210 THE MATHEMATICAL PRINCIPLES BOOK L at the conjunction of the bodies P, S, than at their opposition ; because the generating forces NM and ML are greater. COR. 13. And since the causes and proportions of the errors and varia tions mentioned in these Corollaries do not depend upon the magnitude of the body S, it follows that all things before demonstrated will happen, if the magnitude of the body S be imagined so great as that the system of the two bodies P and T may revolve about it. And from this increase of the body S, and the consequent increase of its centripetal force, from which the errors of the body P arise, it will follow that all these errors, at equal dis tances, will be greater in this case, than in the other where the body S re volves about the system of the bodies P and T. COR. 14. But since the forces NM, ML, when the body S is exceedingly distant, are very nearly as the force SK and the ratio PT to ST conjunctly ; that is, if both the distance PT, and the absolute force of the body 8 be given, as ST3 reciprocally : and since those forces NM, ML are the causes of all the errors and effects treated of in the foregoing Corollaries; it is manifest that all those effects, if the system of bodies T and P con tinue as before, and only the distance ST and the absolute force of the body S be changed, will be very nearly in a ratio compounded of the direct ratio of the absolute force of the body S, and the triplicate inverse ratio of the distance ST. Hence if the system of bodies T and P revolve about a dis tant body S, those forces NM, ML, and their eifl ts, will be (by Cor. 2 and 6, Prop IV) reciprocally in a duplicate ratio c/f the periodical time. And thence, also, if the magnitude of the bodv S be proportional to its absolute force, those forces NM, ML, and their effects, will be directly as the cube of the apparent diameter of the distant body S viewed from T, and so vice versa. For these ratios are the same as the compounded ratio above men tioned. COR. 15. And because if the orbits ESE and PAB, retaining their fig ure, proportions, and inclination to each other, should alter their magni tude ; arid the forces of the bodies S and T should either remain, or be changed in any given ratio ; these forces (that is, the force of the body T, which obliges the body P to deflect from a rectilinear course into the orbit PAB, and the force of the body S, which causes the body P to deviate from that orbit) would act always in the same manner, and in the same propor tion : it follows, that all the effects will be similar and proportional, arid the times of those effects proportional also ; that is, that all the linear er rors will be as tne diameters of the orbits, the angular errors the same as before ; and the times of similar linear errors, or equal angular errors? as the periodical times of the orbits. COR. 16. Therefore if the figures of the orbits and their inclination to each other be given, and the magnitudes, forces, arid distances of the bodies he any how changed, we may. from the errors and times of those errors in