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sing:

a + 2b

46(a+b) The radius of curvature of the epicycloid= the evolute is an epicycloid similar to the original curve, the radii of the base, and generating circle being respectively

a?

ab

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The equations between p and u, the centre of the base being the pole;

u in the epicycloid, pre

where e=a + 2b, - a

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2

hypocycloid, p= (y

e=a2b.

The area contained by the axis, the radius vector, and

the curve

= (a +b)(a +26)(8 – sin o), in the epicycloid,
=> (a - b)(a − 2b)(8 — sin o),

hypocycloid.

If a : b is a finite ratio, the curve may be represented by a finite algebraical equation. The equation of the cardioide, in which a=b, is

y2 + (x − a)]* =a?{y® + (x − a)?}. If the diameter of the curve is the axis of X, and the cusp the origin, the polar equation is

u=a(1 - cos y).

The equation of the epitrochoid in which a=b, the origin being at the first point of contact,

is
æ* + 26,83 (a? — 62%) x2 + 2(x +b)xy® aʻyo +y*=0.

The equation of the epicycloid with two cusps, in which b=ļa, is a*y* = (x2 + yo – aʼ). The equations of the hypotrochoid, in which b=ļa, are

x=( a +b) cos ,

y=(la 6,) sin 0, the equations of an ellipse, of which the semi-axes are ża + b, and la— bı.

(Peacock, Ex. p. 192.) (43.) Spirals. The equation of the spiral of Archimedes, is

u=a0;

u the equation between

р
and u; p=

(a + u)

a?

.

The equation of the Lituus is u*=

0 The first radius produced is an asymptote to the curve, and there is a point of contrary flexure.

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au'

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The équation between p and u; p=

(a? + uo)?
The subtangent =a.
The area = kau.
The length of the arc = a loge

+ (a® +u°)}-a.

(ao +u*)* +a bu

b c+ (co – tuo)

or 0= log (aU?) where c=(12).

[2] p=

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[3] p=

bu

and a <b; then the polar equation is (a’ + u^)}' 0=log,

(u* +co)} – C

U

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b

[5] The logarithmic spiral: p= -u, and

a

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b

log

(a' 6°) The angle contained between the radius vector and the

6 curve is constant, and = tan

(a? 6°)? The evolute is a similar spiral. (45.) The polar equation of the involute of the circle, is

(u? -a)?

= A + sec

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(46.) The tractory: The differential equation is

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The evolute of this curve is the catenary. . The equation of the syntractory is

6 + (° - 04 x=a log

- (b^-^).

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y=

If b<a, this curve has a point of inflexion, corresponding to

b.at

(2a +b)! If b>a, there is no point of inflexion, but when y=(ab), the tangent is perpendicular to the axis. If b<0, there is a point of inflexion, when

-6. y =

(Peacock, Ex. p. 174.)

(2a + b)}" (47.) The catenary. Let the curve be referred to its diameter, and a tangent at the vertex, and let s be the length of the arc; the differential equation is

(c +8°)

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whence &+c=(s* + c), or s=(x* +2cx)}.
The equations between y and 8 are
8 + (so+c^)*

=fc666

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= loge

с

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the equations between w and y are

x+c=hceš +6=6),
Y X++ (202 + 2cv) }

= = logo

C

с

(Whewell, Mech. 111-3.) (48.) The elastic curve. Let the extremities of the elastic line be situated in the axis of y, at equal distances from the origin, and let a be the angle at which the curve passes through the origin; = -a' sin a, and c=abo, then the differential equations of the curve are

a- 0 + 2012
dzy=
(c* — **)(2a – 02 + x)]'

a
dzs=

(c* — **)(2a’ –c? + x^)]

Species [1]. Let , be very small: then, neglecting ca

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[2] : < 1.

In this case the values of y and s can only be

obtained by series.

[3] ; = 1

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In this case d y=

and (a*

a d.8=

(a* — **)] Let l be the length of the curve, and h the distance of the extremity of the curve from the origin, then

121

19.32 1 l=

1+ V2

22.42

12 3 1 12.325 1 h= 1+

+

+ &c. V2 22 2 2.42 3 4

th=na”.

πα

To

+

+ &c.

&c.};

sc.

πα.

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[7] ; =V2. The axis of y is an asymptote, and the

equation of the curve is

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