260 Prof. Barton and Miss Browning on Coupled 



Hence, on inserting the four arbitrary constants, we may- 

 write the general solution and its first derivatives as 

 follows : — 



y = EsinO + e)+Fsm(g* + (£), (m) 



^=jpEcos(p< + e) + 0Fcos(gtf + 0), (68) 



-^^^Ecos(^ + 6)+ ? ^^ 2 gFcos(^ + ^). (69) 



Returning to the comparison of (63) and (64) we see that 



. (70) 



p 2 + q 2 =(2 + u + a, 2 )m 2 = m 2 8say, \ 

 and p 2 q 2 = (l + «) m* = m 4 r) 2 say. i 



whence (p + v) 2 = m 2 (S + 2^), \ 



(p-qf = m\S-2 V ), i * 



^=f{v / (^ + ^)- v /(5-2^)}, J 



and 



^_ v /(S + 27,) + v /(g- 2 ,) 



(71) 

 (72) 



(73) 



An alternative method is to eliminate q 2 between the 

 equations (70), thus obtaining the quadratic in p 2 , 



p 4 -(2 + a + a 2 )m 2 p 2 +(l + a)m 4 = 0. . . (74) 

 calling the larger root of this p 2 and the smaller q 2 y 



^ = m^l+|+^) + ^-V(« 2 + 2« + 5), | 

 gi=m^l+£+ ? J rv /(«H2« + 5), J 



JP _ f 2+« + a 2 + a(a 2 + 2a + 5y V 



^-\2 + a + a 2 -a(« 2 + 2« + 5)U ' * * ^ } 



which agrees with (73). 



Thus by (72) or (75) we see that the superposed vibrations 



whence 



