Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/327

 every ide by everal bodies S, S, S, &c. dipoed about the orbit ESE. For by the actions of thee bodies the action of the body T will be diminihed on every ide, and decreae in more than a duplicate ratio of the ditance.

But ince the progres or regres of the apides depends upon the decreae of the centripetal force, that is, upon its being in a greater or les ratio than the duplicate ratio of the ditance TP, in the paage of the body from the lower apis to the upper; and upon a like increae in its return to the lower apis again; and therefore becomes greatet where the proportion of the force at the upper apis to the force at the lower apis recedes farthet from the duplicate ratio of the ditances inverely; it is plain that when the apides are in the yzygies, they will, by reaon of the ubducting force KL or NM - LM, go forward more wiftly; and in the quadratures by the additional force LM go backward more lowly. When the velocity of the progres or lownes of the regres is continued for a long time, this inequality becomes exceeding great.

If a body is obliged, by a force reciprocally proportional to the quare of its ditance from any centre, to revolve in an ellipis round that centre; and afterwards in its decent from the upper apis to the lower apis, that force by a perpetual acceion of new force is increaed more than a duplicate ratio of the diminihed ditance; it is manifet that the body being impelled always towards the centre by the perpetual acceion of this new force, will incline more towards that centre than if it were urged by that force alone which decreaes in a duplicate ratio of the