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

Cor. 1. Hence if the forces of the points decreaſe in the duplicate ratio of the diſtances, 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 corpuſcle P towards the circle will be as $\scriptstyle 1-{\frac {PA}{PH}}$; that is, as $\scriptstyle {\frac {AH}{PH}}$.

Cor. 2. And univerſally if the forces of the points at the diſtances D be reciprocally as any power $\scriptstyle D^{n}$, of the diſtances; 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 corpuſcle P towards the circle will be as $\scriptstyle {\frac {1}{PA^{n-2}}}-{\frac {PA}{PH^{n-2}}}$.

cor. 3. And if the diameter of the circle be increaſed in infinitum, and the number n be greater than unity; the attraction of the corpuſcle P towards the whole infinite plane will be reciprocally as $\scriptstyle PA^{n-2}$ becauſe the other term $\scriptstyle {\frac {PA}{PH^{n-2}}}$ vaniſhes.