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ATTRACTIONS AND LAPLACE'S

FUNCTIONS.

1. THE Law of Universal Gravitation teaches us, that every particle of matter in the universe attracts every other particle of matter with a force varying directly as the mass of the attracting particle and inversely as the square of the distance between the attracted and the attracting particles. Taking this law as our basis of calculation, we shall investigate the amount of attraction exerted by spherical, spheroidal, and irregular nearly-spherical masses upon a particle, and apply our results in the second part of this Treatise to discover the Figure of the Earth. We shall also show how the attraction of irregular masses lying at the surface of the Earth may be estimated, in order afterwards to ascertain whether the irregularities of mountain-land and the ocean can have any effect on the calculation of this figure.

CHAPTER I.

ON THE ATTRACTION OF SPHERICAL AND SPHEROIDAL

BODIES.

PROP. To find the resultant attraction of an assemblage of particles constituting a homogeneous spherical shell of very small thickness upon a particle outside the shell: the law of attraction of the particles being that of the inverse square.

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The attraction of the whole shell evidently acts in CO.

Let OP revolve about 0 through a small angle de in the plane MOP; then rde is the space described by P. Again, let OPM revolve about OC through a small angle do, then r sin 0 do is the space described by P. And the thickness of the shell is dr. Hence the volume of the elementary portion of the shell thus formed at P equals rdo.r sin Odp. dr ultimately, since its sides are ultimately at right angles to each other.

Then, if the unit of attraction be so chosen, that it equals the attraction of the unit of mass at the unit of distance, the attraction of the elementary mass at P on C in the direction CP

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.. attraction of P on C in CO pr2 sin 0 drdėdo c

=

y2

-r cos e

У

We shall eliminate O from this equation by means of

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being

To obtain the attraction of all the particles of the shell we integrate this with respect to o and y, the limits of O and 2π, those of y being c-r and c+r;

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