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CHAPTER III.

ATTRACTION OF BODIES NEARLY SPHERICAL.

40. As the Earth and other bodies of the Solar System are nearly spherical, and yet may not be precisely of the spheroidal form, it is found necessary in questions of Physical Astronomy to calculate the attraction of bodies nearly spherical. It is in these calculations that the value of the Functions we have been considering in the last Chapter is seen.

If r'e'w' be the co-ordinates to any element of the attracting mass, p' be its density, and cos 0'', then the mass of this element

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and the reciprocal of the distance being R, by Art. 18 and 24, the potential V

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according as r, the distance of the attracted point from the origin, is greater or less than r'. We shall proceed soon to use these formulæ; but we must first find the value of V for a perfect sphere.

PROP. To calculate the value of V for a homogeneous sphere.

41. Let the centre of the sphere be the origin of the polar co-ordinates (r'u'w') to any element of its mass, and the line through the attracted point be that from which the angles are

measured, and p the density. Then - pr'dr'du'do' is the mass of the element: its distance from the attracted point

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Hence, a being the radius of the sphere,

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from μ' = -1 to μ' = 1, = 2πp ] = {(r+r) = (r−r') } dr',

μ'

0

when the attracted point is without, and + when it is within the shell,

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ATTPS:

r

12 r'2 dr'

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when the point is without the sphere.

When the point is within the sphere, the part of V for the shells which enclose the point

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and the part of V for the other shells of the sphere

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PROP. To find the attraction of a homogeneous body, differing little from a sphere in form, on a particle without it.

42. Since the attracted particle is without the attracting mass, we must expand V in a descending series of powers of r, and shall therefore use the first of the expressions for Vin Art. 40. Let the mean radius of the body = a; and let a (1+y) be the variable radius, y' being a function of μ and w', and its square being neglected.

Then, for the excess of the attracting mass over the sphere of which the radius = a, effecting the integration with respect to r' from r'a to r'a (1+ y'), the value of V

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w', be expanded in a series of Laplace's Functions, viz.

Y2+ Y1 + ... + Y; + .....,

then the theorems of Art. 26 and 32 show that

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Hence the value of V for the excess over the sphere becomes

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and the part of V for the sphere, rad. = a, is

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This is the first example in which we see the great value of the properties of Laplace's Functions; they here give us at

once the integrals involved in our expression for V, in terms of the equation to the surface of the attracting mass, without integration.

From the expression for V the attraction can be immediately found by the formula of Art. 20. Thus

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PROP. To find the attraction of a homogeneous body, differing but little from a sphere, on a particle within its mass.

43. We must in this case expand V in an ascending series of powers of r; and shall therefore take the second of the series of Art. 40. By proceeding as in the last Proposition, we find that the part of V which appertains to the excess over the sphere

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Adding to this the part of V which appertains to the sphere of

2 3

radius a, viz, 2πpa2 — — πpr2, for the whole mass,

2

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3a

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We can show that by properly choosing the value of (a) and the origin of the radius of the surface we can make Y, and Y, disappear from the above formulæ.

PROP. To show that by choosing a equal to the radius of the sphere of which the mass equals that of the attracting body we cause Y to vanish, and by taking the centre of gravity of the body as the origin of the radius vector, we cause Y to vanish.

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where r is the radius vector of the surface of the body, and =a (1+y) suppose. Putting this for r mass of body

1

= mass of sphere (rad. = a) +pa3 [*[*ydμdw

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If then a be taken equal to the radius of the sphere of which the mass equals the mass of the body, Y= 0, as was stated.

0

Again, let y be the co-ordinates to the centre of gravity of the body, M its mass: the co-ordinates to the element of which the mass is - pr2 drdudw are

2

r√1-μ2 cos w, r1- sin w, and rμ;

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pr3 √ 1 — μ3 cos w dr dμdw

pr1 √1-μ2 cos wdμdw,

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