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If u = o be the equation to the curve, the following expression for the radius of curvature is frequently convenient,

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or, if u consist of the sum of two parts, the one involving x alone and the other y alone,

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(1)

In the parabola, the equation to which is

y* = 4mx, .

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(4) In all the curves of the second order the radius of curvature varies as the cube of the normal.

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and therefore

All the curves of the second order are included in the equation

y' = 2 px +qx? ;

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Taking the expression for p in which y is the independent variable we find,

pe = (a? – y). (10) In the hypocycloid x} + yi = a), po = 9 (axy)).

If the curve be referred to polar co-ordinates r and 0, then

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or, if it be expressed by the relation between yo and the perpendicular on the tangent (p),

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(12)

In the lemniscate of Bernoulli p = acos 20,

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Sect. 2. Evolutes of Curves. When a curve is referred to rectanglar co-ordinates, the co-ordinates (a, ß) of its centre of curvature are given by the equations

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dir

or, if u = o be the equation to the curve,

a - a B-u du

(du) du du du du duy đu dx dy dy) dw?-? da dy dx dy*(da) dy

To determine the equation to the evolute it is necessary to eliminate æ and y between these equations and that of the given curve; but the complication of the formulæ renders this elimination always very troublesome, and most frequently impracticable. The few cases in which it can be effected we shall give.. (1) In the parabola yö = 4a x, whence

a = 3x + 2a, B=

a - 2a
therefore x = -, y = (- 4 a'ß)*.

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