Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/332

 in this cae, than in the other where the body S revolves about the ytem of the bodies P and T.

But ince the forces NM, ML, when the body S is exceedingly ditant, are very nearly as the force SK and the ratio of PT to ST conjunctly; that is, if both the ditance PT and the abolute force of the body S be given, as ST reciprocally; and ince thoe forces NM, ML are the caues of all the errors and effects treated of in the foregoing corollaries; it is manifet, that all thoe effects, if the ytem of bodies T and P continue as before, and only the ditance ST and the abolute force of the body S be changed, will be very nearly in a ratio compounded of the direct ratio of the abolute force of the body S, and the triplicate invere ratio of the ditance ST. Hence if the ytem of bodies T and P revolve about a ditant body S; thoe forces NM, ML and their effects will be (by cor. 2. and 6. prop. 4.) reciprocally in a duplicate ratio of the periodical time. And thence alo if the magnitude of the body S be proportional to its abolute force, thoe forces NM, ML, and their effects, will be directly as the cube of the apparent diameter of the ditant body S viewed from T, and o vice vera. For thee ratio's are the ame as the compounded ratio above-mentioned.

And becaue if the orbits ESE and PAB, retaining their figure, proportions and inclination to each other, hould alter their magnitude; and the forces of the bodies S and T hould either remain, or be changed in any given ratio; thee forces (that is, the force of the body T which obliges the body P to deflect from a rectilinear