Fig. 2.
B
B

P

18] LEMMA IV. Let BDE, bde, and ẞde (Fig. 2) be concentric spherical surfaces, whose center is C: if the space* Bb is filled with uniform matter, whose particles repel with a force inversely as the square of the distance, a particle placed anywhere within the space Cb, as at P, will be repelled with as much force in one direction as another, or it will not be impelled in any direction. This is demonstrated in Newton, Princip. Lib. I. Prop. 70. It follows also from his demonstration, that if the repulsion is inversely as some higher power of the distance than the square, the particle P will be impelled towards the center; and if the repulsion is inversely as some lower power than the square, it will be impelled from the center.

19] LEMMA V. If the repulsion is inversely as the square of the distance, a particle placed anywhere without the sphere BDE, is repelled by that sphere, and also by the space Bb, with the same force that it would if all the matter therein was collected in the center of the sphere; provided the density of the matter therein is everywhere the same at the same distance from the center. This is easily deduced from Prop. 71, of the same book, and has been demonstrated by other authors.

20] PROP. V. PROBLEM 1. Let the sphere BDE be filled with uniform solid matter, overcharged with electric fluid let the fluid therein be moveable, but unable to escape from it: let the fluid in the rest of infinite space be moveable, and sufficient to saturate the matter therein; and let the matter in the whole of infinite space, or at least in the space BB, whose dimensions will be given below, be uniform and solid; and let the law of the electric attraction and repulsion be inversely as the square of the distance it is required to determine in what manner the fluid will be disposed both within and without the globe.

Take the space Bb such, that the interstices between the particles of matter therein shall be just sufficient to hold a quantity of electric fluid, whose particles are pressed close together, so as to

* By the space Bb or Bß, I mean the space comprehended between the spherical surfaces BDE and bde, or between BDE and Boe: by the space Cb or CB, I mean the spheres bde or ßde.

21]

DISTRIBUTION IN A SPHERE.

9

touch each other, equal to the whole redundant fluid in the globe, besides the quantity requisite to saturate the matter in Bb; and take the space BB such, that the matter therein shall be just able to saturate the redundant fluid in the globe: then, in all parts of the space Bb, the fluid will be pressed close together, so that its particles shall touch each other; the space BB will be intirely deprived of fluid; and in the space Cb, and all the rest of infinite space, the matter will be exactly saturated.

For, if the fluid is disposed in the above-mentioned manner, a particle of fluid placed anywhere within the space Cb will not be impelled in any direction by the fluid in Bb, or the matter in Bß, and will therefore have no tendency to move: a particle placed anywhere without the sphere Bds will be attracted with just as much force by the matter in BB, as it is repelled by the redundant fluid in Bb, and will therefore have no tendency to move: a particle placed anywhere within the space Bb, will indeed be repelled towards the surface, by all the redundant fluid in that space which is placed nearer the center than itself; but as the fluid in that space is already pressed as close together as possible, it will not have any tendency to move; and in the space Bẞ there is no fluid to move, so that no part of the fluid can have any tendency to

move.

Moreover, it seems impossible for the fluid to be at rest, if it is disposed in any other form; for if the density of the fluid is not everywhere the same at the same distance from the center, but is greater near b than near d, a particle placed anywhere between those two points will move from b towards d; but if the density is everywhere the same at the same distance from the center, and the fluid in Bb is not pressed close together, the space Cb will be overcharged, and consequently a particle at b will be repelled from the center, and cannot be at rest: in like manner, if there is any fluid in BB, it cannot be at rest: and, by the same kind of reasoning, it might be shewn, that, if the fluid is not spread uniformly within the space Cb, and without the sphere ẞde, it cannot be at rest.

21] COR. I. If the globe BDE is undercharged, everything else being the same as before, there will be a space Bb, in which the matter will be intirely deprived of fluid, and a space BB, in which the fluid will be pressed close together; the matter in Bb being equal to the whole redundant matter in the globe, and the

redundant fluid in BB, being just sufficient to saturate the matter in Bb: and in all the rest of space the matter will be exactly saturated. The demonstration is exactly similar to the foregoing.

22] COR. II. The fluid in the globe BDE will be disposed in exactly the same manner, whether the fluid without is immoveable, and disposed in such manner, that the matter shall be everywhere saturated, or whether it is disposed as above described; and the fluid without the globe will be disposed in just the same manner, whether the fluid within is disposed uniformly, or whether it is disposed as above described.

