﻿64 
  Prof. 
  Barton 
  and 
  Miss 
  Browning 
  on 
  

  

  Let 
  us 
  now 
  write 
  in 
  the 
  above 
  

  

  §=P, 
  f=™ 
  2 
  0) 
  

  

  Also 
  divide 
  the 
  two 
  equations 
  (1) 
  by 
  P 
  and 
  Q 
  respectively, 
  

   and 
  insert 
  the 
  frictional 
  term 
  2kdy\dt 
  in 
  the 
  first 
  of 
  them. 
  

   We 
  then 
  obtain 
  

  

  dhj 
  I 
  2h 
  dy 
  I 
  1 
  + 
  P+Pp 
  m 
  h 
  s 
  - 
  /3 
  ~- 
  p 
  rn 
  2 
  z 
  (ft 
  

  

  fe 
  l+p 
  + 
  _ 
  _J_ 
  m 
  * 
  v 
  (5 
  ) 
  

  

  ^ 
  + 
  (l 
  + 
  /3)(l 
  + 
  P 
  ) 
  m 
  *-(l+/3)- 
  (1+ 
  P 
  ) 
  m3/ 
  - 
  W 
  

   These 
  may 
  be 
  written 
  

  

  d 
  ^y 
  + 
  2k%+ay=pbz, 
  .... 
  (6) 
  

  

  and 
  dH 
  

  

  dt 
  

  

  Where 
  1+p 
  + 
  Pp 
  . 
  , 
  #»' 
  

  

  + 
  P)' 
  

  

  •-(1+0(1+,)"" 
  J 
  

  

  Solution 
  and 
  Frequencies. 
  — 
  To 
  solve 
  (6) 
  and 
  (7) 
  let 
  us 
  write 
  

  

  (9) 
  

  

  and, 
  on 
  inserting 
  in 
  (7), 
  we 
  have 
  

  

  y 
  

  

  ~\ 
  

  

  _ 
  [ 
  x 
  2 
  + 
  c 
  \ 
  :d 
  

  

  -[-J-)'-} 
  

  

  Then 
  (9) 
  substituted 
  in 
  (6) 
  gives 
  

  

  x 
  2 
  -t-2kx 
  + 
  a)=pb 
  9 
  

  

  m« 
  

  

  or 
  x 
  4 
  + 
  2kx 
  z 
  + 
  (c 
  + 
  a)x 
  2 
  + 
  2kcx 
  + 
  ca-pb 
  2 
  = 
  0, 
  . 
  (10) 
  

  

  which 
  is 
  the 
  auxiliary 
  biquadratic 
  in 
  x. 
  Though 
  this 
  equation 
  

   has 
  the 
  form 
  of 
  the 
  general 
  biquadratic, 
  an 
  approximate 
  

   solution, 
  presenting 
  all 
  the 
  accuracy 
  needed 
  for 
  our 
  purpose, 
  

   may 
  be 
  easily 
  obtained 
  by 
  noting 
  that 
  k 
  is 
  small 
  compared 
  

   with 
  the 
  other 
  constants. 
  For, 
  as 
  appears 
  from 
  the 
  experi- 
  

   ments, 
  k 
  is 
  of 
  the 
  order 
  one-thousandth 
  of 
  the 
  coefficient 
  of 
  

   x 
  2 
  and 
  of 
  the 
  constant 
  term. 
  

  

  Then 
  we 
  may 
  write 
  for 
  the 
  roots 
  of 
  x 
  in 
  the 
  biquadratic 
  

   (10) 
  the 
  values 
  

  

  — 
  r±ip 
  and 
  — 
  s±iq, 
  .... 
  (11) 
  

  

  