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

du du du du (du dul?

- 2
dx2 dx dy dx dy

dy?
du
dc

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+

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2

du 273

dy

2

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

ď u du dul
d.x2

dx) dy
du

dy

{()

+

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1

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2

р

23

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

ye = 4mx,

4 (m + x)

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(3) In the rectangular hyperbola referred to its asymptotes

XY = m?,

(r? + y)

4mo

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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, N2 =

-v {1+63

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

y' = 2 px +qx®;

dy
therefore y = p + qx,

du

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dạy

2

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dy y 3 9

-p?. d x2

dx
Therefore

N6
p

PE

p*
(5) In the cubical parabola 3a'y = id,

(a + c)3

4a® x2

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p2

(6) In the semi-cubical parabola 3 ay = 2x",

(2a + 3x)3 x

p2

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dy

= 0. d x

(9) In the tractory y + (a? – yo)? Taking the expression for p in which y is the independent variable we find,

a?

(ay').

ya

(10) In the hypocycloid xi + = a), p = 9 (a xy)).

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

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

dr

dp
(11) In the cardioid r = a (1 – cos 6),

(8 ar)?
p=

3,

(12) In the lemniscate of Bernoulli me = a' cos 20,
(a* – pt)?

'dri a
d

dr

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-go?

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a (15) The equation to the lituus being po?

ro ( 4 a' + gut)! p:

2a*(4 a' – 904) (16) The equation to the trisectrix being r=a(2 cos 0+1),

(5 + 4 cos 0)|

3 (3 + 2 cos 6) (17) In the logarithmic spiral when referred to p and r,

р p = mr,

p=a

r

p=

m

m?"

(18) In the involute of the circle p' = po- a, and p = p.

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

(a + gol)! r (a® + po?)? P р

a' b (20) In the epicycloid

CP (pu2 - a')

p?

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c - a? p= p

(co a’)(po– aʼ)!

Therefore

с

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

2

2

(dy)

1 +

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a = X

do y

da

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[ocr errors]

dx

du

đ u

d'u

2 đu

()

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

'du) 2

du) 2

+ a

d
du
du

du du
2

+ d x dy dy d x2 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 = 4a x, whence

y? a = 3x + 2a, B

4a2

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Substituting these values in the equation to the parabola,

we find

4 a
(4 a) (a 2a),

3

or 27aß = 4 (a - 2a), the equation to the semi-cubical parabola.

(2) In the rectangular hyperbola referred to its asymp

totes

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whence 2a = 30 + 2ß = 3y +

y Adding

y3 + 2003 Yi + 3x+ y + 3y+ x + r.3 2 (a + b) = 3 (x + y) +

wy

ma or 2m2 (a + B) = (x + y), or x + y = (2 m2)} (a + B). Similarly, subtracting

2 m2 (a - b) = (x - y)", or x - y = (2m?)! (a - ).

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