23] PROP. VI. PROBLEM 2. To determine in what manner the fluid will be disposed in the globe BDE, supposing everything as in the last problem, except that the fluid on the outside of the globe is immoveable, and disposed in such manner as everywhere to saturate the matter, and that the electric attraction and repulsion is inversely as some other power of the distance than the square.

I am not able to answer this problem accurately; but I think we may be certain of the following circumstances.

24] CASE 1. Let the repulsion be inversely as some power of the distance between the square and the cube, and let the globe be overcharged.

It is certain that the density of the fluid will be everywhere the same, at the same distance from the center. Therefore, first, There can be no space as Cb, within which the matter will be everywhere saturated; for a particle at b is impelled towards the center, by the redundant fluid in Bb, and will therefore move towards the center, unless Cb is sufficiently overcharged to prevent it. Secondly, The fluid close to the surface of the sphere will be pressed close together; for otherwise a particle so near to it, that the quantity of fluid between it and the surface should be very small, would move towards it; as the repulsion of the small quantity of fluid between it and the surface would be unable to balance the repulsion of the fluid on the other side. Whence, I think, we may conclude, that the density of the fluid will increase gradually from the center to the surface, where the particles will be pressed close together: whether the matter exactly at the center will be overcharged, or only saturated, I cannot tell,

28]

A PLANE PLATE.

11

25] COR. For the same reason, if the globe be undercharged, I think we may conclude, that the density of the fluid will diminish gradually from the center to the surface, where the matter will be intirely deprived of fluid.

26] CASE 2. Let the repulsion be inversely as some power of the distance less than the square; and let the globe be overcharged.

There will be a space Bb, in which the particles of the fluid will be everywhere pressed close together; and the quantity of redundant fluid in that space will be greater than the quantity of redundant fluid in the whole globe BDE; so that the space Cb, taken all together, will be undercharged: but I cannot tell in what manner the fluid will be disposed in that space.

For it is certain, that the density of the fluid will be everywhere the same, at the same distance from the center. Therefore, let b be any point where the fluid is not pressed close together, then will a particle at b be impelled towards the surface, by the redundant fluid in the space Bb; therefore, unless the space Cb is undercharged, the particle will move towards the surface.

27] COR. For the same reason, if the globe is undercharged, there will be a space Bb, in which the matter will be intirely deprived of fluid, the quantity of matter therein being more than the whole redundant matter in the globe; and, consequently, the space Cb, taken all together, will be overcharged *.

28] LEMMA VI. Let the whole space comprehended between two parallel planes, infinitely extended each way, be filled with uniform matter, the repulsion of whose particles is inversely as the square of the distance; the plate of matter formed thereby will repel a particle of matter with exactly the same force, at whatever distance from it it be placed.

Fig. 3.

For suppose that there are two such plates, of equal thickness, placed parallel to each other, let A (Fig. 3) be any point not placed in or between the two plates: let BCD represent any part of the nearest plate: draw the lines AB, AC, and AD, cutting the furthest plate in

C

B

D

b, c, and d; for it is plain that if they cut one plate, they must, if produced, cut the other: the triangle BCD is to the triangle bcd, as AB2 to Ab2; therefore a particle of matter at A will be repelled with the same force by the matter in the triangle BCD, as by that in bed. Whence it appears, that a particle at A will be repelled with as much force by the nearest plate, as by the more distant; and consequently, will be impelled with the same force by either plate, at whatever distance from it it be placed.

29] COR. If the repulsion of the particles is inversely as some higher power of the distance than the square, the plate will repel a particle with more force, if its distance be small than if it be great; and if the repulsion is inversely as some lower power than the square, it will repel a particle with less force, if its distance be small than if it be great.

30] PROP. VII. PROB. 3. In Fig. 4, let the parallel lines Aa, Bb, &c. represent parallel planes

infinitely extended each way: let the spaces* AD and EH be filled with uniform solid matter: let the electric fluid in each of those spaces be moveable and unable to escape: and let all the rest of the matter in the universe be saturated with immoveable fluid; and let the electric attraction and repulsion be inversely as the square of

Fig. 4.

a

the distance. It is required to determine in what manner the fluid will be disposed in the spaces AD and EH, according as one or both of them are over or undercharged.

Let AD be that space which contains the greatest quantity of redundant fluid, if both spaces are overcharged, or which contains the least redundant matter, if both are undercharged; or, if one is overcharged, and the other undercharged, let AD be the overcharged one. Then, first, There will be two spaces, AB and GH, which will either be intirely deprived of fluid, or in which the particles will be pressed close together; namely, if the whole quantity of fluid in AD and EH together, is less than

* By the space AD or AB, &c. I mean the space comprehended between the planes da and Dd, or between Aa and Bb.