﻿284 
  

  

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

  

  The 
  terms 
  K^ 
  4 
  , 
  K 
  y 
  ij\ 
  , 
  , 
  M 
  XX 
  A^V 
  } 
  M 
  7JX 
  Wh 
  2 
  x 
  2 
  , 
  

  

  (in 
  these 
  latter 
  terms 
  we 
  must 
  first 
  replace 
  A 
  2 
  h 
  2 
  , 
  B 
  2 
  /* 
  2 
  , 
  

  

  by 
  _i_ 
  _JL 
  ? 
  ) 
  give 
  rise 
  to 
  a 
  homogeneous 
  quadratic 
  

  

  n 
  x 
  n 
  y 
  2 
  

  

  function 
  of 
  the 
  a's 
  %2( 
  a 
  i? 
  a 
  2- 
  a 
  »0- 
  

  

  We 
  must, 
  however, 
  make 
  an 
  exception 
  as 
  to 
  the 
  case 
  

  

  8 
  1 
  =2. 
  The 
  terms 
  K** 
  4 
  , 
  K 
  y 
  z/ 
  4 
  , 
  , 
  M 
  xx 
  AVi\ 
  My 
  X 
  Wh\ 
  

  

  do 
  not 
  give 
  any 
  difficulty. 
  The 
  term 
  x 
  2 
  y 
  2 
  gives, 
  besides 
  a 
  

   term 
  with 
  cosine, 
  rise 
  to 
  a 
  constant 
  term. 
  The 
  terms 
  x 
  % 
  y 
  

   and 
  xif 
  give 
  rise 
  resp. 
  to 
  terms 
  with 
  

  

  *x\/o<. 
  x 
  ct.yCO§2(ii 
  x 
  ft 
  x 
  —n 
  } 
  fi 
  y 
  ) 
  and 
  a 
  yS 
  /x 
  x 
  oLyCO$2(n 
  x 
  ft 
  x 
  —n 
  y 
  fS 
  y 
  ). 
  

  

  Therefore 
  B 
  takes 
  in 
  the 
  different 
  cases 
  the 
  following- 
  

   form 
  : 
  

  

  Si=3^ 
  

  

  n 
  x 
  

  

  2n 
  x 
  —n 
  y 
  = 
  p 
  B 
  = 
  

  

  + 
  n 
  v 
  — 
  n=p 
  K 
  = 
  

  

  d.z 
  

  

  cos2(2«A-^/3 
  y 
  ). 
  

  

  2n, 
  

  

  ±n 
  x 
  n 
  y 
  n. 
  

  

  ^/a 
  x 
  CtyC 
  

  

  cos 
  2(n 
  x 
  j3 
  x 
  +ny/3y-n 
  z 
  l3 
  z 
  ). 
  

  

  Si 
  = 
  H 
  

  

  3^ 
  — 
  % 
  = 
  p 
  B 
  = 
  % 
  2 
  (*s> 
  a 
  y> 
  

  

  ±??x+ 
  2n 
  y 
  — 
  n 
  z 
  — 
  p 
  B 
  = 
  % 
  2 
  (a 
  z 
  , 
  a 
  y 
  , 
  

  

  ^ky 
  ay 
  --&y^ 
  /aXCly 
  

  

  cos 
  2 
  (3n 
  x 
  p 
  x 
  — 
  11 
  yfrj). 
  

  

  • 
  «m) 
  + 
  

  

  />' 
  

  

  2n~ 
  z 
  $n 
  T 
  n„ 
  2 
  n 
  ~ 
  y 
  

  

  «yV¥? 
  

  

  cos 
  2 
  ( 
  + 
  n 
  x 
  fix 
  + 
  2?2y/3 
  y 
  — 
  ?? 
  Z 
  /3 
  Z 
  ) 
  . 
  

  

  cos 
  2 
  (?? 
  X 
  /3 
  X 
  + 
  « 
  A 
  ± 
  n 
  z 
  /3 
  z 
  — 
  n 
  u 
  pl 
  . 
  

  

  S 
  1 
  = 
  2 
  

  

  n 
  x 
  — 
  n 
  y 
  — 
  p 
  B 
  = 
  % 
  2 
  (a 
  z 
  , 
  a 
  y 
  , 
  ...a 
  OT 
  )+ 
  f 
  - 
  <* 
  y 
  — 
  

  

  f 
  

  

  a 
  x 
  u 
  y 
  cos 
  4:(n 
  x 
  fi 
  x 
  

  

  2n 
  y 
  ~ 
  y 
  8n 
  x 
  2 
  ny 
  2 
  

  

  — 
  ny/3y)—%^/a 
  x 
  oty(e 
  2 
  'ci 
  x 
  + 
  e±u 
  y 
  ) 
  cos 
  2(v 
  x 
  /3 
  x 
  — 
  %&). 
  

  

  § 
  16. 
  We 
  can 
  make 
  now 
  the 
  following 
  general 
  observation. 
  

   The 
  /3's 
  appear 
  only 
  under 
  the 
  sign 
  cosine 
  ; 
  the 
  coefficients 
  

   of 
  n 
  z 
  /3 
  2 
  , 
  %/3 
  y 
  , 
  ... 
  in 
  the 
  expressions 
  between 
  brackets 
  are 
  the 
  

   same 
  as 
  the 
  coefficients 
  in 
  the 
  relation 
  (5) 
  (p. 
  270). 
  We 
  

   put 
  now 
  : 
  

  

  cj> 
  = 
  2(pn 
  x 
  f3 
  x 
  +qn 
  y 
  l3y 
  + 
  ) 
  (9) 
  

  

  