168 ON THE CHANGING MOTION OF AN OSCILLATING ROD. 



In a similar manner it may be shown that the roots of 

 this equation are all imaginary, for 



( 



L / ImJ l z m 1 



is essentially positive. Hence the solution of (29) will 

 take the form 



x = P cos pt + Q cos qt + R sin pt + S sin qt } 



where 



and 



-vW£-* 



If we suppose the rod to start from a position of rest 

 with velocity V, P and Q, will be each o, and the equation 

 will take the form 



<27 = Hsin/?^ + S sin<^, 



with the condition 



Y = np + Sq. 



The other equation necessary to determine R and S is 

 quite arbitrary, and depends upon what hypothesis we 

 make regarding the initial velocities of the rings. The 

 subsidence of motion due to internal friction in elastic 

 solids has not been taken into account. 



