Page:Newton's Principia (1846).djvu/184

 ellipsis, in descending from the upper to the lower apsis, describes an angle, if I may so speak, of 180 deg., the other body in a movable ellipsis, and therefore in the immovable orbit we are treating of, will in its descent from the upper to the lower apsis, describe an angle VCp of $$\scriptstyle \frac{180}{\sqrt{3}}$$ deg. And this comes to pass by reason of the likeness of this orbit which a body acted upon by an uniform centripetal force describes, and of that orbit which a body performing its circuits in a revolving ellipsis will describe in a quiescent plane. By this collation of the terms, these orbits are made similar; not universally, indeed, but then only when they approach very near to a circular figure. A body, therefore revolving with an uniform centripetalforce in an orbit nearly circular, will always describe an angle of $$\scriptstyle \frac{180}{\sqrt{3}}$$ deg., or 103 deg., 55 m., 23 sec., at the centre; moving from the upper apsis to the lower apsis when it has once described that angle, and thence returning to the upper apsis when it has described that angle again; and so on in infinitum.

. 2. Suppose the centripetal force to be as any power of the altitude A, as, for example, An-3 3, or $$\scriptstyle \frac{A^{n}}{A^{3}}$$; where n - 3 and n signify any indices of powers whatever, whether integers or fractions, rational or surd, affirmative or negative. That numerator An or $$\scriptstyle \overline{T-X}|^{n}$$ being reduced to an indeterminate series by my method of converging series, will become $$\scriptstyle T^{n}-nXT^{n-1}+\frac{nn-n}{2}XXT^{n-2}$$, &c. And conferring these terms with the terms of the other numerator RGG - RFF + TFF - FFX, it becomes as RGG - RFF + TFF to Tn, so - FF to $$\scriptstyle -nT^{n-1}+\frac{nn-n}{2}XT^{n-2}$$, &c. And taking the last ratios where the orbits approach to circles, it becomes as RGG to Tn, so - FF to -nTn-1, or as GG to Tn-1, so FF to nTn-1; and again, GG to FF, so Tn-1 to nTn-1, that is, as 1 to n; and therefore G is to F, that is the angle VCp to the angle VCP, as 1 to $$\scriptstyle\sqrt{n}$$. Therefore since the angle VCP, described in the descent of the body from the upper apsis to the lower apsis in an ellipsis, is of 180 deg., the angle VCp, described in the descent of the body from the upper apsis to the lower apsis in an orbit nearly circular which a body describes with a centripetal force proportional to the power An-3, will be equal to an angle of $$\scriptstyle \frac{180}{\sqrt{n}}$$ deg., and this angle being repeated, the body will return from the lower to the upper apsis, and so on in infinitum. As if the centripetal force be as the distance of the body from the centre, that is, as A, or $$\scriptstyle \frac{A^{4}}{A^{3}}$$, n will be equal to 4, and $$\scriptstyle\sqrt{n}$$ equal to 2; and therefore the angle