﻿66 
  

  

  Prof. 
  Barton 
  and 
  Miss 
  Browning 
  on 
  

  

  Thus, 
  using 
  (11) 
  in 
  (9) 
  and 
  introducing 
  the 
  usual 
  constants, 
  

   the 
  general 
  solution 
  may 
  be 
  written 
  in 
  the 
  form 
  

  

  z=e-«(A^ 
  + 
  Be- 
  p 
  *) 
  + 
  e- 
  tt 
  (C<F 
  i 
  + 
  'De-* 
  it 
  ), 
  . 
  (25) 
  

   and, 
  omitting 
  r 
  2 
  and 
  s 
  2 
  , 
  

  

  y= 
  (~ 
  P 
  l 
  + 
  c) 
  e- 
  rt 
  (Ae 
  pit 
  + 
  Be- 
  pit 
  )+ 
  (~^ 
  + 
  c) 
  e 
  - 
  st 
  (Ce 
  qit 
  + 
  De~ 
  qit 
  ) 
  

  

  + 
  

  

  2pri 
  

  

  ■ 
  rt 
  (-Ae 
  pit 
  -hBe- 
  pit 
  ) 
  4- 
  ^^(-C^' 
  + 
  Dr 
  9 
  *). 
  

  

  (26) 
  

  

  Or, 
  by 
  transformation 
  of 
  (25) 
  and 
  (26) 
  and 
  use 
  of 
  (21)- 
  

   (24), 
  we 
  may 
  write 
  the 
  general 
  solution 
  in 
  the 
  form 
  

  

  and 
  

  

  £ 
  = 
  E*" 
  p 
  "sin 
  (mt 
  + 
  €) 
  + 
  Fe~ 
  9t 
  sm( 
  

  

  mt 
  

  

  mt 
  

  

  y=-pWe-**nn 
  (mt+e 
  1 
  ) 
  + 
  FV- 
  s 
  'sin(- 
  

  

  Where 
  (rY 
  - 
  ^™ 
  2 
  +Hi 
  + 
  l3W 
  -] 
  

  

  + 
  

  

  f), 
  (2S) 
  

  

  and 
  

  

  also 
  

  

  and 
  

  

  tan(e' 
  — 
  e) 
  

  

  J3W 
  

  

  2(l+fi)k 
  

   /3m 
  

  

  tan( 
  f 
  _^)==^i±^; 
  

  

  (29) 
  

  

  (30) 
  

  

  the 
  exponential 
  coefficient 
  5 
  being 
  given 
  by 
  (24), 
  and 
  E, 
  e, 
  

   F, 
  and 
  <f> 
  being 
  the 
  arbitrary 
  constants 
  dependent 
  on 
  the 
  

   initial 
  conditions. 
  In 
  many 
  of 
  the 
  experimental 
  cases 
  E' 
  

   may 
  be 
  assimilated 
  to 
  E 
  and 
  F' 
  to 
  F 
  without 
  appreciable 
  

   error. 
  The 
  changes 
  (e'— 
  e) 
  and 
  (</>' 
  — 
  <f>) 
  of 
  the 
  phase 
  angles 
  

   may 
  be 
  distinctly 
  appreciable 
  for 
  very 
  small 
  values 
  of 
  /3. 
  

   But 
  in 
  these 
  cases 
  the 
  vibrations 
  show 
  a 
  slow 
  waxing 
  and 
  

   waning 
  of 
  amplitude 
  and 
  the 
  phase 
  is 
  of 
  very 
  little 
  importance. 
  

   On 
  the 
  other 
  hand, 
  for 
  /3 
  equal 
  to 
  unity, 
  we 
  have 
  

  

  tan 
  (e 
  f 
  — 
  e) 
  =4Jc/m 
  and 
  tan 
  (<£' 
  — 
  <£)== 
  —2</2k/m. 
  

  

  And 
  the 
  numerical 
  values 
  of 
  these 
  are 
  of 
  the 
  order 
  0*020 
  

   and 
  0-014, 
  hence 
  e'-e=l° 
  10' 
  and 
  </>' 
  -<j> 
  = 
  0° 
  48' 
  nearly. 
  

   Hence 
  for 
  all 
  our 
  present 
  experimental 
  cases, 
  we 
  may 
  drop 
  

   the 
  four 
  accents 
  in 
  equation 
  (28). 
  

  

  