SPHERICAL PENDULUM. 



22, 23] 



Since 



we have, equating the coefficients of #, 



49 



from which 



ft<y 



y = - 



-I 



Since a and ft both lie 

 between I and I the 

 numerator is positive 

 irrespective of the sign 

 of either, and since y is 

 negative, the denominator 

 must be positive, or 

 a + ft > 0. Since between 

 ft and y (&) is negative, 

 z cannot in the motion 

 lie in this region (for ~\/<l>(z) must be real). Now since, 66) 



Fig. 13. 



when <P(#) vanishes # is a maximum or minimum, hence the motion 

 takes place between two horizontal circles at depths Z = K and z = ft 

 below the origin. Although ft may be negative, yet since ft -f- a > 

 the mean position of the particle is below the center of the sphere. 



Since by 50) ^ 



_ 



cp always varies in the same sense 



and when z equals a or ft the path has a horizontal tangent, for 

 - 



- = 0, while -~ is not equal to zero. 



If # is a root, that is if the particle was originally on one of 

 the limiting circles, we must take the positive sign for the radical 



in the integrals if # = ft (so that ^| may be positive and e increase), 

 the negative sign if = a. 



The time of passing from the highest to the lowest point is 



T= r ids m 

 J V*' 



The meridian planes passing through the points of tangency 

 with the parallels a and ft are planes of symmetry for the path. 



WEBSTER, Dynamics. 4 



