164 DR. J. BOTTOMLEY ON THE CHANGING 



whence, by (23), 



0=V(q z -p z ) (26) 



Since q*—p z cannot vanish, the only solution of this equa- 

 tion is P=o. Therefore also 



Q=o, 



and the equation becomes 



x = B, sinpt + S sin qt. 



To determine the two constants R and S we have only 

 one equation (24) . To obtain a second equation we must 

 know the initial velocity of the ring. This is quite arbi- 

 trary, for evidently, when the rod is set in motion, we 

 may simultaneously impress any velocity we please upon 

 the ring. Suppose that the rod is set in motion by a blow 

 and that the ring is initially at rest. Differentiating the 

 last equation three times we get 



-rpj = — Rjo 3 cos pt — S<7 ? cos qt. 



d^X 



Differentiate (18), substitute for x and -yy from the two 



til 



dx 

 last equations, make -~^ = o } and put initial values for the 



other quantities ; then we get this equation, 



From this equation, combined with (24), we derive the 

 following values : — 



V 



mp (q z —p z ) 



mq{q z —p z ) 



s = -CT){"^- 2 (l + 7)}- 



