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

In a plane curve the radius of curvature ${\displaystyle r}$, is expressed by

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

and in a curve of double curvature it is

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

${\displaystyle ds}$ being the constant element of the curve. Let the angle ${\displaystyle com}$ be represented by ,${\displaystyle \theta }$, then if Am be the indefinitely small but constant clement of the curve MN, the triangles com and ADm are similar ; hence mA : mD :: om : mcj or <ft : dx::l : sin O, and sin = — In the same manner cos := ^, ds dM Butd.cose=:— (2dBine,andde=:— ^l2^; also i2 . sin 9 = sin 9 do cos 0, and dB = — I — -_ ; but these evidently become cos 9 dy. de= + ^ . d^anddd= -^ . dlS; or dy da dx da dB^ +i^andde=:^i:V . dy dx Now if om the radius of curvature be represented by r, then moA being the indefinitely small increment dO of the angle com^ we have r I da III I dB) for the sine of the infinitely small angle is to be considered as coinciding with the arc: hence dO = _, whence r . .^ da .dy dadx -n^,.!!. ,. ,

• " — - J, = -;--• But d2r + tfy* = cfa", and as <2« is constant

d^x d^y 1 ^ » dr.d^x + dyd^ = 0. Whence ^ = - ^, or ( -^^ = ^^^ ~ dx d*y J d'y dx"