Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/334

 body P as the quare of the periodical time of the body P directly, and the quare of the periodical time of the body T inverely. And therefore the mean motion of the line of the apides will be in a given ratio to the mean motion of the nodes; and both thoe motions will be as the periodical time of the body P directly, and the quare of the periodical time of the body T inverely. The increae or diminution of the eccentricity and inclination of the orbit PAB makes no enible variation in the motions of the apides and nodes, unles that increae or diminution be very great indeed.

Since the line LM becomes ometimes greater and ometimes les than the radius PT, let the mean quantity of the force LM be expreed by that radius PT; and then that mean force will be to the mean force SK or SN (which may be alo expreed by ST) as the length PT to the length ST. But the mean force SN or ST, by which the body T is retained in the orbit it decribes about S, is to the force with which the body P is retained in its orbit about T, in a ratio compounded of the ratio of the radius ST to the radius PT and the duplicate ratio of the periodical time of the body P about T, to the periodical time of the body T about S. And ex æquo, the mean force LM is to the force by which the body P is retained in its orbit about T (or by which the ame body P might revolve at the ditance PT in the ame periodical time about any immovable point T) in the ame duplicate ratio of the periodical times. The periodical times therefore being given, together with the ditance PT; the mean force LM is alo given; and that force being given; there is given alo the force MN