Page:Somerville Mechanism of the heavens.djvu/108

32 Radius of Curvature.

83. The circle $$\text{A}m\text{B}$$, fig. 22, which coincides with a curve or curved surface through an indefinitely small space on each side of $$m$$ the point of contact, is called the curve of equal curvature, or the oscillating circle of the curve $$\text{MN}$$, and $$om$$ is the radius of curvature.

In a plane curve the radius of curvature $$r$$, is expressed by

$r=\frac{ds^2}{\sqrt{(d^2x)^2+(d^2y)^2}}$|undefined

and in a curve of double curvature it is

$r=\frac{ds^2}{\sqrt{(d^2x)^2+(d^2y)^2+(d^2z)^2}}$,|undefined

$$ds$$ being the constant element of the curve.

Let the angle $$com$$ be represented by $$\theta$$, then if $$Am$$ be the indefinitely small but constant element of the curve $$MN$$, the triangles $$com$$ and $$ADm$$ are similar; hence $$mA : mD :: om : mc$$ or $$ds : dx :: l : \sin \theta$$, and $$\sin\theta= \frac{dx}{ds}$$. In the same manner $$\cos\theta = \frac{dy}{ds}$$,

But $$d.\cos \theta = -d\theta \sin\theta$$, and $$d\theta = -\frac{d.\cos\theta}{\sin\theta}$$; also $$d.\sin\theta = d\theta\cos\theta$$, and $$d\theta = \frac{d.\sin\theta}{\cos\theta}$$; but these evidently become

$d\theta = + \frac{ds}{dy}.d\frac{dx}{ds}$ and $d\theta = -\frac{dw}{dx}.d\frac{dy}{ds}$; or

$d\theta = + \frac{d^2x}{dy}$ and $d\theta = -\frac{d^2y}{dx}.$

Now if $$om$$ the radius of curvature be represented by $$r$$, then $$moA$$ being the indefinitely small increment $$d\theta$$ of the angle $$com$$, we have $$r : ds :: l : d\theta$$; for the sine of the infinitely small angle is to be considered as coinciding with the arc: hence $$d\theta = \frac{ds}{r}$$, whence $$ = -\frac{ds.dy}{d2s} = \frac{dsdx}{d^2y}$$. But $$dx^2 + dy^2 = ds^2$$, and as $$ds$$ is constant $$dx.d^2x + dyd^2y = 0$$. Whence $$\frac{d^2s}{d^2y} = -\frac{dy}{dx}$$, or $$\left(\frac{d^2x}{d^2y}\right)^2 = \frac{dy^2}{dx^2}$$,