$$ which is the equation of the evolute.''

Find the equation of the evolute of the parabola $$y^2 = 4px$$.


 * [[Image:Wag 119-2 evolute of parabola.jpg|right|173px|Evolute of a parabola.]]


 * Solution.
 * style="text-align: right;"|$$\frac{dy}{dx}$$
 * $$=\ \frac{2p}{y},\ \frac{d^2 y}{dx^2} = -\frac{4p^2}{y^3}.$$
 * First step.
 * style="text-align: right;"|$$\alpha$$
 * $$= 3x + 2p,\ \beta = -\frac{y^3}{4p^2}.$$
 * Second step.
 * style="text-align: right;"|$$x$$
 * $$= \frac{\alpha - 2p}{3},\ y = -(4p^2 \beta)^{\frac{1}{3}}.$$
 * Third step
 * style="text-align: right;"|$$(4p^2 \beta)^{\frac{2}{3}}$$
 * $$= 4p \left( \frac{\alpha - 2p}{3} \right);$$
 * or,
 * style="text-align: right;"|$$p\beta^2$$
 * $$= \frac{4}{27} (\alpha - 2p)^3.$$
 * }
 * or,
 * style="text-align: right;"|$$p\beta^2$$
 * $$= \frac{4}{27} (\alpha - 2p)^3.$$
 * }


 * Remembering that $$\alpha$$ denotes the abscissa and $$\beta>$$ the ordinate of a rectangular system of coordinates, we see that the evolute of the parabola AOB is the semi cubical parabola DC'E; the centers of curvature for $$O, P, P_1, P_2$$ being at $$C', C, C_1, C_2$$ respectively.