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

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  III. THE third part of the inequali- 

ty, anſwering to the trilinear ſpace OKQ, being the difference of the elliptic ſector OFQ and the triangle OFK.

  THE ſector OQF is proportional to

an angle, which is the difference of two angles, whoſe tangents are in the gi- ven proportion of the ſemi-latus rectum FB and the ſemi-tranſverſe FN, or in the duplicate proportion of the leſſer axis to the axis of the orbit. So that this ſector, when at a maximum, is as an angle, whoſe ſine is to the radius, as the difference of the latus rectum and tranſverſe to their ſum ; or as the diffe- rence of the ſquares of the ſemi-axes to their ſum.

  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 coſine, in the circle on the radius FN; or as the ſine of the double of that angle, whoſe ſine is OH ; that is, the double of the angle, whoſe tangent is to the tangent of the angle QFL, in the given ratio of the greater to the leſſer axis ; or whoſe tangent is the tangent of the angle of mean motion anſwering to the elliptic ſector QFL, in the duplicate of the ſaid ratio