. Carnegie Institution of Washington publication. SPECIAL CASE. 91 It is seen from this equation that when the rotation and revolution are both in the same direction P and D either both increase or both decrease, if they change at all. The necessary condition for a maximum or a minimum of E is, from (26), 1/-»3 27t dP P*-MP + Wi = 0 (28) The corresponding condition from (27) is 2;rWi dD The only D having a physical meaning is real. Since no real negative D satisfies this equation, it follows that when D is negative E has no finite maximum or minimum. In this case by fig. 10, dotted curve, the
. Carnegie Institution of Washington publication. SPECIAL CASE. 91 It is seen from this equation that when the rotation and revolution are both in the same direction P and D either both increase or both decrease, if they change at all. The necessary condition for a maximum or a minimum of E is, from (26), 1/-»3 27t dP P*-MP + Wi = 0 (28) The corresponding condition from (27) is 2;rWi dD The only D having a physical meaning is real. Since no real negative D satisfies this equation, it follows that when D is negative E has no finite maximum or minimum. In this case by fig. 10, dotted curve, the period of revolution must always decrease and the two bodies ultimately fall together. Taking the last four terms to the right and extracting the cube root, we have, D*^=MD-mi Since the roots of this equation are the same as those of (28), it follows that when E is a maximum or minimum D = Pandthe system moves as a rigid body. The real roots of (28) are the abscissas of the intersections of the curves y = Pi y = MP — mi (29) It is evident from fig. 11 that there are two, or no, intersections of these curves. Y. Fio. 11. For a given m, the value of M may always be taken so great that there will be two real roots; or, so small that there will be no real roots. The limiting value of M, as it decreases, for which the real roots exist is that value for which they are equal. The condition that (28) shall have equal roots is 4p^-M = 0 (30). Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original Carnegie Institution of Washington. Washington, Carnegie Institution of Washington
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