Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/314

 will decribe an ellipis round the ame centre C; and the point D, becaue the lines CS and CD are proportional, will decribe a like ellipis over-againt: it. But the bodies T and L, attracted by the motive forces SD x T and SD x L, the firt by the firt, and the lat by the lat equally and in the direction of the parallel lines TI and LK as was aid before. will (by cor. 5. and 6. of the laws of motion) continue to decribe their ellipes round the moveable centre D as before. Q. E. I.

Let there be added a fourth body V, and by the like reaoning it will be demontrated that this body and the point C will decribe ellipes about the common centre of gravity B; the motions of the bodies T, L and S round the centres D and C remaining the ame as before; but accelerated. And by the ame method one may add yet more bodies at pleaure. Q. E. I.

This would be the cae, though the bodies T and L artract each other mutually with accelerative forces either greater or les than thoe with which they attract the other bodies in proportion to their ditance. Let all the mutual accelerative attractions be to each other as the ditances multiplyed into the attracting bodies; and from what has gone before it will eaily be concluded that all the bodies will decribe different ellipes with equal periodical times about their common centre of gravity B, in an immoveable plane. Q. E. I.