298 Mr. L. C. Jackson on 



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

 solution 



(m? — q 2 )u . . . (p 2 — m 2 )u . \ 



y— i « -A sin»H-V-= wv~ sin at, 



(p—q 2 )p r (p 2 -Q 2 )q I 0) 



— (p 2 — m 2 )(m 2 — q 2 )u . (m 2 — q 2 )(p 2 — m 2 )u . 



z — — - 2 , / - 2 , - sin pt + 2/2 2^ - sm Q t '> 



m 2 u(p~ — q' i )p nroi(p' i — q i )q * 



so 



■-f=f=j2* and ° =i, . . . (21) 



where G and H are the quick and slow s vibrations. 



(ii.) Upper bob struck. 

 Here we may write 



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

 satisfied by 



e=0, <j> = 0, t=^^ y F.-^p^. . (23) 



So inserting these values in (9) and (10) we have the 

 special solution 



— m 2 uv . m 2 a.v . /,,, N 



- m >«npf+ ( ^-fi v «n0i . . (25) 



so 



(f-q 2 )p K {p 2 -q 2 )q 



E —Q j G ( p 2 — m 2 ) q 



F p H (m 2 -q J )p 



Note the contrast between (21) and (26). 



(26) 



(iii.) Lower bob displaced ; upper free. 



Let the displacement a of the lower bob M be produced 

 by a horizontal force. Then the corresponding displace- 

 ment z of N when at rest can be found statically. We thus 

 obtain 



»— ' >=W t = °> gpOfar.-O.;. (27) 



