Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/268

 there will arie RGC - RFF + TFF to $$\scriptstyle bT^m + cT^n$$ as -FF to $$\scriptstyle - mbT^{m - 1} - ncT^{n - 2} + \frac {mm - m}{2}bXT^{m - 2} + \frac {nn - n}{2}cXT^{n - 2}$$ &c. And taking the lat ratio's that arie when the orbits come to a circular form, there will come forth GG to $$\scriptstyle bT^{m - 1} + cT{n - 1}$$ as FF to $$\scriptstyle bT^{m - 2} + ncT^{n - 1}$$ and again GG to FF as $$\scriptstyle bT^{m - 1} + cT{n - 1}$$ to $$\scriptstyle bT^{m - 2} + ncT^{n - 1}$$. This proportion. by expreing the greatet altitude CV or T arithmetically by unity, becomes, GG to FF as b + c to mb + nc, and therefore as 1 to $$\textstyle \frac {mb + nc}{b + c}$$. Whence G becomes to F, that is the angle VCp to the angle VCP as 1 to $$\textstyle \sqrt \frac {mb + nc}{b + c}$$. And therefore ince the angle VCP between the upper and the lower apis, in an immovable ellipis, is of 180 deg. the angle VCp between the ame apides in an orbit which a body decribes with a centripetal force. that is as $$\textstyle \frac {bA^m + cA^n}{A^2}$$ will be equal to an angle of $$\textstyle 180 \sqrt {\frac {b + c}{mb + nc}}$$ deg. And by the ame reaoning if the centripetal force be as $$\textstyle \frac {bA^m - cA^n}{A^3}$$ the angle between the apides will be found equal to $$\textstyle 180 \sqrt {\frac {b + c}{mb + nc}}$$ deg. After the ame manner the problem is olved in more difficult caes. The quantity to which the centripetal force is proportional. mut always be reolved into a converging