﻿Oscillations 
  about 
  a 
  Position 
  oj 
  Eqitilibrium. 
  269 
  

  

  where 
  

  

  (2) 
  (2) 
  

  

  £=(n 
  x 
  + 
  cr)* 
  + 
  X 
  ; 
  ^r 
  = 
  (w 
  y 
  + 
  T)* 
  + 
  /*; 
  .... 
  

  

  W 
  (*) 
  r- 
  7 
  ) 
  . 
  , 
  . 
  

  

  3. 
  cr, 
  7, 
  . 
  . 
  . 
  . 
  are 
  functions 
  of 
  // 
  or 
  this 
  form 
  : 
  

  

  «... 
  

  

  (r) 
  (r) 
  r 
  (r+1) 
  (r 
  + 
  1) 
  

  

  a 
  = 
  (i 
  h 
  -+- 
  <i 
  h 
  

  

  j_ 
  

  

  (2) 
  (2) 
  

  

  a 
  = 
  s 
  //- 
  + 
  • 
  • 
  • 
  • 
  

  

  (2) 
  (2) 
  

  

  T 
  = 
  t 
  h- 
  a- 
  . 
  . 
  . 
  . 
  

  

  A7i, 
  BA, 
  .... 
  \, 
  /x,v 
  . 
  . 
  . 
  . 
  are 
  constants 
  of 
  integration; 
  

   A. 
  B, 
  C 
  .... 
  are 
  of 
  moderate 
  greatness; 
  li 
  is 
  a 
  quantity 
  

   which 
  is 
  small 
  in 
  respect 
  to 
  a 
  certain 
  greatness 
  /, 
  depending 
  

   e. 
  o. 
  on 
  the 
  dimensions 
  of 
  the 
  mechanism. 
  

  

  S 
  is 
  the 
  sum 
  of 
  the 
  absolute 
  values 
  of 
  p, 
  q, 
  r 
  . 
  . 
  . 
  

  

  In 
  order 
  to 
  prove 
  this 
  proposition, 
  the 
  expansions 
  (2) 
  were 
  

   substituted 
  in 
  the 
  differential 
  equations 
  ; 
  the 
  products 
  and 
  

   powers 
  of 
  the 
  cosines 
  and 
  sines 
  were 
  written 
  in 
  the 
  form 
  of 
  

   sums 
  of 
  cosines 
  and 
  constant 
  terms 
  ; 
  then 
  the 
  sum 
  of 
  the 
  

   terms 
  with 
  the 
  same 
  expression 
  cos 
  (/:></> 
  + 
  '/"^ 
  + 
  . 
  • 
  .) 
  having 
  

   as 
  coefficient 
  a 
  function 
  of 
  h, 
  the 
  coefficient 
  of 
  each 
  power 
  of 
  

   li 
  may 
  he 
  equalized 
  to 
  zero. 
  

  

  By 
  substituting 
  as 
  before 
  we 
  find 
  for 
  the 
  coefficient 
  of 
  

   cos 
  (/'</> 
  + 
  </^ 
  + 
  • 
  . 
  • 
  •) 
  the 
  following 
  expression 
  : 
  

  

  (2) 
  (2) 
  (S) 
  

  

  r_ 
  Upn 
  x 
  +qn 
  4 
  j+ 
  (/«7-f 
  7 
  r 
  + 
  . 
  . 
  . 
  .) 
  } 
  1 
  + 
  n 
  i 
  » 
  + 
  2AW 
  + 
  ] 
  * 
  

  

  + 
  other 
  terms 
  containing 
  products 
  of 
  a's, 
  (3 
  s, 
  etc. 
  

   (the 
  terms 
  — 
  A 
  2 
  /r 
  originate 
  from 
  terms 
  as 
  e. 
  <j. 
  

  

  9 
  

  

  cos 
  2 
  </> 
  cos 
  (/><f> 
  + 
  fjyjr+ 
  . 
  . 
  . 
  .)). 
  

  

  (2) 
  (2) 
  

  

  Supposing 
  the 
  coefficients 
  in 
  the 
  expressions 
  <r, 
  r 
  . 
  . 
  . 
  . 
  

   to 
  have 
  been 
  calculated 
  already, 
  we 
  may 
  write 
  the 
  coefficient 
  

  

  of 
  a 
  cos 
  (j><j) 
  + 
  >jyp* 
  + 
  . 
  . 
  .) 
  in 
  the 
  form 
  : 
  

  

  2 
  

  

  f 
  = 
  -{(/>-l)fl 
  ■ 
  + 
  '/".,+ 
  • 
  • 
  •} 
  [(p 
  + 
  lK 
  + 
  ^%+ 
  • 
  • 
  .} 
  + 
  !)/<> 
  ... 
  (3) 
  

  

  The 
  remaining 
  terms 
  which 
  appear 
  and 
  which 
  we 
  suppose 
  

   to 
  have 
  been 
  arranged 
  according 
  to 
  powers 
  of 
  h, 
  do 
  noi 
  

  

  8) 
  f 
  

   contain 
  a 
  in 
  their 
  coefficients, 
  but 
  products 
  of 
  a's, 
  fire, 
  . 
  . 
  . 
  ., 
  

  

  •• 
  

   which 
  may 
  have 
  been 
  calculated 
  from 
  preceding 
  equal 
  i 
  

  

  