of the common centre of gravity, and the motion of the ſpace which moves along with this centre uniformly in a right line, and alſo the very firſt, or beginning motions of the bodies in reſpect of this ſpace. Then (by cor. 5. of the laws, and the laſt theorem) the ſubſequent motions will be performed in the ſame manner in that ſpace, as if that ſpace together with the common centre of gravity were at reſt, and as if the bodies did not attract each other; but were attracted by a third body placed in that centre. The motion therefore in this moveable ſpace of each body going off from a given place, in a given direction, with a given velocity, and acted upon by a centripetal force tending to that centre, is to be determined by prob. 9. and 26. and at the ſame time will be obtained the motion of the other round the ſame centre. With this motion compound the uniform progreſſve motion of the entire ſyſtem of the ſpace and the bodies revolving in it, and there will be obtained the abſolute motion of the bodies in immovable ſpace. Q. E. I.
Proposition LXIV. Problem XL.
Suppoſing forces with which bodies mutually attract each other to increaſe in a ſimple ratio of their diſtances from the centres; it it required to find the motions of ſeveral bodies among themſelves.
Suppoſe the two firſt bodies T and L (Pl. 21. Fig. 1.) to have their common centre of gravity
