Page:Newton's Principia (1846).djvu/244

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 1 - $\scriptstyle {\frac {PA}{PH}}$; 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 Dⁿ of the distances; that is, if FK be as $\scriptstyle {\frac {1}{D^{n}}}$ and therefore the area AHIKL as $\scriptstyle {\frac {1}{PA^{n-1}}}-{\frac {1}{PH^{n-1}}}$; the attraction of the corpuscle P towards the circle will be as $\scriptstyle {\frac {1}{PA^{n-2}}}-{\frac {1}{PH^{n-1}}}$.

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 PA^n-2, because the other term $\scriptstyle {\frac {PA}{PA^{n-1}}}$ vanishes.

PROPOSITION XCI. PROBLEM XLV.

To find the attraction of a corpuscle situate in the axis of a round solid, to whose several points there tend equal centripetal forces decreasing in any ratio of the distances whatsoever.

Let the corpuscle P, situate in the axis AB of the solid DECG, be attracted towards that solid. Let the solid be cut by any circle as RFS, perpendicular to the axis: and in its semi-diameter FS, in any plane PALKB passing through the axis, let there be taken (by Prop. XC) the length FK proportional to the force with which the corpuscle P is attracted towards that circle. Let the locus of the point K be the curve line LKI, meeting the planes of the outermost circles AL and BI in L and I; and the attraction of the corpuscle P towards the solid will be as the area LABI.   Q.E.I.

Cor. 1. Hence if the solid be a cylinder described by the parallelogram ADEB revolved about the axis AB, and the centripetal forces tending to the several points be reciprocally as the squares of the distances from the points; the attraction of the corpuscle P towards this cylinder will be as AB - PE + PD. For the ordinate FK (by Cor. 1, Prop. XC) will be as 1 - $\scriptstyle {\frac {PF}{PR}}$. The part 1 of this quantity, drawn into the length AB, describes