If a fluid be composed of particles mutually flying each other, and the density be as the compression, the centrifugal forces of the particles will be reciprocally proportional to the distances of their centres. And, vice versa, particles flying each other, with forces that are reciprocally proportional to the distances of their centres, compose an elastic fluid, whose density is as the compression.
Let the fluid be supposed to be included in a cubic space ACE, and then to be reduced by compression into a lesser cubic space ace; and the distances of the particles retaining a like situation with respect to each other in both the spaces, will be as the sides AB, ab of the cubes; and the densities of the mediums will be reciprocally as the containing spaces AB3, ab3. In the plane side of the greater cube ABCD take the square DP equal to the plane side db of the lesser cube: and, by the supposition, the pressure with which the square DP urges the inclosed fluid will be to the pressure with which that square db urges the inclosed fluid as the densities of the mediums are to each other, that is, as ab3 to AB3. But the pressure with which the square DB urges the included fluid is to the pressure with which the square DP urges the same fluid as the square DB to the square DP, that is, as AB2 to ab2. Therefore, ex æquo, the pressure with which the square DB urges the fluid is to the pressure with which the square db urges the fluid as ab to AB. Let the planes FGH, fgh, be drawn through the middles of the two cubes, and divide the fluid into two parts. These parts will press each other mutually with the same forces with which they are themselves pressed by the planes AC, ac, that is, in the proportion of ab to AB: and therefore the centrifugal forces by which these pressures are sustained are in the same ratio. The number of the particles being equal, and the situation alike, in both cubes, the forces which all the particles exert, according to the planes FGH, fgh, upon all, are as the forces which each exerts on each. Therefore the forces which each exerts on each, according to the plane FGH in the greater cube, are to the forces which each exerts on each, according to the plane fgh in the lesser cube, as ab to AB, that is, reciprocally as the distances of the particles from each other. Q.E.D.
And, vice versa, if the forces of the single particles are reciprocally as the distances, that is, reciprocally as the sides of the cubes AB, ab; the sums of the forces will be in the same ratio, and the pressures of the sides DB, db as the sums of the forces; and the pressure of the square DP to the pressure of the side DB as ab2 to AB2. And, ex æquo, the pressure of the square DP to the pressure of the side db as ab3 to AB3; that is, the force of compression in the one to the force of compression in the other as the density in the former to the density in the latter. Q.E.D.
SCHOLIUM.
By a like reasoning, if the centrifugal forces of the particles are reciprocally in the duplicate ratio of the distances between the centres, the cubes of the compressing forces will be as the biquadrates of the densities. If the centrifugal forces be reciprocally in the triplicate or quadruplicate ratio of the distances, the cubes of the compressing forces will be as the quadrato-cubes, or cubo-cubes of the densities. And universally, if D be put for the distance, and E for the density of the compressed fluid, and the centrifugal forces be reciprocally as any power Dn of the distance, whose index is the number n, the compressing forces will be as the cube roots of the power Eⁿ⁺², whose index is the number n + 2; and the contrary. All these things are to be understood of particles whose centrifugal forces terminate in those particles that are next them, or are diffused not much further. We have an example of this in magnetical bodies. Their attractive virtue is terminated nearly in bodies of their own kind that are next them. The virtue of the magnet is contracted by the interposition of an iron plate, and is almost terminated at it: for bodies further off are not attracted by the magnet so much as by the iron plate. If in this manner particles repel others of their own kind that lie next them, but do not exert their virtue on the more remote, particles of this kind will compose such fluids as are treated of in this Proposition. If the virtue of any particle diffuse itself every way in infinitum, there will be required a greater force to produce an equal condensation of a greater quantity of the fluid. But whether elastic fluids do really consist of particles so repelling each other, is a physical question. We have here demonstrated mathematically the property of fluids consisting of particles of this kind, that hence philosophers may take occasion to discuss that question.