Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/333

 coure into the orbit PAB, and the force of the body S, which caues the body P to deviate from that orbit) would act always in the ame manner, and in the ame proportion; it follows that all the effects will be imilar and proportional, and the times of thoe effects proportional alo; that is, that all the linear errors will be as the diameters of the orbits, the angular errors the ame as before; and the times of imilar linear errors, or equal angular errors as the periodical times of the orbits.

Therefore if the figures of the orbits and their inclination to each other be given, and the magnitudes, forces, and ditances of the bodies be any how changed; we may, from the errors and times of thoe errors in one cae, collect very nearly the errors and times of the errors in any other cae. But this may be done more expeditiouly by the following method. The forces NM, ML, other things remaining unaltered, are as the radius TP; and their periodical effects (by cor. 2. lem. 10.) are as the forces, and the quare of the periodical time of the body P conjunctly. Thee are the linear errors of the body P; and hence the angular errors as they appear from the centre T (that is the motion of the apfides and of the nodes, and all the apparent errors as to longitude and latitude) are in each revolution of the body P, as the quare of the time of the revolution very nearly. Let thee ratio's be compounded with the ratio's in cor. 14. and in any ytem of bodies T, P, S, where P revolves about T very near to it, and T revolves about S at a great ditance, the angular errors of the body P, oberved from the centre T, will be in each revolution of the