Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/351

  is proportional to the emi-diameter of the sphere.

For conceive two corpucles to be everally attracted by two pheres, one by one the other by the other, and their ditances from the centres of the pheres to be proportional to the diameters of the pheres repectively; and the pheres to be reolved into like particles dipoed in a like ituation to the corpucles. Then the attractions of one corpucle towards the everal particles of one phere, will be to the attractions of the other towards as many analogous particles of the other phere in a ratio compounded of the ratio of the particles directly and the duplicate ratio, of the diŧances inverely. But the particles are as the pheres, that is in a triplicate ratio of the diameters, and the ditances are as the diameters; and the firt ratio directly with the lat ratio taken twice inverely, becomes the ratio of diameter to diameter. Q. E. D.

Hence if corpucles revolve in circles about pheres compoed of matter equally attracting; and the ditances from the centres of the pheres be proportional to their diameters; the periodic times will be equal.

And vice versa, if the periodic times are equal, the ditances will be proportional to the diameters. Thee two corollaries appear from cor. 5. prop. 4.

If to the everal points of any two olids whatever, of like figure and equal denity, there tend equal centripetal forces decreaing in a duplicate ratio of the ditances from thoe points; the forces with which corpucles placed in a like