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Sir ISAAC N E W T O N’s Book IV.

10. To fix the limit of these two cafes, we are to consider,that the velocities in an orbit, at a and b, are in the invertedproportion of the distances there from the centre of gravitation ;and that, if the gravity increase as the cubes of the distancesdecrease, the velocities necessary to describe circles at a and bare in the fame inverted proportion of the distances at a and bfrom s. In this cafe, therefore, the velocities in circles,and in the orbit at a and b, vary in the fame proportion, andthe fame which exceeds at the one distance must exceed at theother ; so that, for the fame reason for which the body ap-proached to s at a, it would approach to it at b, and if it re-ceded from it at b, it must recede from it at a ; that is, if itonce begin to approach, it must always approach to s, and ifit once begin' to recede, it must always recede from it. Thisalso appears from what we said of the centrifugal force, which,in the fame orbit, increases as the cube of the distance de-creases ; and consequently in the same proportion in which thegravity is supposed to increase in this cafe ; so that, of thesetwo, which ever is supposed to prevail in any one apsid, thesame must prevail in any other apsid, if such could be assigned ;and the body must either descend continually to the centre, orrise from it for ever.

11. Is the gravity increase in a higher proportion than asthe cubes of the distances from the centre of attraction de-crease, then the velocities necessary to carry bodies in circlesabout that centre, in approaching to it, will increase in ahigher proportion than the distances decrease ; that is, in ahigher proportion than the velocity in an orbit increases froma to b j so that as the velocity in a circle at a exceeded thevelocity in the orbit there, it will much more exceed it at B ;and therefore the body, acted on by a gravity varying in such a

manner,