﻿Coupled 
  Vibrations 
  : 
  Unequal 
  Masses 
  or 
  Periods. 
  71 
  

  

  Solution 
  and 
  Frequencies. 
  — 
  In 
  equation 
  (61) 
  try 
  

  

  z 
  = 
  e 
  xt 
  ~\ 
  

  

  then 
  we 
  have 
  / 
  (C 
  , 
  .. 
  

  

  _ 
  x\(Z 
  + 
  /3 
  V 
  + 
  2rj) 
  + 
  (£ 
  + 
  2 
  V 
  ) 
  m 
  \ 
  % 
  \ 
  ■ 
  • 
  C 
  64 
  ) 
  

  

  y 
  fa? 
  J 
  

  

  And, 
  by 
  (64) 
  in 
  (60), 
  we 
  obtain 
  

  

  \x 
  2 
  (/3 
  + 
  f3 
  v 
  + 
  2 
  v 
  ) 
  + 
  {/3 
  + 
  2 
  v 
  )m 
  2 
  } 
  {x 
  2 
  (0 
  +fa 
  + 
  2n) 
  

   + 
  {2 
  + 
  /3)m 
  2 
  \=l3 
  2 
  m'. 
  

  

  This 
  reduces 
  to 
  the 
  auxiliary 
  biquadratic 
  in 
  x, 
  

  

  x\/3 
  + 
  /3v 
  + 
  2r)) 
  + 
  2(l 
  + 
  /3 
  + 
  v 
  )m 
  2 
  x 
  2 
  + 
  2m*=0. 
  . 
  (65) 
  

  

  Solving 
  this 
  as 
  a 
  quadratic 
  in 
  x 
  2 
  , 
  we 
  have 
  

  

  ^_ 
  m 
  ,i^ 
  + 
  v±ya-vy 
  + 
  w. 
  . 
  . 
  (66) 
  

  

  P 
  + 
  /3r} 
  + 
  2?j 
  v 
  y 
  

  

  Or, 
  let 
  us 
  write 
  

  

  x=±pi 
  or 
  +qi 
  (67) 
  

  

  Then, 
  for 
  the 
  sake 
  of 
  brevity 
  putting 
  A 
  2 
  for 
  (1 
  — 
  ^) 
  2 
  + 
  /3 
  2 
  , 
  

   we 
  have 
  

  

  , 
  2 
  _ 
  l 
  + 
  fl 
  + 
  i? 
  + 
  A 
  _, 
  

   /3 
  + 
  ^ 
  + 
  2t7 
  

  

  jtr 
  = 
  

  

  and 
  

  

  j 
  tl 
  + 
  /3 
  + 
  7;-AJ 
  

  

  Thus, 
  using 
  (67) 
  in 
  (64) 
  and 
  introducing 
  the 
  usual 
  con- 
  

   stants, 
  we 
  obtain 
  

  

  s 
  = 
  Esin(p* 
  + 
  e) 
  + 
  JTsiii(g* 
  + 
  0), 
  . 
  . 
  . 
  (69) 
  

   and 
  

  

  y 
  1 
  ~^ 
  +A 
  Esin(p< 
  + 
  e)+ 
  A 
  ~ 
  ( 
  ^~^ 
  Fsin( 
  g 
  t 
  + 
  <^),(70) 
  

  

  p 
  and 
  q 
  being 
  defined 
  by 
  (68). 
  

  

  Initial 
  Conditions. 
  — 
  Consider 
  the 
  case 
  of 
  pulling 
  aside 
  the 
  

   bob 
  Q 
  of 
  the 
  pendulum 
  of 
  length 
  I 
  whose 
  vibrations 
  are 
  

   denoted 
  by 
  2, 
  the 
  other 
  bob 
  hanging 
  at 
  rest 
  in 
  a 
  more 
  or 
  

   less 
  displaced 
  position 
  according 
  to 
  the 
  magnitude 
  of 
  the 
  

   coupling. 
  

  

  