﻿290 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillat 
  

  

  ions 
  

  

  to 
  given 
  values 
  o£ 
  C 
  2 
  and 
  C 
  3 
  the 
  curves 
  have 
  contracted 
  into 
  

   an 
  isolated 
  point 
  (periodic 
  form 
  of 
  motion). 
  For 
  k 
  = 
  de- 
  

   generation 
  takes 
  place 
  into 
  the 
  point 
  f 
  = 
  0, 
  the 
  circles 
  f 
  = 
  C 
  2 
  

   and 
  f=C 
  3 
  and 
  the 
  straight 
  line 
  cos 
  $ 
  = 
  (asymptotic 
  form 
  

   of 
  motion). 
  For 
  2 
  = 
  Og 
  then 
  of 
  necessity 
  k 
  = 
  0. 
  (Fig. 
  3.) 
  

   Now 
  we 
  suppose 
  p^fiO. 
  Therefore 
  

  

  A/£ 
  , 
  cf-c 
  s 
  )(c 
  3 
  -r)cos^= 
  / 
  )'^-?). 
  

  

  We 
  may 
  imagine 
  p' 
  to 
  be 
  positive. 
  We 
  give 
  to 
  C 
  2 
  and 
  C 
  3 
  

   constant 
  values 
  and 
  we 
  find 
  for 
  a 
  certain 
  value 
  of 
  p 
  the 
  

   forms 
  of 
  the 
  curves 
  satisfying 
  the 
  different 
  possible 
  values 
  of 
  

   k. 
  We 
  then 
  see 
  how 
  this 
  system 
  of 
  curves 
  varies 
  when 
  p' 
  

   passes 
  through 
  all 
  values 
  from 
  very 
  little 
  to 
  very 
  large. 
  

  

  For 
  every 
  value 
  of 
  p 
  three 
  cases 
  can 
  be 
  distinguished 
  : 
  

  

  k>C 
  s 
  . 
  As 
  f 
  remains 
  smaller 
  than 
  C 
  3 
  , 
  the 
  second 
  member, 
  

   so 
  also 
  cos 
  cj), 
  remains 
  positive. 
  Curves 
  on 
  the 
  right 
  of 
  

   (Case 
  of 
  libration). 
  

  

  k<C 
  2 
  . 
  Curves 
  on 
  the 
  left 
  of 
  (Case 
  of 
  libration). 
  

  

  C 
  2 
  <Aj<C 
  3 
  . 
  The 
  second 
  member, 
  therefore 
  also 
  cos 
  (/>, 
  be- 
  

   comes 
  zero 
  for 
  ^—k. 
  Curves 
  which 
  surround 
  (General 
  

   form 
  of 
  motion). 
  

  

  The 
  domains 
  of 
  the 
  plane 
  occupied 
  by 
  these 
  different 
  kinds 
  

   of 
  curves 
  are 
  bounded 
  by 
  the 
  curves 
  which 
  correspond 
  to 
  

   k 
  = 
  C 
  3 
  and 
  & 
  = 
  C 
  2 
  . 
  For 
  these 
  values 
  of 
  k 
  a 
  degeneration 
  

   takes 
  place. 
  

  

  For 
  jfe= 
  C 
  a 
  in 
  

  

  £=C 
  2 
  and 
  ^(CW) 
  cos-* 
  = 
  -p\/f=(V 
  

  

  The 
  latter 
  curve 
  lies 
  on 
  the 
  left 
  of 
  0, 
  it 
  begins 
  and 
  ends 
  in 
  

  

  7T 
  

  

  the 
  points: 
  £==G 
  2 
  , 
  </>=±x. 
  

   For 
  & 
  = 
  C 
  3 
  in 
  

  

  r=C 
  3 
  and 
  v/?(f-C 
  3 
  ) 
  cos 
  cj>=p 
  '</<J 
  3 
  -£ 
  

   The 
  latter 
  curve 
  lies 
  on 
  the 
  right 
  of 
  ; 
  it 
  begins 
  and 
  ends 
  

  

  7T 
  

  

  in 
  the 
  points: 
  f=C 
  3 
  , 
  d>= 
  + 
  ~> 
  

  

  z 
  

  

  The 
  degenerated 
  curves 
  point 
  to 
  asymptotic 
  forms 
  of 
  

   motion. 
  

  

  To 
  investigate 
  how 
  the 
  system 
  of 
  curves 
  varies 
  when 
  p' 
  is 
  

   changed, 
  it 
  is 
  sufficient 
  to 
  investigate 
  the 
  variation 
  of 
  the 
  

   degenerated 
  curves. 
  The 
  result 
  is, 
  that 
  the 
  domain 
  of 
  the 
  

  

  