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properties belong exclusively or not to the law of the inverse square of the distance. To determine this is the object of the four following Propositions.

PROP. To find the attraction of a homogeneous spherical shell on a particle without it; the law of attraction being represented by $(y), y being the distance.

7. The calculation is exactly analogous to that given above: we have only to alter the law of attraction. Then attraction on C in CO nprdr

*+&9)

(y + c72) $ (y) dy (integrated by parts) ca uppdr y

[(y® + c* *) S$ (y) dy 2/{yS$(y) dy} dy] προrdr

{(+¢— ) $(y) 241 (y) + const.} suppose, =2wprdr {** ®: (+r) – 4 (c+r) - - * 0,60-r) + 4/0=r}}

d Syr (c+r) – 2 (c = 27 prdr dol

= ,

} this latter form being introduced merely as an analytical artifice to simplify the expression.

PROP. To find the attraction of the shell on an internal particle, with the same law.

ca

8. The calculation is the same as in the last Article, except that the limits of y are r -c and r+c:

:. attraction = 2#prder (0.4 (r +c) – 4 (v +c)

+*8. (—c) + 3x (r –c)} = 2wprds is {y +0), prof}.

+

PROP. To find what laws of attraction allow us to suppose a spherical shell condensed into its centre when attracting an external point.

9. Let (r) be the law of force; then if c be the distance of the centre of the shell from the attracted point and r the radius of the shell, and

7 (r) = 1 {rso (r) dr} dr, then the attraction of the shell

dx (c+r) – 4 (C – ro)

= 27 prdrão

(6-1)}.

с

But if the shell be condensed into its centre, the attraction

= 47 pri dr $ (C); dy r— 4 C

:. 2ro (c) =āc 1

==

.) = 0, whatever r be; de la care -o,

d (defcr
fac 78

1
- 2
+

+
de \ dc c dos C 1.2.3

d id&c go3 1 2c

+ dc I doo

C 1.2.3

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dc dc3

d (1 dupe + ...

)

d (1 d.)

..

= 0, ...

de

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doo

d d2fc

dfc But

=cs& (c) dc S$ (c) dc + (c), dc

dca difc

dфс 20c + c ; dcs

dc therefore by the first of the above equations of condition

2 doc

=const. = 34, dc

фс +

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and multiplying by c and integrating

c* $ (c) = Ac + B, A and B being independent of c,

B

$ (c) = Ac + This is the most general solution of the first of the equations of condition for ti (e), and it satisfies all the rest. Hence the only laws of attraction which have the property in question are those of the direct distance, the inverse square, and a law compounded of these.

PROP. To find for what laws the shell attracts an internal point equally in every direction. 10. When this is the case

Sf r+o) – (r –c)
de l
deur d'fr ca

+
dr drs 1.2.3

+ ... =- A, whatever c is, A being a constant independent of c;

defr

d'un dr

= 0, ...

drs These conditions are all satisfied if the first is: this gives

A 15$ (7) dr =- A, 0(-) = , and therefore the inverse square is the only law which possesses this property.

11. The form of the Earth and of the other bodies of the Solar System differing from the spherical, and more resembling the spheroidal, it is desirable to find the attraction of a spheroid upon an external and an internal point.

PROP. To find the attraction of a homogeneous oblate spheroid upon a particle within its mass; the law of attraction being that of the inverse square of the distance.

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12. Let a, c be the semi-axes; the minor axis 2c coinciding with the axis of z: then the equation to the spheroid from the centre is

20* + y2

1. a?

ca

Let fgh be the co-ordinates to the attracted particle, which we shall take as the origin of polar co-ordinates,

p=radius vector of any particle of the attracting mass, O= angle which r makes with a line parallel to z,

$= angle which the plane in which 0 is measured makes with the plane xz; .. x=f+r sin 0 cos , y=9+r sin @ sin $, z=h +r cos 0, and the equation to the spheroid becomes (f+rsin 0 cos ®)+(9+r sin o sin o) (h+r cos 0)*,

+

= 1, a'

c?

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The volume of the attracting element = gol sin drdo do as in Art. 2: let p be the density of the spheroid. Then the attraction of this element on the attracted particle is

psin O drdo do: and the resolved parts of this parallel to the axes of xyz are p sin? 0 cos odrdodo, psin o sin pdr de ,

p sin 0 cos O dr dodo. Let A, B, C be the attractions of the whole spheroid in the directions of the axes, estimated positive towards the centre of the spheroid. Then these equal the integrals of the attractions of the element; the limits of r being — g' and 7", of being 0 and , of being 0 and . Hence

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Now it is easily seen that if R (sin a, cosa a) be a rational function of sin a and cos' a, then

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