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If u = 0 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 a alone and the other y alone,

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(1) In the parabola, the equation to which is

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

If N be the length of the normal, N = y2 1 +

dy

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All the curves of the second order are included in the

equation

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(5) In the cubical parabola 3a2y = x3,

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(6) In the semi-cubical parabola 3ay2

= = 2 x3,

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(9) In the tractory y + (a2 - y3)}

dy

dx

= 0.

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

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then

(10) In the hypocycloid x + y = a3, p2 = 9 (a xy)3.

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

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

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(12) In the lemniscate of Bernoulli 2= a2 cos 20,

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(13) In the spiral of Archimedes r = a0,

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(15) The equation to the lituus being r2 =

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(16) The equation to the trisectrix being r=a (2 cos 0±1),
(5 ± 4 cos 0)
= a 3 (3 ± 2 cos 0)

p =

(17) In the logarithmic spiral when referred to p and r,

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(18) In the involute of the circle p2 = r2 — a3, and p = p.

(19) The equation to Cotes' spirals is p =

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br

(a2 + m2)}'

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When a curve is referred to rectanglar co-ordinates, the co-ordinates (a, ẞ) of its centre of curvature are given by the

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or, if u = 0 be the equation to the curve,

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2

dx

dy

+

du\?

dx dy

du du d'u

+

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2 d'u

dy dx2 da dy dx dy dx dy

To determine the equation to the evolute it is necessary to eliminate 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 y=4ax, whence

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Substituting these values in the equation to the parabola, we find

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the equation to the semi-cubical parabola.

(2) In the rectangular hyperbola referred to its asymp

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2 m2 (a − ẞ) = (x − y)3, or x - y = (2 m2)1 (a − ẞ)1.

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