Cavendish had advanced so far, he was beginning to see his way to the form of the theory which he finally published, and that he did not care to finish the manuscript of the imperfect theory. The general theory of fluids repelling according to any inverse power of the distance is given much more fully than in the paper of 1771, and the remarks on the constitution of air are very interesting. I have therefore printed this paper, but in order to avoid interrupting the reader with a repetition of much of what he has already seen, I have placed it at the end of this Note. CAVENDISH'S FIRST MATHEMATICAL THEORY *. Let a fluid whose particles mutually repel each other be spread uniformly through infinite space. Let a be a particle of that fluid; draw the cone baß con- B force from the particles between a and bẞ than from all those situated beyond it, but if their repulsion is inversely as any less power than the cube, then the repulsion of the particles placed beyond bẞ is infinitely greater than that of those between a and bẞ. If the repulsion of the particles is inversely as the n power of the distance, n being greater than 3, it would constitute an elastic fluid of the same nature as air, except that its elasticity would be inversely as the n+2 power of the distance of the particles, or directly as the n+ 2 power of the density of the fluid. 3 But if n is equal to, or less than 3, it will form a fluid of a very different kind from air, as will appear from what follows. B b COR. 1. Let a fluid of the above-mentioned kind be spread uniformly through infinite space except in the hollow globe BDE, and let the sides of the globe be so thin that the force with which a particle placed contiguous to the sides of the globe would be repelled by so much of the fluid as might be lodged within the space occupied by the sides of the globe should be trifling in respect of the repulsion of the whole quantity of fluid in the globe. If the fluid within the globe was of the same density as without, the particles of the fluid adjacent to either the inside or outside P surface of the globe would not press against those surfaces with any sensible force, as they would be repelled with the same force by the fluid on each side of them. But if the fluid within the globe is denser * From MS. bundle 17. than that without, then any particle adjacent to the inside surface of the globe will be pressed against by the repulsion of so much of the fluid within the globe as exceeds what would be contained in the same space if it was of the same density as without, and consequently will be greater if the globe be large than if it be small. Consequently the pressure against a given quantity (a square inch for example) of the inside surface of the globe will be greater if the globe is large than if it is small. If the particles of the fluid repel each other with a force inversely as their distance, the pressure against a given quantity of the inside surface would be as the square of the diameter of the globe. So that it is plain that air cannot consist of particles repelling each other in the above-mentioned manner. If the repulsion of the particles was inversely as some higher power of the distance than the cube, then any particle of the fluid would not be sensibly affected except by the repulsion of those particles which were almost close to it, so that the pressure of the fluid against a given quantity of the inside surface would be the same whatever was the size of the globe, but then the elasticity [would] be in a greater proportion than that of the power of the density. If the repulsion of the particles is inversely as some less power than the cube of the distance, and the density of the fluid within the globe is less than it is without, then the particles on the outside of the globe will press against it, and the force will be greater if the globe is large than if it be small. If the density of the fluid within the globe be greater than without, then the density will not be the same in all parts of the globe, but will be greater near the surface and less near the middle, for if you suppose the density to be everywhere the same, then any particle of the fluid, as d, would be pressed with more force towards a, the nearest part of the surface of the sphere, than it would [be] in the contrary direction. If the repulsion of the particles is inversely as the square of the distance, I think the inside of the sphere would be uniformly coated with the fluid to a certain thickness, in which the density would be infinite, or the particles would be pressed close together, and in all the space within that, the density would be the same as on the outside of the sphere. The pressure of a particle adjacent to the inside surface against it is equal to the repulsion of all the redundant matter in the sphere collected in the center, and the force with which a particle is pressed towards the surface of the sphere diminishes in arithmetical progression in going from the inside surface to that point at which its density begins to be the same as without, therefore the whole pressure against the inside of the sphere is equal to that of half the redundant matter in the sphere pressed by the repulsion of all the redundant matter collected in the center of the sphere. Therefore, if the quantity of fluid in the sphere is such that its density, if uniform, would be 1+d, and the radius of the sphere be called r, the whole pressure against the inside surface will be as dr.3 dr.3 X and the pressure against a given space of the inside surface will be as d2r2. JP , If this pressure be called P, d is as and dr3 is as r2 √P. Consequently, supposing the fluid to be pumped into different sized globes, the quantity of fluid pumped in will be as the square root [of the force] with which it is pumped, multiplied by the square of the diameter of the globe. If the density within the sphere is less than without, then the density within the sphere will not be uniform, but will be greater towards the middle and less towards the outside, and if the repulsion of the particles is inversely as the square of the distance, there would be a sphere concentric to the hollow globe in which the density would be the same as on the outside of the globe, and all between that and the inside surface of the globe would be a vacuum. From these corollaries it follows that if the electric fluid is of the nature here described, and is spread uniformly through bodies, except when they give signs of electricity, that then if two similar bodies of different sizes be equally electrified, the larger body will receive much less additional electricity in proportion to its bulk than the smaller one, and moreover when a body is electrified, the additional electricity will be lodged in greater quantity near the surface of the body than near the middle. Let us now suppose the fluid within the globe BDE to be denser than without, and let us consider [in what manner] the fluid without will be affected thereby. 