. Applied calculus; principles and applications . s. III. The squares of the periodic times of the planets areas the cubes of the major axes of their orbits. The statement of these laws marked an epoch in thedevelopment of mechanics, for the investigations of Newtonas to the nature of the attractive force led to his discovery ofthe law of universal gravitation. The conclusions deducedby Newton from Keplers three laws will be briefly shown. 244. Nature of the Force which Acts upon the Planets. —(1) From the second of Keplers Laws, it follows that the planets are retained in their orbitsby an at
. Applied calculus; principles and applications . s. III. The squares of the periodic times of the planets areas the cubes of the major axes of their orbits. The statement of these laws marked an epoch in thedevelopment of mechanics, for the investigations of Newtonas to the nature of the attractive force led to his discovery ofthe law of universal gravitation. The conclusions deducedby Newton from Keplers three laws will be briefly shown. 244. Nature of the Force which Acts upon the Planets. —(1) From the second of Keplers Laws, it follows that the planets are retained in their orbitsby an attraction tending towardsthe Sun. Let (x, y) be the position of aplanet at the time t, referred torectangular axes through the Sunin the plane of the motion ofthe planet; X, 7, the componentaccelerations due to the attraction acting on it, resolvedparallel to the axes; then the equations of motion are, ^ = Y ^ = y- By Keplers second law, if A be the area described by theradius vector, dA /dt is constant, •■• i - \ft=\ (from (5), Art. 242). =^ o\^-^ — y-n) = ^ constant, fromZ\ dt at/ (3)^ Art. 242. FORCE WHICH ACTS UPON THE PLANETS 479 Differentiating gives dx ■0; . xY -yX = 0 (from (1)) ) • X Y X ■ -, or y y = X which shows that the axial components of the acceleration,due to the attraction acting on the planet, are proportionalto the coordinates of the planet; and therefore by theparallelogram of forces, the resultant of X and Y passesthrough the origin. Hence, the forces acting on the planetsall pass through the Suns center. (2) From the first of Keplers laws it follows that thecentral attraction varies inversely as the square of thedistance. The polar equation of the ellipse, referred to its focus, is a(l-e^) 1 l+ecosB P = r~i n or - = u= —jz ^> 1 + e cos 0 p a (1 — ^2) which by differentiation gives, fu ^ 1 d^2 + ^ a (1 - e) and, therefore, if ¥ is the attraction to the focus, by (8),Art. 242, a(l -62) p2 Hence, ij the orbit be an ellipse, described a
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