Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/267

 therefore the angle between the upper and lower apis will be of deg. or 127 deg. 16 min. 45 ec. and therefore a body revolving with uch a force, will, by a perpetual repetition of this angle, move alternately from the upper to the lower, and from the lower to the upper apis for ever. So alo if the centripetal force be reciprocally as the biquadrate root of the eleventh power of the altitude, that is reciprocally as $$\textstyle A \frac {11}4$$ and therefore directly as $$\textstyle \frac 1{A^{\frac {11}4}}$$ or as $$\textstyle \frac {A ^{\frac 34}}{A^3}$$, as will be equal to $$\textstyle \frac 14$$ and $$\textstyle {180}{\sqrt n}$$ deg. will be equal to 360 deg. and therefore the body parting from the upper apis, and from thence perpetually decending will arrive at the lower apis when it has compleated one entire revolution; and thence acending perpetually, when it has compleated another entire revolution it will arrive again as the upper apis; and o alternately for ever.

. Taking m and n for any indices of the powers of the altitude, and b and c for any given numbers, uppoe the centripetal force to be as $$\textstyle \frac {b \ into \ \overline {T - X} \vert ^m c \ into \ \overline {T - X} \vert ^n}{A^3}$$ or (by the method of converging eries above-mentioned) as $$\textstyle \frac {bT^m + cT^n - mbXT^{m - 1}ncXT^{n - 2}}{A^3} + \frac {mm - m}{2}bXXT^{m - 2} + \frac {nn - n}{2}cXXT^{n - 2}$$ &c. and comparing the terms of the numerators,