Page:Newton's Principia (1846).djvu/187

the centre either decreases in a less than a triplicate ratio of the altitude, or increases in any ratio of the altitude whatsoever, the body will never descend to the centre, but will at some time arrive at the lower apsis; and, on the contrary, if the body alternately ascending and descending from one apsis to another never comes to the centre, then either the force increases in the recess from the centre, or it decreases in a less than a triplicate ratio of the altitude; and the sooner the body returns from one apsis to another, the farther is the ratio of the forces from the triplicate ratio. As if the body should return to and from the upper apsis by an alternate descent and ascent in 8 revolutions, or in 4, or 2, or 1½; that is, if m should be to n as 8, or 4, or 2, or 1½ to 1, and therefore $\scriptstyle {\frac {nn}{mm}}-3$, be $\scriptstyle {\frac {1}{64}}$ - 3, or $\scriptstyle {\frac {1}{16}}$ - 3, or $\scriptstyle {\frac {1}{4}}$ - 3, or $\scriptstyle {\frac {4}{9}}$ - 3; then the force will be as $\scriptstyle A^{{\frac {1}{64}}-3}$; or $\scriptstyle A^{{\frac {1}{16}}-3}$; or $\scriptstyle A^{{\frac {1}{4}}-3}$; or $\scriptstyle A^{{\frac {4}{9}}-3}$; that is, it will be reciprocally as $\scriptstyle A^{3-{\frac {1}{64}}}$, or $\scriptstyle A^{3-{\frac {1}{16}}}$, or $\scriptstyle A^{3-{\frac {1}{4}}}$, or $\scriptstyle A^{3-{\frac {4}{9}}}$. If the body after each revolution returns to the same apsis, and the apsis remains unmoved, then m will be to n as 1 to 1, and therefore $\scriptstyle A^{{\frac {nn}{mm}}-3}$ will be equal to A^-2, or $\scriptstyle {\frac {1}{AA}}$; and therefore the decrease of the forces will be in a duplicate ratio of the altitude; as was demonstrated above. If the body in three fourth parts, or two thirds, or one third, or one fourth part of an entire revolution, return to the same apsis; m will be to n as ¾ or ⅔ or ⅓ or ¼ to 1, and therefore $\scriptstyle A^{{\frac {nn}{mm}}-3}$ is equal to $\scriptstyle A^{{\frac {16}{9}}-3}$, or $\scriptstyle A^{{\frac {9}{4}}-3}$, or $\scriptstyle A^{9-3}$, or $\scriptstyle A^{16-3}$; and therefore the force is either reciprocally as $\scriptstyle A^{\frac {11}{9}}$ or $\scriptstyle A^{\frac {3}{4}}$, or directly as A⁶ or A¹³. Lastly if the body in its progress from the upper apsis to the same upper apsis again, goes over one entire revolution and three deg. more, and therefore that apsis in each revolution of the body moves three deg. in consequentia; then m will be to n as 363 deg. to 360 deg. or as 121 to 120, and therefore $\scriptstyle A^{{\frac {nn}{mm}}-3}$ will be equal to $\scriptstyle A^{-{\frac {29523}{14641}}}$, and therefore the centripetal force will be reciprocally as $\scriptstyle A^{\frac {29523}{14641}}$, or reciprocally as $\scriptstyle A^{2{\frac {4}{243}}}$ very nearly. Therefore the centripetal force decreases in a ratio something greater than the duplicate; but approaching 59¾ times nearer to the duplicate than the triplicate.

Cor. 2. Hence also if a body, urged by a centripetal force which is reciprocally as the square of the altitude, revolves in an ellipsis whose focus is in the centre of the forces; and a new and foreign force should be added to or subducted from this centripetal force, the motion of the apsides arising from that foreign force may (by the third Example) be known; and so on the contrary. As if the force with which the body revolves in the ellipsis