Chap. 5- PHILOSOPHICAL DISCOVERIES. 339

the sun ; and it has likewise two points os contrary flexure inevery such part *.

By considering this path, we shall arrive at the same con-clusions which Sir Isaac Newton derived, more briefly, fromthe laws of motion ; that if the solar action was the same onthe satellite and on the primary, and in the same direction,the motion of the satellite around the primary, would be thesame as is the fun was away. This will appear from the fol-lowing propositions, where we suppose the orbits of the primaryabout the fun, and of the satellite about the primary, to beboth circular, and the motions in these orbits to be uniformand in the same plane.

. PROPOSITION I. Fig. jo. PI. VI.

The path of the satellite , on an immoveable plane , is the epi-cycloid that is described by a given point in the plane of a circle ,which revolves on a circular base , having its centre in the centreof the fun , and its diameter in the fame proportion to the dia-meter of the revolving circle , as the periodic time of the pri-mary about the fun, to the-time of the fynodic revolution of thesatellite about the prh?iary : the tangent of the path is perpe?i-dicular to the right line that joins the satellite to the co?itacl ofthe two circles : and the absolute velocity of the satellite is alwaysas its diflance from that contaSl.

Let t denote the periodic time of the primary about thefun, t the periodic time of the satellite about the primary. Lets represent the sun, a a the orbit of the primary ; upon theradius a s, take a e to a s as / is to t. From the centre s de-

* Sce the note to Corol, i. Prop. II. below.

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