Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/342

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at reſt are not attracted, or were much more or much leſs attracted or much lire or much leſs agitated.

This may be demoſtrated after the ſame mamner as prop. 66. but by a more prolix reaſoning, which I therefore paſs over. It will be ſufficient to conſider it after this manner. From the demonſtration of the laſt propoſition it is plain, that the centre, towards which the body S is urged by the two forces conjunctly, is very near to the common centre of gravity of thoſe two other bodies. If this centre were to coincide with that common centre, and moreover the common centre of gravity of all the three bodies were at reſt; the body S on one ſide, and the common centre of gravity of the other two bodies on the other ſide. would deſcribe true ellipſes about that quieſcent common centre. This appears from cor. 2. prop. 58. compared with what was demonſtrated in prop. 64 and 65. Now this accurate elliptical motion will be diſtributed a little by the diſtance of the centre of the two bodies from the centre towards which the third body S is attracted. Let there be added moreover a motion to the common centre of the three, and the perturbation will be increaſed yet more. Therefore the perturbation is leaſt when the common centre of the three bodies is at reſt; that is, when the innermoſt and greateſt body T is attracted according to the ſame law as the reſt are; and is always greateſt, when the common centre of the three, by the diminution of the motion of the body T; begins to be moved, and is more and more agitated.