SEC. V.] OF NATURAL PHILOSOPHY. 301 de isity ; or the triplicate ratio of tlie force the same with the quadruplicate ratio of the density : in which case, if the gravity he reciprocally as the square of the distance from the centre, the density will be reciprocally at the cube of the distance. Suppose that the cube of the compressing force be as the quadrato-cube of the density ; and if the gravity be reciprocally as the square of the distance, the density will be reciprocally in a sesquiplicate ratio of the distance. Suppose the compressing force to be in a du plicate ratio of the density, and the gravity reciprocally in a duplicate ra tio of the distance, and the density will be reciprocally as 'the distance. To run over all the cases that might bo offered would be tedious. But as to our own air, this is certain from experiment, that its density is either accurately, or very nearly at least, as the compressing force ; and therefore the density of the air in the atmosphere of the earth is as the weight of the whole incumbent air, that is, as the height of the mercury in the ba rometer. PROPOSITION XXIII. THEOREM XVIII. If a fluid be composed of particles mutually flying each other, and the drnsity be as the compression, the centrifugal forces of the particles 'will be reciprocally proportional to tlie distances of their centres. And, vice versa, particles flying each otli,er, 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 par- F tides 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 re ciprocally as the containing spaces AB3, ab3. In the plane side of the greater cube A BCD 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 me diums are to each other, that is, asa/>3 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 abz. Therefore, ex cequo, 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,/°V?, U drawn through the middles of the two cubes, and divide the fluid into tw^/ parts, These parts will press each other mutually with the same forces with which they A