﻿316 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  to 
  the 
  axes 
  of 
  coordinates. 
  This 
  angle 
  must 
  be 
  found 
  from 
  

   cot 
  2a> 
  = 
  2 
  v 
  /V 
  (§ 
  48). 
  According 
  to 
  § 
  46 
  we 
  have 
  

  

  f' 
  = 
  f 
  cos 
  2 
  «+ 
  (1 
  — 
  f)sin 
  2 
  o> 
  — 
  2 
  v 
  /f(l 
  — 
  f) 
  cos 
  <£ 
  sin 
  eacosw 
  

   = 
  f 
  cos 
  2 
  g>+(1 
  — 
  f) 
  sin 
  2 
  o)— 
  (f— 
  1) 
  sin 
  2a> 
  cot 
  2a> 
  

  

  — 
  1 
  

  

  The 
  envelope 
  consists 
  of 
  two 
  squares. 
  In 
  each 
  of 
  these 
  

   squares 
  we 
  have 
  a 
  system 
  of 
  ellipses 
  exactly 
  as 
  in 
  the 
  cases 
  

   ^ 
  = 
  or 
  q=zcc>. 
  These 
  systems 
  contain 
  both 
  the 
  circle 
  with 
  

   radius 
  \ 
  y/2. 
  Therefore 
  we 
  are 
  in 
  the 
  case 
  of 
  the 
  asymptotic 
  

   form 
  of 
  motion 
  ; 
  the 
  motion 
  approaches 
  asymptotically 
  to 
  

   the 
  motion 
  in 
  that 
  circle. 
  The 
  two 
  squares 
  coincide 
  for 
  

   q 
  = 
  — 
  4?* 
  =0, 
  and 
  for 
  q 
  = 
  —Ar=co 
  . 
  

  

  § 
  51. 
  The 
  case 
  of 
  the 
  degeneration 
  into 
  conic 
  sections 
  

   (/>= 
  — 
  . 
  I, 
  § 
  45) 
  occurs 
  here 
  for 
  q=l. 
  If 
  we 
  examine 
  the 
  

   signification 
  of 
  the 
  coefficients, 
  it 
  becomes 
  clear 
  that 
  the 
  

   surface 
  for 
  these 
  values 
  of 
  the 
  coefficients 
  is 
  a 
  surface 
  of 
  

   revolution. 
  The 
  relation 
  between 
  f 
  and 
  </> 
  may 
  be 
  written 
  

  

  £(l-f) 
  sin 
  2 
  <£=-n 
  

  

  From 
  the 
  formulae 
  of 
  § 
  41 
  it 
  follows 
  that 
  the 
  axes 
  of 
  the 
  

   ellipse 
  are 
  of 
  invariable 
  length 
  ; 
  therefore 
  the 
  shape 
  of 
  the 
  

   ellipses 
  is 
  invariable. 
  In 
  order 
  to 
  prove 
  that 
  the 
  ellipse 
  turns 
  

   with 
  an 
  uniform 
  angular 
  velocity, 
  and 
  in 
  order 
  to 
  calculate 
  

  

  this 
  angular 
  velocity, 
  it 
  is 
  necessary 
  to 
  calculate 
  -j- 
  from 
  

   the 
  system 
  of 
  equations 
  : 
  

  

  —2 
  == 
  m 
  x 
  f 
  (1 
  — 
  f 
  ) 
  sin 
  cf> 
  cos 
  <£, 
  

  

  f(l-f)sin 
  2 
  >=-r, 
  

  

  tan2*=^£g3W 
  

  

  It 
  will 
  be 
  found 
  that 
  the 
  results 
  agree 
  with 
  those 
  Prof. 
  

   Korteweg 
  arrives 
  at. 
  

  

  The 
  envelope 
  consists 
  of 
  two 
  concentric 
  circles 
  ; 
  this 
  

   might 
  also 
  be 
  deduced 
  from 
  the 
  equation 
  (37, 
  § 
  46), 
  where 
  

   0=0. 
  

  

  The 
  two 
  enveloping 
  circles 
  coincide 
  for 
  r= 
  — 
  ^ 
  ; 
  then 
  we 
  

   are 
  in 
  the 
  case 
  of 
  the 
  uniform 
  motion 
  in 
  a 
  parallel 
  circle 
  of 
  

   the 
  surface. 
  Another 
  special 
  case 
  we 
  have 
  when 
  r=0 
  ; 
  the 
  

   motion 
  takes 
  place 
  in 
  an 
  arbitrary 
  meridian 
  of 
  the 
  surface 
  ; 
  

   one 
  of 
  the 
  enveloping 
  circles 
  has 
  a 
  radius 
  of 
  unit 
  length 
  ; 
  

   the 
  other 
  has 
  contracted 
  into 
  a 
  point. 
  

  

  