﻿2SQ 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  If 
  now 
  R 
  does 
  not 
  contain 
  t 
  explicitly 
  (this 
  condition 
  is 
  

   satisfied 
  in 
  the 
  different 
  cases 
  we 
  have 
  to 
  consider), 
  then 
  

  

  K 
  = 
  constant 
  (14) 
  

  

  is 
  an 
  integral 
  o£ 
  system 
  (7) 
  (p. 
  283). 
  

  

  By 
  substitution 
  of 
  (11) 
  or 
  (13) 
  in 
  (14), 
  we 
  get 
  a 
  relation 
  

   between 
  f 
  and 
  <j>, 
  which 
  in 
  the 
  different 
  cases 
  has 
  this 
  

   form 
  : 
  

  

  S 
  i 
  = 
  3 
  i 
  n 
  x 
  +n 
  y 
  -n 
  z 
  =p, 
  \A(?-0 
  2 
  )(C 
  3 
  -f) 
  cos<£ 
  = 
  &+/£ 
  

  

  I 
  !_ 
  2y/n 
  x 
  n 
  y 
  n 
  z 
  p 
  \ 
  

  

  \ 
  p 
  ~ 
  Rop'h 
  r 
  

  

  3n 
  x 
  —n 
  y 
  =p, 
  £\/f(l 
  — 
  f)cos(£ 
  = 
  pf 
  2 
  +2f+?\ 
  

  

  Si=4 
  <j 
  ± 
  n 
  x 
  +2n 
  y 
  -n 
  z 
  =p, 
  J^ 
  (C~£( 
  C 
  3 
  - 
  £) 
  cos 
  =££*+<£+»•. 
  

  

  Lnx+ny+n^-^rrrp, 
  -x/ftCi- 
  f)(0,- 
  g)(C 
  8 
  ± 
  ?) 
  cos 
  =p? 
  2 
  + 
  #+ 
  * 
  

  

  Si 
  = 
  2 
  w*-% 
  = 
  /o, 
  r(l-?)cos 
  2 
  ^>+/ 
  v 
  / 
  ^l-?)cos^=X 
  2 
  +^f+^ 
  

  

  § 
  18. 
  When, 
  in 
  the 
  first 
  equation 
  of 
  system 
  (7) 
  (§ 
  14), 
  

  

  we 
  replace 
  the 
  as 
  by 
  the 
  expressions 
  (11) 
  or 
  (13) 
  of 
  § 
  1G, 
  

   then 
  according 
  to 
  the 
  form 
  of 
  R 
  (§ 
  15), 
  the 
  equation 
  takes 
  

   this 
  form 
  

  

  g^/UJsin^ 
  (15) 
  

  

  This 
  regards 
  the 
  cases 
  S 
  2 
  = 
  3 
  and 
  Si 
  = 
  4 
  ; 
  for 
  each 
  of 
  the 
  

   relations 
  /(f) 
  has 
  another 
  form 
  . 
  

  

  In 
  the 
  case 
  S 
  x 
  = 
  2 
  it 
  takes 
  this 
  form 
  : 
  

  

  J=/i(£) 
  sin 
  2*+/»(© 
  sin 
  <f>. 
  . 
  . 
  . 
  (16) 
  

  

  If 
  now 
  we 
  eliminate 
  cj> 
  between 
  (14) 
  and 
  (15) 
  or 
  (16), 
  

  

  then 
  -~ 
  is 
  known 
  as 
  a 
  function 
  of 
  f, 
  and 
  f 
  may 
  be 
  found 
  

  

  as 
  a 
  function 
  of 
  t 
  ; 
  from 
  (11) 
  or 
  (13) 
  we 
  may 
  find 
  the 
  

  

  a's, 
  and 
  from 
  (14) 
  (p. 
  Thereupon 
  ~-=j?, 
  -j^, 
  in 
  (7) 
  

  

  (p. 
  283) 
  are 
  known 
  as 
  functions 
  off, 
  and 
  fi 
  x 
  , 
  /3 
  y 
  , 
  may 
  be 
  

  

  