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.

The attraction of + Let OP revol plane MOP; tl let OPM revo! r sin 0 do is t the shell is d of the shell mately, sinc other.

Then, if the attrac

the attra direction

... at

T

the distance.

an is exactly analogous to that given i to alter the law of attraction. Then

со

́ ̧‚' + c2— r3) 4 (y) dy (integrated by parts)

__2+c2—r2) sp (y) dy — 2ƒ{yƒ¢ (y) dy} dy]

̧y2 + c2 — r2) Þ, (3) − 2↓ (y) + const.} suppose,

prdr

↓ (c − r) Į

− r)},

is 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.

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

c

1

.. attraction = 2πprdr {r+° 4, (r + c) − a ¥ (r+c)

с

This result shows that the shell attracts the particle at C in the same manner as if the mass of the shell were condensed into its centre.

3. It follows also that a sphere, which is either homogeneous or consists of concentric spherical shells of uniform density, will attract the particle C in the same manner as if the whole mass were collected at its centre.

PROP. To find the attraction of a homogeneous spherical shell of small thickness on a particle situated within it.

4. We must proceed as in the last Proposition; but the limits of y are in this case r― с and r+c; hence,

therefore the particle within the shell is equally attracted in every direction.

5. This result may easily be arrived at geometrically in the following manner. Through the attracted point suppose an elementary double cone to be drawn, cutting the shell in two places. The inclinations of the elementary portions of the shell, thus cut out, to the axis of the cone will be the same, the thickness the same, but the other two dimensions of the elements will each vary as the distance from the attracted point; and therefore the masses of the two opposite elements of the shell will vary directly as the square of the distance from that point, and consequently their attractions will be exactly equal, and being in opposite directions will not affect the point. The whole shell may be thus divided into pairs of equal attracting elements and in opposite directions, and therefore the whole shell has no effect in drawing the point in any one direction more than in another.

6. The results of these two Propositions are so simple and beautiful, that it is interesting to enquire whether these

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. attraction on C in CO

ctr

"prdr [*** (y2 + c2 — r2) 4 (y) dy (integrated by parts)

c2

πрrdr

c2

πρrdr

=

c2

[(y2+c2 — r2) S$ (y) dy — 2ƒ{yfþ(y) dy} dy]

{ (y2 + c2 — r2) ¤, (y) — 24 (y) + const.} suppose,

Then

(c+r) — ©=” $,(c—r) + 32 ¥ (c−r)}

с

с

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.

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

1

.. attraction = 2πprdr {”+ © Þ1(r+c) — — ↓ (r+ c)

d

r-c

C

➡ 2 wprdr £ {↓ (r + c) — * (r− e) } .

=

dc

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

† (r) = S {r Sø (r) dr} dr,

then the attraction of the shell

γ

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

therefore by the first of the above equations of condition.