﻿282 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  In 
  all 
  the 
  cases 
  there 
  is 
  at 
  least 
  one 
  of 
  the 
  frequencies 
  

   which 
  appears 
  in 
  the 
  relation 
  (5) 
  (p. 
  270) 
  with 
  a 
  coefficient 
  

   — 
  1. 
  Let 
  it 
  be 
  n 
  v 
  , 
  then 
  the 
  relation 
  may 
  be 
  written 
  as 
  

   follows 
  : 
  

  

  n 
  v 
  =p 
  1 
  n 
  x 
  + 
  q 
  1 
  n 
  I/ 
  + 
  — 
  p. 
  

  

  We 
  now 
  introduce 
  a 
  frequency 
  nj 
  in 
  such 
  a 
  way 
  that 
  

  

  n 
  v 
  = 
  n 
  v 
  ' 
  — 
  p. 
  

  

  Between 
  the 
  frequencies 
  n 
  X3 
  n 
  y 
  nj 
  exists 
  the 
  strict 
  

  

  relation 
  : 
  

  

  Pin 
  x 
  + 
  q 
  x 
  n 
  y 
  + 
  — 
  n 
  v 
  ' 
  = 
  0. 
  

  

  In 
  the 
  differential 
  equations 
  we 
  may 
  in 
  the 
  terms 
  of 
  

   higher 
  order 
  replace 
  n 
  v 
  by 
  nj 
  . 
  In 
  the 
  terms 
  of 
  order 
  h 
  this 
  

   is 
  not 
  permitted. 
  The 
  frequency 
  n 
  v 
  appears 
  in 
  the 
  equation 
  

  

  v 
  + 
  n 
  v 
  z 
  v 
  — 
  ^ 
  — 
  =U. 
  

   dv 
  

  

  Now 
  

  

  n 
  v 
  2 
  =n 
  v 
  ' 
  2 
  — 
  2pn 
  v 
  ' 
  + 
  p 
  2 
  . 
  

  

  By 
  substitution 
  of 
  this 
  in 
  the 
  equation, 
  we 
  may 
  neglect 
  the 
  

   term 
  p 
  2 
  . 
  The 
  equation 
  takes 
  the 
  form 
  

  

  v 
  + 
  n 
  v 
  ' 
  2 
  v-2pn 
  v 
  'v-]^=0. 
  

  

  If 
  we 
  now 
  admit 
  in 
  II 
  a 
  term 
  pn 
  v 
  'v 
  2 
  , 
  then 
  the 
  equations 
  of 
  

   motion 
  hold 
  the 
  simple 
  form, 
  which 
  is 
  written 
  down 
  at 
  the 
  

   beginning 
  of 
  this 
  section. 
  However 
  n 
  v 
  is 
  everywhere 
  replaced 
  

   by 
  n 
  v 
  ' 
  ; 
  between 
  the 
  frequencies 
  exists 
  a 
  strict 
  relation. 
  

  

  § 
  14. 
  In 
  order 
  to 
  integrate 
  this 
  system 
  of 
  equations, 
  we 
  

   make 
  use 
  of 
  the 
  method 
  of 
  the 
  variation 
  of 
  the 
  canonical 
  

   constants. 
  This 
  means, 
  as 
  is 
  known, 
  that 
  the 
  equations, 
  

  

  arising 
  when 
  the 
  terms 
  -c 
  — 
  , 
  =r 
  — 
  , 
  etc 
  , 
  are 
  omitted, 
  first 
  are 
  

  

  solved, 
  in 
  which 
  solution 
  2m 
  arbitrary 
  constants 
  appear 
  (as 
  

   the 
  mechanism 
  has 
  m 
  degrees 
  of 
  freedom). 
  We 
  then 
  

   investigate 
  what 
  functions 
  of 
  the 
  time 
  must 
  be 
  the 
  quantities 
  

   just 
  now 
  regarded 
  as 
  constants, 
  so 
  that 
  the 
  expressions 
  for 
  

   the 
  coordinates, 
  taken 
  in 
  this 
  way, 
  represent 
  the 
  solution 
  of 
  

  

  the 
  complete 
  equations 
  containing 
  ^— 
  , 
  ^— 
  , 
  The 
  

  

  equations 
  in 
  which 
  ^— 
  , 
  ^ 
  — 
  , 
  are 
  lacking, 
  are 
  solved 
  

  

  according 
  to 
  the 
  method 
  of 
  Hamilton-Jacobi, 
  in 
  order 
  that 
  

   the 
  constants 
  we 
  obtain 
  may 
  form 
  a 
  canonical 
  system. 
  

  

  If 
  a 
  x 
  , 
  oLy 
  } 
  j 
  J3 
  X 
  , 
  fty 
  are 
  the 
  canonical 
  constants, 
  

  

  