34 o Sir f S A A C NEWT 0 N’s Book IV.

scribe the circle e«z, and from the centre a the circle e m f.Let this circle e m f revolve on the other e e z, as its- base rthen a point l, taken on the plane os the circle e m f, at thedistance a l, equal to the distance of the satellite from theprimary, shall describe the path of the satellite.

For suppose the circle e m f to move into the situation e m f.the point A to s, l to /, arid let a l and a /, produced, meete m f and e inf \ in m and m. Upon the arc em take e r= e m,then the angle ear — e am. Let a r meet the circle c I d>described from the centre a with the distance a /, in q ; and'because eaq — e a l, the angle e a q represents the elongation-of the satellite from the sun at its first place l. Because e m(= e r +. r m) = e e + e m and er — em, it follows that r m — e e ;consequently the angle ram : esE : : se : a E : : t — / : /or, as the angular velocity of the satellite from the sun, to theangular velocity os the primary about the sun. But e s e isthe angle described by the primary about the sun, consequentlyr a m> or q a /, is the simultaneous increment of the elonga-tion of the satellite from the sun ; / is its place when theprimary comes to a \ and the epicycloid described by I is thepath of the satellite.

I

Because the circle emf moves on the point e, the directionof the motion of any point l is perpendicular to e l ; or thetangent of the path at any point l is perpendicular to e l. Tfievelocity of any point l is as its distance e l ; and, the mo-tion of the primary a being supposed uniform and representedby e a, the velocity of the satellite shall be represented by el.

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