Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/297

 thereof proportional to the centripetal force with which the body tends towards the centre C; TM a right line perpendicular to the curve uperficies; TI a part thereof proportional to the force of preure with which the body urges the uperficies, and therefore with which it is again repelled by the uperficies towards M; PTF a right line parallel to the axis and paing through the body, and GF, IH right lines let fall perpendicularly from the points G and I upon that parallel PHTF. I ay now that the area AOP, decribed by the radius OP from the beginning of the motion is proportional to the time. For the force TG (by cor. 2. of the laws of motion) is reolved into the forces TF, FG; and the force TI into the forces TH, HI; but the forces TF, TH acting in the direction of the line PF perpendicular to the plane AOP, introduce no change in the motion of the body but in a direction perpendicular to that plane. Therefore its motion o far as it has the ame direction with the poition of the plane, that is, the motion of the point P, by which the projection AP of the trajectory is decribed in that plane, is the ame as if the forces TF, TH were taken away, and the body were acted on by the forces FG, HI alone; that is, the ame as if the body were to decribe in the plane AOP the curve AP by means of a centripetal force tending to the centre O, and equal to the um of the forces FG and HI. But with uch a force as that (by prop. 1.) the area AOP will be decribed proportional to the time. Q. E. D.

. By the ame reaoning if a body, acted on by forces tending to two or more centres in