Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/272

 force ubducted as cA, and therefore the remaining force as $$\textstyle \frac {}{}$$; then (by the third exam.) 6 will be equal to 1, m equal to 1, and n equal to 4; and therefore the angle of revolution between the apides is equal to $$\textstyle 180 \sqrt {\frac {1-c}{1-4c}}$$ deg. Suppoe that foreign force to be 357.45 parts les than the other force with which the body revolves in the ellipis; that is c to be $$\textstyle \frac {100}{35745}A$$ or T being equal to 1, and then $$\textstyle 180 \sqrt {\frac {1-c}{1-4c}}$$ will be 1$$\textstyle 180 \sqrt {\frac {35645}{35345}}$$ or 180.7623, that is, 180 deg. 45 min. 44 ec. Therefore the body parting from the upper apis, will arrive at the lower apis with an angular motion of 180 deg. 45 min. 44 ec. and this angular motion being repeated will return to the upper apis; and therefore the upper apis in each revolution will go forward 1 deg. 31 m. 28 ec. The apis of the Moon is about twice as wift.

So much for the motion of bodies in orbits whoe planes pas through the centre of force. It now remains to determine thoe motions in eccentrical planes. For thoe authors who treat of the motion of heavy bodies ue to conider the acent and decent of uch bodies, not only in a perpendicular direction, but at all degrees obliquity upon any given planes; and for the ame reaon we are to conider in this place the motions of bodies tending to centres by means of any forces whatoever. when thoe bodies move in eccentrical planes. Thee planes are uppoed to be perfectly mooth and polihed o as not to retard the motion of the bodies in the leat. Moreover in the demontrations