Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/391

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Hence if the forces of the points decreae in the duplicate ratio of the ditances, that is, if FK be as $$\scriptstyle \frac {1}{PF^2}$$, and therefore the area AHIKL as $$\scriptstyle \frac {1}{PA} - \frac {1}{PH}$$; the attraction of the corpucle P towards the circle will be as $$\scriptstyle 1- \frac {PA}{PH}$$; that is, as $$\scriptstyle \frac {AH}{PH}$$.

And univerally if the forces of the points at the ditances D be reciprocally as any power $$\scriptstyle D^n$$, of the ditances; that is, if FK be as $$\scriptstyle \frac 1{D^n}$$, and therefore the area AHIKL as $$\scriptstyle \frac {1}{PA^{n - 2}} - \frac {1}{PH^{n-2}}$$; the attraction of the corpucle P towards the circle will be as $$\scriptstyle \frac {1}{PA^{n - 2}} - \frac {PA}{PH^{n-2}}$$.

. And if the diameter of the circle be increaed in infinitum, and the number n be greater than unity; the attraction of the corpucle P towards the whole infinite plane will be reciprocally as $$\scriptstyle PA^{n - 2}$$ becaue the other term $$\scriptstyle \frac {PA}{PH^{n - 2}}$$ vanihes.