Variably Coupled Vibrations. 299 



These conditions inserted in (9) to (12) give equations 

 satisfied by 



- *" V _ P*~£ + (> 2 -g 2 )a 2 -(2a 3 + *)m 2 1 



(i+« 2 )(p 2 -^) V"' L (28) 



tt (2a» + l)(m»«) 



* r ~ (l + a^p'-qr 



These values put in (9) and (10) give the special 

 solution : 



p 2 -q 2 + (p 2 -q 2 )cc 2 -(2cc^u)m 2 



, (2a 2 +l)(7n 2 a) # , 0Q . 



+ (>^^«^^ ' • ( 29 > 



-(p 2 -m 2 ){p 2 -q 2 +(p 2 -q 2 )a 2 -(2c<? + cc)m 2 \ 

 Z - (1 + a 2 ) O 2 - ? 2 ) (m 2 a) ~ a C ° S pt 



+ (l + ^V-rt g0 ° 8gt (30) 



So the ratio of the amplitudes of the quick and slow 

 vibrations of the y and z motions are given by 



E _ p 2 -q 2 +(p 2 -q 2 )a 2 -(2oc* + u)m * 



F~ (2a 2 +l)(m 2 a) ' ^ 6L) 



(32) 



q_-(j P »-m s ){jP > -g > +(p'--g 8 )« , -(2« , + a)m 2 } 

 H""~ (2a* + l)(m s a) 



(iv.) Upper bob displaced ; lower bob free. 

 This case may be represented by 



y= ah, z = b, d £ = 0, ^ = for * = 0. . (33) 



These put in (9) to (12) give equations satisfied by 



T , t r, —a.q 2 b v ocp 2 b 



.- 5 , *=j, E=^- ? , F = ? ^ ? . . (34) 



