166 DR. J. BOTTOMLEY ON THE CHANGING 



and also when 



inir 



t — j 



q-p 



n being any positive integer including o . 



If among the physical constants we have the following 



relation 



X' X \m _ 

 L / lm I ~~ ' 



the equation of motion of the rod takes the form 



V /s'mpt smqt\ 

 x — — 1 — — ■ H ), 



2 \ p q J 



and the equation of motion of the ring takes the form 



V (sin pt sin^A 



Jb j 



The equation of motion of the rod before attaching the 

 ring would be 



V 



x — 



J- 



/2\' 



V Lm 



and if the ring were rigidly attached to the rod, the 

 equation of motion would be 



mV 



x = 



. \ / , . Sin \ / . ty 



V 2\' V li{m + mA 



Vm + m l v 2X v Hm + mJ 



the impulsive force setting the system in motion being 

 supposed to be the same in each case. 



Instead of a single ring situated at the middle of the 

 rod, we might have a number of rings symmetrically 

 disposed about the middle. Let x xi x Zi &c. be the 

 distances of the rings from their position of rest, and 



