Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/376

288 ariing from the ame particle in the centre, that is, in the ubduplicate ratio of the ditances SI, SP to each other reciprocally. Thee two ubduplicate ratio's compoe the ratio of equality, and therefore the attractions in I and P produced by the whole phere are equal. By the like calculation if the forces of the particles of the phere are reciprocally in a duplicate ratio of the ditance, it will be found that the attraction in I is to the attraction in P as the diŧance SP to the emi-diameter SA of the phere. If thoe forces are reciprocally in a triplicate ratio of the ditances, the attractions in I and P will be to each other as $$\scriptstyle SP^2$$ to $$\scriptstyle SA^2$$; if in a quadruplicate ratio as $$\scriptstyle SP^3$$ to $$\scriptstyle SA^3$$. Therefore ince the attraction in P was found in this lat cae to be reciprocally as $$\scriptstyle PS^2 \times PI$$, the attraction in I will be reciprocally as $$\scriptstyle SA^3 \times PI$$, that is, becaue $$\scriptstyle SA^3$$ is given, reciprocally as $$\scriptstyle SA^3 \times PI$$. And the progreion is the ame in infinitum. The demontration of this theorem is as follows.

The things remaining as above contructed and a corpucle being in any place P, the ordinate DN was found to be as $$\scriptstyle {DE^2 \times PS}{PE \times V}$$. Therefore if IE be drawn, that ordinate For any other place of the corpucle as I, will become (mutatis mutandis) as $$\textstyle \frac {DE^2 \times IS}{IE \times V}$$. Suppoe the centripetal forces flowing from any point of the phere as E, to be to each other at the diŧances IE and PE, as $$\scriptstyle PE^n$$ to $$\scriptstyle IE^n$$, (where the number in n denotes the index of the powers of PE and IE) and thoe ordinates will become as $$\textstyle \frac {DE^2 \times PS}{PE \times PE^n}$$ and $$\textstyle \frac {DE^2 \times IS}{IE \times IE^n}$$ whoe ratio