Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/519

      [ 55 ] III. THE third part of the inequali- ty, anwering to the trilinear pace OKQ, being the difference of the elliptic ector OFQ and the triangle OFK. THE ector OQF is proportional to an angle, which is the difference of two angles, whoe tangents are in the gi- ven proportion of the emi-latus rectum FB and the emi-tranvere FN, or in the duplicate proportion of the leer axis to the axis of the orbit. So that this ector, when at a maximum, is as an angle, whoe ine is to the radius, as the difference of the latus rectum and tranvere to their um ; or as the diffe- rence of the quares of the emi-axes to their um. THE triangle OFK is proportional to the rectangle of the co-ordinates OH and HF ; that is, as the rectan- gle of the line OH and its coine, in the circle on the radius FN; or as the ine of the double of that angle, whoe ine is OH ; that is, the double of the angle, whoe tangent is to the tangent of the angle QFL, in the given ratio of the greater to the leer axis ; or whoe tangent is the tangent of the angle of mean motion anwering to the elliptic ector QFL, in the duplicate of the aid                                           ratio