﻿278 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  If 
  we 
  take 
  again 
  as 
  a 
  first 
  approximation 
  : 
  

  

  x 
  = 
  Ah 
  cos 
  (nJ 
  + 
  X), 
  ?/ 
  = 
  B7tcos 
  (ii 
  y 
  t 
  4- 
  /jl), 
  , 
  

  

  In 
  which 
  A, 
  B, 
  . 
  .A. 
  //,, 
  ... 
  are 
  functions 
  of 
  t 
  in 
  such 
  a 
  manner 
  

   that 
  A, 
  B, 
  ...X, 
  fl, 
  ... 
  are 
  of 
  order 
  h 
  2 
  or 
  smaller, 
  then 
  in 
  the 
  

   terms 
  of 
  higher 
  order 
  we 
  may 
  substitute 
  — 
  n 
  2 
  x 
  for 
  'x, 
  

  

  —n/y 
  for 
  y, 
  ..., 
  A 
  2 
  h 
  2 
  — 
  n 
  x 
  2 
  x 
  2 
  for 
  x 
  2 
  , 
  B 
  2 
  h 
  2 
  — 
  n 
  y 
  2 
  y 
  2 
  for 
  y 
  2 
  , 
  

  

  We 
  then 
  retain 
  as 
  disturbing 
  terms 
  of 
  the 
  first 
  kind 
  in 
  the 
  

   first 
  equation 
  only 
  those 
  with 
  

  

  A 
  2 
  h 
  2 
  x, 
  B 
  2 
  h 
  2 
  x, 
  , 
  x 
  3 
  , 
  xy 
  2 
  , 
  

  

  B 
  2 
  h 
  2 
  

   .Now 
  we 
  replace 
  xy 
  2 
  by 
  —^—x, 
  etc., 
  which 
  is 
  permitted 
  

  

  because 
  they 
  give 
  rise 
  to 
  the 
  same 
  disturbing 
  terms. 
  

  

  If 
  we 
  reduce 
  in 
  the 
  same 
  way 
  the 
  disturbing 
  terms 
  of 
  the 
  

   first 
  kind 
  in 
  the 
  remaining 
  equations, 
  we 
  shall 
  find 
  that 
  

   these 
  terms 
  are 
  in 
  the 
  different 
  equations 
  the 
  derivatives 
  

   resp. 
  according 
  to 
  x, 
  y, 
  of 
  : 
  

  

  K*** 
  + 
  K 
  y 
  y± 
  + 
  4- 
  M, 
  Z 
  A 
  2 
  /^ 
  2 
  + 
  M 
  1JX 
  B 
  2 
  h\v 
  2 
  + 
  M^ 
  A 
  2 
  h 
  2 
  y 
  2 
  -f 
  

  

  where 
  the 
  K's 
  and 
  M's 
  are 
  coefficients. 
  

  

  About 
  the 
  disturbing 
  terms 
  of 
  the 
  second 
  kind 
  we 
  may 
  

   make 
  the 
  following 
  general 
  observation. 
  They 
  appear 
  only 
  

   in 
  the 
  equations 
  for 
  those 
  coordinates 
  which 
  enter 
  in 
  the 
  

   relation, 
  and 
  they 
  contain 
  no 
  other 
  coordinates. 
  So 
  we 
  may, 
  

   in 
  order 
  to 
  determine 
  these 
  terms, 
  restrict 
  ourselves 
  respec- 
  

   tively 
  to 
  mechanisms 
  with 
  2, 
  3, 
  or 
  4 
  degrees 
  of 
  freedom. 
  We 
  

   now 
  shall 
  discuss 
  these 
  terms 
  for 
  the 
  different 
  relations 
  

   separately. 
  

  

  § 
  9. 
  Zn 
  x 
  — 
  n 
  y 
  =p. 
  — 
  In 
  this 
  case 
  : 
  

  

  T 
  = 
  \x 
  2 
  + 
  \f 
  + 
  ±P 
  xx 
  x 
  2 
  + 
  B 
  xy 
  xy 
  + 
  1P^ 
  2 
  , 
  

  

  w 
  

  

  here 
  

  

  Further 
  

  

  Pxx 
  = 
  \a 
  xx 
  x 
  2 
  + 
  b 
  xx 
  xy 
  + 
  \c 
  xx 
  y 
  2 
  , 
  

  

  *- 
  xy 
  - 
  = 
  "2 
  / 
  Q'xyffi 
  i 
  X 
  yXy 
  T 
  '^-njV 
  ' 
  

  

  *-yy~2 
  a 
  yy 
  x 
  + 
  Ar*$ 
  ' 
  + 
  2 
  c 
  yyV 
  • 
  

  

  U 
  = 
  J-^V 
  + 
  ln/z/ 
  2 
  + 
  U 
  4 
  , 
  

  

  where 
  U 
  4 
  represents 
  a 
  homogeneous 
  function 
  of 
  degree 
  4 
  in 
  

   x 
  and 
  y. 
  

  

  When 
  in 
  the 
  same 
  way 
  as 
  is 
  done 
  for 
  the 
  former 
  cases 
  

   the 
  disturbing 
  terms 
  in 
  the 
  different 
  equations 
  are 
  reduced, 
  

  

  