54] EXAMPLES OF CYCLIC MOTIONS. 195 



216) d v 'A = d 9 'P r dr = 

 or by 213), 



217) d (p 'A = mr*d<p'*, 



which is positive, illustrating Theorem II, 53. 



A further example is found in the motion of the following 

 system. Two particles of equal masses m are fastened to a rod of 

 length 2 a pivoted at its central point upon an axis fastened to the 

 horizontal rod of the previous example at a distance & from the axis 

 of rotation in such a way that the two masses can move in the 

 vertical plane containing the axis of rotation. The inclination of the 

 pivoted rod to the vertical being ft, the distances of the particles 

 from the axis of rotation are respectively 



r^ = b -f a sin #, r 2 = 1} a sin #. 



The system is fully specified by the coordinates # and qp, the latter 

 having the same meaning as before. 



It is evident that the kinetic energy is given by 



218) T = KV 2 + W + 2 2 #' 2 } 



= m 



' 2 



so that cp is again the cyclic coordinate. 1 ) 

 To find the change of # we have 



giving us the differential equation, 



- 2ma 2 y' 2 sin # cos # = 0, 



If the motion is isocyclic <p is constant, and since the angular 

 acceleration -j-y vanishes when & equals zero or > we see that the 

 rod carrying the particles will remain at rest relatively to the hori- 

 zontal rod in either a vertical or horizontal position. It is easy to 

 see that the vertical position is one of unstable equilibrium, for, 

 writing the equation 209) 



220 ) 



we see that if # be slightly different from zero, # will tend to become 

 still greater in absolute value. Writing however # = -y &' the 

 equation becomes 



221) 



1) The system is cyclic if we neglect 



13* 



