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.

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.

b

B

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 [Note 2.]

с

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 AB 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 bcd. 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.

B

D

Fig. 4.

a

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

H

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 Aa and Dd, or between Aa and Bb.

sufficient to saturate the matter therein, they will be intirely deprived of fluid; the quantity of redundant matter in each being half the whole redundant matter in AD and EH together: but if the fluid in AD and EH together is more than sufficient to saturate the matter, the fluid in AB and GH will be pressed close together; the quantity of redundant fluid in each being half the whole redundant fluid in both spaces. Secondly, In the space CD the fluid will be pressed close together; the quantity of fluid therein being such, as to leave just enough fluid in BC to saturate the matter therein. Thirdly, The space EF will be intirely deprived of fluid; the quantity of matter therein being such that the fluid in FG shall be just sufficient to saturate the matter therein consequently, the redundant fluid in CD will be just sufficient to saturate the redundant matter in EF; for as AB and GH together contain the whole redundant fluid or matter in both spaces, the spaces BD and EG together contain their natural quantity of fluid; and therefore, as BC and FG each contain their natural quantity of fluid, the spaces CD and EF together contain their natural quantity of fluid. And fourthly, The spaces BC and FG will be saturated in all parts.

For, first, If the fluid is disposed in this manner, no particle of it can have any tendency to move: for a particle placed anywhere in the spaces BC and FG, is attracted with just as much force by EF, as it is repelled by CD; and it is repelled or attracted with just as much force by AB, as it is in a contrary direction by GH, and, consequently, has no tendency to move. A particle placed anywhere in the space CD, or in the spaces AB and GH, if they are overcharged, is indeed repelled with more force towards the planes Dd, Aa and Hh, than it is in the contrary direction; but as the fluid in those spaces is already as much compressed as possible, the particle will have no tendency to move.

Secondly, It seems impossible that the fluid should be at rest, if it is disposed in any other manner: but as this part of the demonstration is exactly similar to the latter part of that of Problem the first, I shall omit it.

31] COR. I. If the two spaces AD and EH are both overcharged, the redundant fluid in CD is half the difference of the redundant fluid in those spaces: for half the difference of the

redundant fluid in those spaces, added to the quantity in AB, which is half the sum, is equal to the whole quantity in AD. For a like reason, if AD and EH are both undercharged, the redundant matter in EF is half the difference of the redundant matter in those spaces; and if AD is overcharged, and EH undercharged, the redundant fluid in CD exceeds half the redundant fluid in AD, by a quantity sufficient to saturate half the redundant matter in EH.

32] COR. II. It was before said, that the fluid in the spaces AB and GH (when there is any fluid in them) is repelled against the planes Aa and Hh; and, consequently, would run out through those planes, if there was any opening for it to do so. The force with which the fluid presses against the planes Aa and Hh, is that with which the redundant fluid in AB is repelled by that in GH; that is, with which half the redundant fluid in both spaces is repelled by an equal quantity of fluid. Therefore, the pressure against Aa and Hh depends only on the quantity of redundant fluid in both spaces together, and not at all on the thickness or distance of those spaces, or on the proportion in which the fluid is divided between the two spaces. If there is no fluid in AB and GH, a particle placed on the outside of the spaces AD and EH, contiguous to the planes Aa or Hh, is attracted towards those planes by all the matter in AB and GH, id est, by all the redundant matter in both spaces; and, consequently, endeavours to insinuate itself into the space AD or EH; and the force with which it does so depends only on the quantity of redundant matter in both spaces together. The fluid in CD also presses against the plane Dd, and the force with which it does so is that with which the redundant fluid in CD is attracted by the matter in EF.

33] COR. III. If AD is overcharged, and EH undercharged: and the redundant fluid in AD is exactly sufficient to saturate the redundant matter in EH, all the redundant fluid in AD will be collected in the space CD, where it will be pressed close together: the space EF will be intirely deprived of fluid, the quantity of matter therein being just sufficient to saturate the redundant fluid in CD, and the spaces AC and FH will be everywhere saturated. Moreover, if an opening is made in the planes Aa or Hh, the fluid