Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/315



Bodies, whoe forces decreae in a duplicate ratio of their differences from their centres, may move among themelves in ellipis; and by radii drawn to the foci may decribe area's proportional to the time very nearly.

In the lat propoition we demontrated that cae in which the motions will be performed exactly in ellipes. The more ditant the law of the forces is from the law in that cae, the more will the bodies diturb each others motions; neither is it poible that bodies attracting each other mutually according to the law uppoed in this propoition hould move exactly in ellipes unles keeping a certain proportion of ditances from each other. However in the following caes the orbits will not much differ from ellipes.

Imagine everal leer bodies to revolve about ome very great one at different ditances from it, and uppoe abolute forces tending to every one of the bodies, proportional to each. And becaue (by cor. 4. of the laws) the common centre of gravity of them all is either at ret or moves uniformly forward in a right line, uppose the leer bodies o mall that the great body may be never at a enible ditance from that centre; and then the great body will, without any enible