﻿Coupled 
  Vibrations 
  : 
  Unequal 
  Masses 
  or 
  Periods. 
  65 
  

  

  where 
  i 
  denotes 
  \/( 
  — 
  1), 
  and 
  r 
  and 
  s 
  (being 
  comparable 
  to 
  k) 
  

   are 
  to 
  be 
  treated 
  as 
  small 
  quantities 
  whose 
  squares 
  or 
  products 
  

   are 
  negligible 
  in 
  comparison 
  with 
  p 
  and 
  q 
  which 
  depend 
  

   upon 
  the 
  larger 
  constants 
  o£ 
  the 
  equation. 
  

  

  Thus, 
  with 
  the 
  roots 
  from 
  (11) 
  we 
  may 
  write 
  instead 
  of 
  

   (10) 
  the 
  equivalent 
  equation 
  

  

  (x 
  + 
  r 
  — 
  ip) 
  (x 
  -f 
  r 
  + 
  ip) 
  (x 
  + 
  s—iq)(x 
  + 
  s 
  + 
  iq) 
  = 
  0, 
  

   or 
  x 
  4 
  + 
  2 
  (r 
  + 
  s) 
  x 
  z 
  + 
  (p 
  2 
  + 
  q 
  2 
  + 
  r 
  2 
  + 
  s 
  2 
  + 
  4rs> 
  2 
  

  

  + 
  2(p 
  2 
  s 
  + 
  q 
  2 
  r 
  + 
  r 
  2 
  s 
  + 
  rs 
  2 
  )x+(p 
  2 
  + 
  r 
  2 
  )(q 
  2 
  + 
  s 
  2 
  ) 
  = 
  0. 
  (12) 
  

  

  This, 
  on 
  omitting 
  the 
  negligible 
  quantities, 
  becomes 
  the 
  

   approximate 
  equation 
  sufficiently 
  accurate 
  for 
  our 
  purpose, 
  

   x* 
  + 
  2(r 
  + 
  s)x 
  d 
  + 
  (p 
  2 
  + 
  q 
  2 
  )x 
  2 
  + 
  2(p 
  2 
  s 
  + 
  q 
  2 
  r)x+p 
  2 
  q 
  2 
  = 
  0. 
  (13) 
  

   The 
  comparison 
  of 
  coefficients 
  in 
  (10) 
  and 
  (13) 
  yields 
  

  

  r+s 
  = 
  k 
  } 
  (14) 
  

  

  p* 
  + 
  q* 
  = 
  c 
  + 
  a, 
  (15) 
  

  

  p 
  2 
  s 
  + 
  q 
  2 
  r 
  = 
  ck, 
  (16) 
  

  

  pY 
  = 
  ca-pb- 
  (17) 
  

  

  From 
  (15) 
  and 
  (17) 
  we 
  may 
  eliminate 
  q 
  2 
  and 
  obtain 
  a 
  

   quadratic 
  in 
  p 
  2 
  whose 
  roots 
  may 
  be 
  called 
  p 
  2 
  and 
  q 
  2 
  . 
  We 
  

   thus 
  find 
  

  

  2p* 
  = 
  c 
  + 
  a+V{(a-c)* 
  + 
  4pb*},1. 
  

  

  and 
  2gW+«-./{(a-c)*+4p&*}.j" 
  " 
  ' 
  (18) 
  

  

  Again, 
  from 
  (14) 
  and 
  (16) 
  we 
  obtain 
  

   r 
  

  

  A 
  oh 
  and 
  s= 
  2 
  % 
  &. 
  

  

  p 
  £ 
  —q 
  z 
  P 
  9 
  

  

  And 
  by 
  use 
  of 
  (18) 
  these 
  become 
  

   and 
  

  

  

  i_ 
  2v 
  , 
  {(a-c) 
  2 
  + 
  V 
  2 
  } 
  ' 
  ' 
  ' 
  ( 
  j 
  

  

  Then, 
  inserting 
  the 
  values 
  of 
  a, 
  &, 
  and 
  c 
  from 
  (8) 
  in 
  

   (18), 
  (19), 
  and 
  (20), 
  we 
  obtain 
  

  

  ^ 
  =m 
  - 
  q= 
  V^TW) 
  (21) 
  

  

  £=v/(l 
  + 
  /3), 
  (22) 
  

  

  *=rf, 
  < 
  24 
  > 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  35. 
  No. 
  205. 
  Jan. 
  1918. 
  F 
  

  

  