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:: the whole moving force or tension of the string is

(y + uta - r). Hence the accelerating force is

9 + 1 + a - 2

a

F =

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the velocity of the chain, when quitting the quadrant, is given by V2 =

{8r + 4 (2a – r)* }

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4a

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632. Let y, y, be the radii of the required annulus, and I the distance of its centre from the pole; also let u, b, be the semiaxis of the generating ellipse; then it may easily be shewn that

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Again, if u denote the radius of any of the concentric circles which compose the ring, the attraction of any particle in its circumference upon the pole is

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22 +32 and by the resolution of forces the attraction in the direction of the axis

1

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Hence the whole force of the circle in this direction is

2TNI

(u? +12) and that of the annulus is ...

udu 2πα S

(u2 +2?}} taken between u = y' and u = y; that is

1

1 2πα

(y2 + x2 N

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or 2

(26x– x?)+r?

– )

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= max. m

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Hence, and by the question,

bWx

(2ba a? - 62 . x)
dm
1

b

1
Х
dx

2 26 N 3 2V * (2ba' - a - b. x) (a* – 60)

= 0
2ba" - a - b.x)}
:. (9bao (4 8*) =)} = 98% / 90
1

(865 a4)}
as 69

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and x =

X 2ba?

2 (ba®

3'5 a)

6 which gives the position of the annulus for spheroids of all eccentricities.

633.

Let the distance of the particle attracted from the centre of the sphere be a, r the radius of the sphere, and suppose any circular section whose radius is y to be made by a plane I to a, and distant from the point by the interval x; then if y' denote the radius of any o concentric with the former, the attraction of any particle in the circumference of this circle is

1

*+2) and this resolved into the direction of x is

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Hence the attraction of the whole circumference is

2ncy'

(y^2 + x2)
and that of the area of the section is

y'dy'
27X
(y'? +x")

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ATTRACTIONS.

taken between ý' = y, and y' = 0; that is

1

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3

3
23

(yo + x)}
Hence the attraction of the whole sphere is
2 dx

dx
X 3
3

(y +x*)! taken between the limits of x =a + 1, x = a - r. But by the equation to the circle

ya = gole – (a - x)
dx

d.x
(y + xo)} (pe – a* + 2ax)!
1

; and we have for the whole attraction
(på - a* + 2ax)}
1

S1
3
(a+r) 3a

atr

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634. If x be the altitude of the cone, and s its slant side, the whole attraction is (Vince, p. 142)

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= U.

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Again, supposing the given quantity of matter to be a3, and the density to be constant, we have

7. (s* - **) x

2

a

3

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whence by substituting for x &c. &c. I may be found exactly or by approximation.

635. Let c be the distance of P from the surface of the sphere, r the radius of the sphere, then ctor is the radius of the

2

generated sphere. Let any circular section of the sphere be made by a plane I to c or to the axis, and let y be the radius of that section, and y' that of any circle concentric with it; also let x be the distance of P from the plane of this section; then the attraction of any particle in the circumference of the circle whose radius is y', is

1

y'a + rs and the attraction of the whole circumference is

2πυ

y' + x89

Hence the attraction of the whole section is

2y'dy

= al. (y'? + ?) + C

y + taken between yʻ = 0 and y' = y; it is ::

y' + x2

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.cdr + 2.3 2.(c + r)

2.(c

+ r)x - c(c + 2r) and we finally obtain, after finding the correction on the supposi

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