1st. There will be a certain space surrounding the globe, as ßde, which will be a perfect vacuum, for first let us suppose that the density without the globe is uniform, then any particle would be repelled with more force from the globe than in the contrary direction. 2ndly. Let us suppose that the space ßde, BDE is not a vacuum, but rarer than the rest of the fluid; still a particle placed close to the surface of the globe would be repelled from it with more force than in the contrary direction. 3rdly. Let us [suppose that] the density in the space between BDE and ẞde is greater than without, then according to some hypothesis of the law of repulsion a particle placed at B might be in equilibrium, but one placed at ẞ could by no means be so. So that there is no way by which the particles can be in equilibrium, unless there is a vacuum all round the globe to a certain distance. How the density of the fluid will be affected beyond this vacuum I cannot exactly tell, except in the following case: If the repulsion of the particles is inversely as the square of the distance, there will be a perfect vacuum between BDE and ẞde, and beyond that the density will be perfectly uniform, Boe being a sphere concentric to BDE, and of such a size, that if the matter in BDE was spread uniformly all over the sphere Boe, its density would be the same as beyond it. For any quantity of matter spread uniformly over the globe ẞde or BDE affects a particle of matter placed without that sphere just in the same manner as if the whole fluid was collected in the center of the sphere, so that any particle of matter placed without the sphere ẞde will be in perfect equilibrio. In like manner if the fluid within BDE is rarer than without, there will be a certain space surrounding the globe, as that between BDE and Bde, in which the density will be infinite, or in which the particles will be pressed close together, and if the repulsion of the particles is inversely as the square of the distance, the density of the fluid beyond that will be uniform: the diameter of ẞde being such that if all the matter within it was spread uniformly, its density would be the same as without. Let a fluid of the above-mentioned kind be spread uniformly through infinite space except in the canal acdef of any shape whatsoever, except that the ends aghb and mden are straight canals of an equal diameter, and of such a length that a particle placed at a or d shall not be sensibly affected by the repulsion of the matter in the part gemnfh, and let there be a greater quantity of the fluid in this canal than in an equal space without. Then the density of the fluid in different parts of the canal will be very different, but I imagine the density will be just the same at a as at d. For suppose ab and de to be joined, as in the figure, by a canal of an uniform diameter and regular shape, nowhere approaching near enough to gcmnfh to be affected by the repulsion of the particles within it. If the matter was not of the same density [at a and d] the matter therein could not be at rest, but there would be a continual current through the canal, which seems highly improbable. COR. Let C be a conductor of electricity of any shape, em and fn wires extending from thence to a great distance. Let a and b be two equal bodies placed on those wires at such a distance from C as not to be sensibly affected by the electricity thereof, and let the conductor or wires be electrified by any part: the quantity of electric fluid in the bodies a and b will not be sensibly different, or they will appear equally electrified. A B a Case 1. Let the parallel planes Aa, Bb, &c., be continued infinitely. Let all infinite space except the space contained between Aa and Cc, and between Ee and Hh, be filled uniformly with particles repelling inversely as the square of their distance; let the space between Ee and Hh be filled with fluid of the same density, the particles of which can move from one part to another; and let the space between Aa and Cc be filled with matter whose density is to [that in] the rest of space as AD to AC. H h Take EF = CD, and GH such that the matter between Ee and Ff when pressed close together, so that the particles touch each other, shall occupy the space between Gg and Hh. The space between Ee and Ff will be a vacuum, that between Ff and Gg of the same density as the rest of space; and between Gg and Hh the particles will touch one another. Case 2. Let everything be as in case the first, except that there is a canal opening into the plane Hh, by which the matter in the space EH is at liberty to escape; part of the matter will then run out, and the density therein will be everywhere the same as without, except in the space EF, which will be a vacuum, EF being equal to CD. Case 3. Suppose now that a canal opens into the plane Aa by which the fluid in the space AC may escape. It will have no tendency to do so, for the repulsion of the redundant fluid in AC on a particle at a will be exactly equal to [the] want of repulsion of EH. the space Case 4. Let now the space between Aa and Cc be filled with matter whose density is to the rest of space as AB to AC. Then the space between Hh and Gg will be a vacuum, GII being equal to BC. In the space EF the particles of matter will be pressed together so as to touch each other, the quantity of matter therein exceeding what is naturally contained in that space by as much as is driven out of the space GH; and in the space between Ff and Gg the matter will be of the same density as without. Case 5. Suppose now that a canal opens into the plane Hh as in Case 2, then will matter run into the space EH, and the density will be everywhere the same as without, except in the space EF, where the particles will be pressed close together, the quantity of matter therein |