MOTION OF AN OSCILLATING ROD. 167 



suppose all the rings to have the same mass; then the 

 equations of motion are 



d z x V X, N X. N f . 



m -rji = —2yX—2j{X—X l )—2j(X—X z ) — &C., \2j) 



d x n of ^ 



m i 1.2. — 2 T \ x X n)' 



Bv addition we obtain 



d z x 



m w 



l Co Ou i . tt Ou f_ Cb X„ \ /Y / rt\ 



Differentiating (33) twice, we obtain 

 d*x 



m 



dt 4 



d z x/X ! nX\ X/d z x 1 d z x 2 d z x n \ 



~ 2 W\L + T/ + 2 T\dt z ~ + ~dF ~cW/ ; 



substituting from (34), we obtain an equation of the form 



Ct X . U X -^ N 



_ + A ^ + B*, (29) 



where B is written for 



V X 



L lmm x 

 and A for 



(X 1 nX Xm\ 

 L I ImJ' 



If we proceed to solve this equation in the same manner 

 as equation (22), we obtain as the auxiliary equation, 



