From what has been said may be de duced the velocity and periodic time of a body revolving in a circle, at any given distance from the earth's center, by means of its own gravity. feet, the space described by gravity, at the surface, in the first second of time, wiz. = A M ; then putting r = the radius A C ; it is A 13 X A M=V2gr the velocity in a circle at its surface in one second of time ; and hence, putting c = 3.14159 &c. the circumference of the earth being 2 cr -- 25,000 miles, or 132,000,000 2r feet, it will be V 2 g r : 2 cr :: 1" : = 5078 seconds nearly, or lh 24m 38s, the periodic time at the circumference : also the velocity there, or V2 g r is —26,000 feet per second nearly. Then, since the force of gravity varies in the inverse cate ratio of the distance, by the rules laid down, it Vr::v or 26,000: 26,000 r the velocity of a body re • it volving about the earth at the distance R ; and %,/ r3: ‘,/ R3 5078 t, or 5078" : — r3 • T, the time of revolution in the same.
So if, for instance, it be the moon revolv ing about the earth at the distance of 60 semi-diameters ; then It = 60 r, and the above expressions become V = 26,000 = 3357 feet per second, or miles per minute, for the velocity of the moon in her orbit ; and T = 5078 = r3 2,360,051 seconds, or days nearly, for the periodic time of the moon in her orbit at that distance.
Thus, also, the ratio of the forces of gravitation of the moon towards the sun and earth may be estimated. For one year, or days, being the periodic time of the earth and moon about the sun, and 27 3 days, the periodic time of the moon about the earth, also 60 being the distance of the moon from the earth, and 23,920 the distance from the sun, in semi-diameters of the earth, it is 60 23920 27.3= 27.3' 365.25' 6U 365.25' = 22. that is, the proportion of the moon's gravitation towards the sun is to that to wards the earth as 24 to 1 nearly.
Again, we may hence compute the cen trifugal force of a body at the equator, arising from the earth's rotation. For the periodic time, when the centrifugal force is equal to the force of gravity, it has been shown above, is 5078 seconds ; and 23 hours, 56 minutes, or 86,160 se conds, is the period of the earth's rota tion on its axis ; therefore, as 86,160= : 5078= :: 1: I the centrifugal force re quired, which therefore is the 289th part of gravity at the earth's surface. See Simpson's Fluxions, vol. i.