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 = 



(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 = (30) 



