Variably Coupled Vibrations. 297 



iv 



j = pK cos (j>t + e)+qF cos (^ + <£), (11) 



dt m' 2 ot J *~ " v/ " 7 7?i 2 a 



rj~ lit lit Of 



- -2 — j»E cos (/>£ + e) + j-4- <?F cos (f « + $ ) . 



... (12) 



It will be seen tlmt these are the equations of the Cord and 

 Lath Pendulums *, but the values of the various quantities 

 are different, as will be seen below ; but this difference might 

 be expected, since the elastic pendulum oscillates entirely on 

 one side of the vertical. 



From the comparison of (6) and (7), and putting M = N 

 and m=7L, we see that 



p 2 + q 2 = 2m 2 + mV 

 p 2 g 2 = m\ 



' J . . . . (13) 



Eliminate q from equations (13), thus obtaining the 

 quadratic in p 2 , 



]> A - {2m 2 + m 2 cc 2 ) p 2 -f m 4 = 0. . . (14) 



Thus, calling the larger root pr and the smaller q 2 , we 

 have 



_ ,»'t(2 + ,') + [(2 + «')'-4]'} 



V — .) , . . . [lO) 



whence 



ga _ , n '{(2 + q»)-[(2 + a ')'-4]*K _ _ _ (16) 



P = i(2 + *')+[(2 + c Sf-i] i l 17) 



9 {(2+^-[(2 + «^-4]*}** ■'■.'." 



Initial Conditions. 

 (i.) Lower bob struck. 

 AVe ma) r here write 



y = 0, z = 0, || = «, J = for t = 0. . (18) 

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

 , = 0, </> = 0, E=^I^, F = (^-^ . (19) 



(p 2 -g 2 )p ip 2 -q 2 )q 



* Phil. Mag. vol. xxxiv. no. 202, p. 260. 



