CURVATURE AND EVOLUTES.

200000000

117.

To find the radius R of curvature to the common

118.

a

(1

x2

dx2

(1 + 2a – †) 3

Required the chord of curvature parallel to the axis,

of the common parabola, whose equation is y2 = px.

dx2

dy?

R =

X +

-dey

dx2

Now the chord of curvature parallel to the axis = 2R ..

a, Bare the co-ordinates of the centre of curvature, and .. of the evolute of a curve; and if we can eliminate from them y and X, the co-ordinates of the curve, the result will be the equation to the evolute.

120.

By the question, the equation to the curve is

x = r. vers. y = r r cos. y,

121.

bola is

Hence R =

dy d2x d2x

= r cos. y (1 + r2 sin.o y)*.

By problem 118 the radius of curvature of the para

(4 x + p), and the equation to the parabola

being y2 =px the normal (N) =

.. at the vertex of the parabola, where x = 0,

R =

123.

equation is

To find the evolute of the common cycloid, whose

..B= vers.

(a

4r)

√2r (a

equation to the evolute, which is therefore a cycloid equal to the

curve itself, but having its baɛe

either extremity.

base of the given cycloid at

CONTRARY FLEXURE.

124.

To find the points of contrary flexure of a curve,

(2-ly) 0 or , by the rule, which gives

2, or x=(ly)3 = 8,

and y = e2, which are the co-ordinates of the point required.

125.

If R be the radius of the wheel, r the distance of the generating point from its centre, a the abscissa of the trochoid, measured from the vertex or highest point of it; then the equation to the curve is

and we have two points of contrary flexure, whose abscissæ are

spectively, the latter point being a ceratoid, as we learn by sub

stituting in equation 2.