Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/361

Rh equiditant, one on one ide, the other on the other, from the centre of the phere; G and g the interections of the planes and the axis; and H any point in the plane EF. The centripetal force of the point H upon the corpucle P, exerted in the direction of the line PH is as the ditance PH; and (by cor. 2. of the laws) the ame exerted in the direction of the line PG, or towards the centre S, is at the length PG. Therefore the force of all the points in the plane EF (that is of that whole plane) by which the corpucle P is attracted towards the centre S is as the ditance PG multiplied by the number of thoe points, that is as the olid contained under that plane EF and the ditance PG. And in like manner the force of the plane ef by which the corpucle P is attracted towards the centre S, is as at plane drawn into its ditance Pg, or as the equal plane EF drawn into that ditance P; and the um of the forces of both planes as are plane EF drawn into the um of the ditances PG + Pg, that is as that plane drawn into twice the ditance PS of the centre and the corpucle; that is, as twice the plane EF drawn into the ditance PS, or as the um of the equal planes EF + ef drawn into the ame ditance. And by a like reaoning the forces of all the planes in the whole phere, equiditant on each ide from the centre of the phere, are as the um of thoe planes drawn into the ditance PS, that is, as the whole phere and the diŧance PS conjunctly. Q. E. D.

Let now the corpucle P attract the phere AEBF. And by the ame reaoning it will appear that the force with which the phere is attracted is as the ditance PS. Q. E. D.