But we have evidently by means of the binomial theorem,

If now we make x = ra, the same quantity may be written

a

Having thus the value of the repulsion due to the inner sphere B on an exterior point p, it remains to determine that due to the fluid on A's surface. But this last is represented by

Σπαρ d. (a+r)3* — (a − r) 3 *

1-n.3

-n dr

..(17).

(Mec. Cel. Liv. II. No. 12.) Now by expanding this function there results

=

2.

-n

1+.

3

n.n+12
4.5

2-n

3

n.n+1.n+2.n+3

4.5.6.73 +&a

-.Σn. n + 1. h +2......+28-1 (+1)

4 5

6

28 +3

}

The last of these expressions may readily be exhibited under a finite form, by remarking that

= S'x"dx (1 − x') = Σ " . n + 2 ...... n + 28 → 2

n.n+2.n+4

() ()

Hence, since г()=√, the value of the repulsion arising from A's surface becomes

Now by adding the repulsion due to the inner sphere which is given by the formula (16), we obtain, (since it is evidently indifferent what variable enters into a definite integral, provided each of its limits remain unchanged)

for the value of the total repulsion upon a particle p of positive fluid situate within the sphere A and exterior to B. We thus see that when P' is positive the particle p is always impelled by a force which is equal to zero at B's surface, and which continually increases as p recedes farther from it. Hence, if any particle of positive fluid is separated ever so little from B's surface, it has no tendency to return there, but on the contrary, it is continually impelled therefrom by a regularly increasing force; and consequently, as was before observed, the equilibrium can not be permanent until all the positive fluid has been gradually abstracted from B and carried to the surface of A, where it is retained by the non-conducting medium with which the sphere A is conceived to be surrounded.

Let now q represent the total quantity of fluid in the inner sphere, then the repulsion exerted on p by this will evidently be qr, when r is supposed infinite. Making therefore r infinite in the expression (15), and equating the value thus obtained to the one just given, there arises

When the equilibrium has become permanent, q is equal to the total quantity of that kind of fluid, which we choose to consider negative, originally introduced into the sphere A; and if now q, represent the total quantity of fluid of opposite name contained within A, we shall have, for the determination of the two unknown quantities P' and b, the equations

and hence we are enabled to assign accurately the manner in which the two fluids will distribute themselves in the interior of A; q and 9, the quantities of the fluids of opposite names originally introduced into A being supposed given.

9. In the two foregoing articles we have determined the

manner in which our hypothetical fluids will distribute themselves in the interior of a conducting sphere A when in equilibrium and free from all exterior actions, but the method employed in the former is equally applicable when the sphere is under the influence of any exterior forces. In fact, if we conceive them all resolved into three X, Y, Z, in the direction of the co-ordinates x, y, z of a point p, and then make, as in Art. 1,

we shall have, in consequence of the equilibrium,

which, multiplied by dx, dy and de respectively, and integrated,

where Xdx + Ydy +Zdz is always an exact differential.

We thus see that when X, Y, Z are given rational and entire functions V will be so likewise, and we may thence deduce (Art. 5)

p = (1 − x'3 — y′′ — z'3) *.ƒ (x', y', z'),

where f is the characteristic of a rational and entire function of the same degree as V.

The preceding method is directly applicable when the forces X, Y, Z are given explicitly in functions of x, y, z. But instead of these forces, we may conceive the density of the fluid in the exterior bodies as given, and thence determine the state which its action will induce in the conducting sphere A. For example, we may in the first place suppose the radius of A to be taken as the unit of space, and an exterior concentric spherical surface, of which the radius is a, to be covered with fluid of the density U" U being a function of the two polar co-ordinates " and " of any element of the spherical surface of the same kind as those considered by Laplace (Mec. Cel. Liv. III.). Then it is

easy to perceive by what has been proved in the article last cited, that the value of the induced density will be of the form

"

r', 0, being the polar co-ordinates of the element du, and U'") what U" becomes by changing ", " into e', '.

Still continuing to follow the methods before explained, (Art. 4 and 5) we get in the present case

f(r) = B2+ Br® +B ̧‚'+B ̧»'" + &c.

Hence, fBU'"), and

Then, by giving to t all the values 1, 2, 3, &c. of which it is susceptible, and taking the sum of all the resulting quantities, we shall have, since in the present case V reduces itself to the single term V,

the sign S belonging to the unaccented letter t.

2i+26+1

;

If now V' represents the function analogous to V and due to the fluid on the spherical surface, we shall obtain by what has been proved (Art. 3)

() representing the same function as in the article just cited.