Page:Somerville Mechanism of the heavens.djvu/108

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

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

and in a curve of double curvature it is

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

$ds$ 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

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

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 $=-{\frac {ds.dy}{d2s}}={\frac {dsdx}{d^{2}y}}$. But $dx^{2}+dy^{2}=ds^{2}$, and as $ds$ is constant $dx.d^{2}x+dyd^{2}y=0$. Whence ${\frac {d^{2}s}{d^{2}y}}=-{\frac {dy}{dx}}$, or $\left({\frac {d^{2}x}{d^{2}y}}\right)^{2}={\frac {dy^{2}}{dx^{2}}}$,