﻿68 
  Prof. 
  Barton 
  and 
  Miss 
  Browning 
  on 
  

  

  Hence, 
  for 
  the 
  special 
  solution 
  with 
  these 
  initial 
  conditions, 
  

   we 
  have 
  

  

  where 
  s 
  = 
  

  

  Thus 
  the 
  ratios 
  of 
  the 
  amplitudes 
  of 
  the 
  quick 
  and 
  slow 
  

   components 
  in 
  the 
  y 
  and 
  z 
  vibrations 
  are 
  respectively 
  given 
  by 
  

  

  -p 
  -(p-i)st 
  d 
  1 
  -( 
  P 
  -i)st 
  / 
  43 
  v 
  

  

  (l 
  + 
  p)l3 
  e 
  and 
  (l+p)P 
  e 
  ' 
  ' 
  { 
  } 
  

  

  Case 
  II. 
  — 
  Suppose 
  now 
  that 
  the 
  heavy 
  bob 
  (of 
  mass 
  

   Q=pP) 
  is 
  pulled 
  aside 
  while 
  the 
  light 
  one 
  (of 
  massP) 
  is 
  held 
  

   undisplaced. 
  Then 
  we 
  have 
  : 
  

  

  For 
  f=0, 
  *=/, 
  y=0, 
  

  

  at 
  at 
  

  

  Putting 
  (44) 
  in 
  (27), 
  (28) 
  without 
  accents, 
  (32) 
  and 
  (33), 
  

   and 
  omitting 
  small 
  quantities 
  as 
  before, 
  we 
  find 
  

  

  / 
  = 
  E 
  sin€ 
  + 
  Fsin<£, 
  -| 
  

  

  O=-pEsin6 
  + 
  Fsin0; 
  J 
  ' 
  

  

  = 
  E?7lCOS6 
  + 
  F?2COS</>, 
  i 
  

  

  0=— 
  pEmcose 
  + 
  Frccos^. 
  J 
  

   Then 
  (46) 
  is 
  satisfied 
  by 
  

  

  e=^ 
  and 
  <£ 
  = 
  !", 
  

   and 
  putting 
  these 
  values 
  in 
  (45) 
  , 
  we 
  obtain 
  

  

  E 
  =T 
  ^- 
  and 
  F 
  = 
  ^-. 
  

   1+p 
  1+p 
  

  

  Hence, 
  for 
  the 
  special 
  solution 
  with 
  these 
  initial 
  conditions, 
  

   we 
  have 
  

  

  ■- 
  -Pf-^oosmt 
  + 
  ^Le-'cos-yj^-^, 
  (47) 
  

  

  1 
  + 
  P 
  1+P 
  WQ- 
  + 
  &Y 
  

  

  -f^^-^+jf^-cos-^. 
  . 
  (48) 
  

  

  