﻿270 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  obtained 
  by 
  equalizing 
  to 
  zero 
  the 
  coefficients 
  or! 
  smaller 
  

   powers 
  of 
  /tin 
  terms 
  containing 
  factors 
  cos 
  {p'(f> 
  + 
  q'^r 
  + 
  • 
  . 
  . 
  .)• 
  

   We 
  shall 
  call 
  the 
  sum 
  of 
  the 
  coefficients 
  of 
  these 
  remaining 
  

   terms 
  P. 
  Then 
  we 
  have 
  : 
  

  

  pqr 
  - 
  

  

  (S) 
  

  

  ? 
  * 
  = 
  P. 
  

  

  pqr... 
  pqr... 
  pqr... 
  

  

  Therefore 
  : 
  

  

  (S) 
  P 
  

  

  " 
  = 
  ^ 
  (4) 
  

  

  pqr... 
  £ 
  

  

  pqr... 
  

  

  From 
  (3) 
  it 
  follows 
  that 
  it 
  is 
  possible 
  that 
  £ 
  takes 
  an 
  

  

  pq... 
  

  

  abnormally 
  small 
  value 
  ; 
  in 
  this 
  case 
  according 
  to 
  (4) 
  a 
  takes 
  

  

  pq... 
  

  

  an 
  abnormally 
  great 
  value. 
  This 
  will 
  happen 
  when 
  one 
  of 
  

   the 
  expressions 
  

  

  (p-X)n 
  x 
  +qn 
  y 
  + 
  or 
  

  

  0+ 
  1 
  K+^V 
  + 
  

  

  becomes 
  small, 
  or 
  in 
  other 
  words 
  when 
  there 
  exists 
  a 
  relation 
  

   between 
  n 
  x 
  , 
  n 
  , 
  etc. 
  of 
  the 
  form 
  : 
  

  

  Pin 
  x 
  + 
  qin 
  y 
  + 
  = 
  P, 
  (5) 
  

  

  where 
  p 
  l9 
  q 
  l9 
  etc. 
  are 
  positive 
  or 
  negative 
  integers, 
  and 
  where 
  

   p 
  is 
  with 
  respect 
  to 
  n 
  x 
  , 
  n 
  , 
  etc. 
  a 
  small 
  quantity, 
  which 
  we 
  

   call 
  residue 
  of 
  relation. 
  

  

  When 
  k 
  is 
  an 
  integer, 
  then 
  

  

  kp 
  l 
  n 
  x 
  + 
  kq 
  1 
  n 
  y 
  + 
  

  

  is 
  also 
  small 
  (we 
  suppose 
  k 
  not 
  to 
  be 
  very 
  great) 
  . 
  

   If 
  now 
  in 
  (3) 
  

  

  p 
  — 
  l=*kpu 
  q 
  = 
  kq 
  u 
  .... 
  

   or 
  if 
  

  

  p 
  + 
  l 
  = 
  kp 
  u 
  q 
  = 
  kq 
  l} 
  .... 
  

  

  then 
  £ 
  becomes 
  very 
  small. 
  

  

  pqr... 
  

  

  We 
  see 
  that 
  there 
  are 
  in 
  that 
  case 
  among 
  the 
  terms 
  of 
  

   higher 
  order 
  terms 
  with 
  an 
  abnormally 
  great 
  coefficient 
  in 
  

   consequence 
  of 
  the 
  relation 
  (5) 
  ; 
  they 
  represent 
  vibrations 
  

   of 
  abnormally 
  great 
  intensity, 
  which 
  are 
  called 
  vibrations 
  of 
  

   relation. 
  By 
  vibrations 
  of 
  relation 
  of 
  the 
  first 
  kind 
  we 
  

   understand 
  such 
  as 
  correspond 
  to 
  

  

  o 
  = 
  kpi 
  — 
  l, 
  q 
  = 
  kq 
  l9 
  

  

  