Chap. 3. PHILOSOPHICAL DISCOVERIES. 311

force prevailing in the lower apsis makes the body recede fromit; and, by their actions, the body for ever revolves from the oneto the other.

9. It is easy to fee from what we have said, that the bodycan descend from the higher apsis to the lower, and ascendagain from the lower apsis to the higher, when the velocitiesnecessary to carry bodies in circles about the centre of attractionincrease, in approaching to that centre, in a less proportionthan the velocity of a body moving in an orbit amb increases.For tho’ the velocity in a circle in the greater distances exceedthe velocity in the orbit, this latter, by increasing more quicklyas the distance decreases, gets the better of the other in thelower part of the orbit, and carries the body off again. Butif the velocities by which circles can be described about thecentre of attraction increase, in approaching to that centre, ina higher proportion, or in the same proportion, as the velocityin the orbit increases, then this latter having been supposed ata less than the former, it must always continue less than it,and never get the better of it, so as to be able to carry off' thebody ; and therefore, in all such cafes, the body can neverrecede from the centre after it has once begun to approach to it,but must descend to distances less and less, till it fall into thecentre. It approaches at a, because its velocity is less than thatwhich is requisite to carry it in a circle there : its velocity in-deed increases as it descends to lesser distances, but the velo-cities which would carry bodies in circles at these distancesabout s, increasing also in as great a proportion, the velocityin the orbit must still continue to be less than in these circles,and the body must still continue to approach to the centre.

10. To