﻿

  about 
  a 
  Position 
  of 
  Equilibrium.* 
  317 
  

  

  § 
  52. 
  Another 
  interesting 
  case 
  occurs 
  when 
  r 
  = 
  (if 
  we 
  

   turn 
  the 
  axes 
  through 
  an 
  angle 
  of 
  45°, 
  then 
  according 
  to 
  

   § 
  49,$' 
  = 
  1-4/). 
  

  

  The 
  relation 
  between 
  f 
  and 
  <p 
  becomes 
  : 
  

  

  £(W) 
  cos 
  2 
  = 
  tf(l-f). 
  

  

  If 
  not 
  £ 
  = 
  or 
  £ 
  = 
  1 
  continually, 
  then 
  

  

  cos 
  </>= 
  ± 
  vq. 
  

  

  All 
  ellipses 
  of 
  one 
  of 
  these 
  systems 
  correspond 
  to 
  the 
  same 
  

   value 
  of 
  <£. 
  For 
  this 
  case 
  (20, 
  § 
  30) 
  takes 
  this 
  form 
  : 
  

  

  f 
  sin 
  t 
  cos(t 
  -(/>) 
  + 
  (1 
  — 
  cos 
  t 
  sin(£ 
  -</>)= 
  0. 
  

   Therefore 
  

  

  _sin((£ 
  — 
  £)cos£ 
  -, 
  c,__, 
  cos 
  (<f> 
  — 
  t) 
  si 
  n 
  * 
  

   ' 
  ~ 
  sin 
  <£ 
  ' 
  sin 
  9 
  

  

  x 
  and 
  z/ 
  may 
  be 
  written 
  as 
  functions 
  of 
  t 
  : 
  

  

  /cos 
  6 
  t 
  sin 
  (<£ 
  — 
  t) 
  /cos 
  2 
  (t 
  — 
  <£) 
  sin 
  t 
  

  

  X 
  -\/ 
  sin</> 
  ^-"V^ 
  srn^ 
  - 
  " 
  ; 
  

  

  r 
  sin2(£-c£) 
  

  

  ^ 
  V 
  

  

  sin 
  2£ 
  

  

  The 
  shape 
  of 
  the 
  envelope 
  as 
  it 
  is 
  represented 
  in 
  fig. 
  36 
  

   (PL 
  VIII.) 
  may 
  be 
  deduced 
  from 
  these 
  equations. 
  It 
  

   consists 
  of 
  two 
  closed 
  parts, 
  one 
  having 
  four 
  summits, 
  the 
  

   other 
  four 
  cusps. 
  For 
  decreasing 
  values 
  of 
  q 
  the 
  two 
  parts 
  

   move 
  towards 
  each 
  other; 
  for 
  q 
  — 
  they 
  coincide 
  into 
  a 
  

   square 
  (fig. 
  33). 
  For 
  increasing 
  values 
  of 
  q 
  one 
  part 
  moves 
  

   to 
  the 
  circle 
  with 
  radius 
  unity 
  ; 
  the 
  other 
  contracts 
  into 
  the 
  

   origin 
  ; 
  this 
  has 
  taken 
  place 
  for 
  q=l 
  (fig. 
  41). 
  

  

  The 
  envelope 
  of 
  fig. 
  36 
  (we 
  include 
  the 
  axes 
  of 
  coordinates) 
  

   represents 
  the 
  limit 
  form 
  of 
  the 
  envelope 
  of 
  fig. 
  35 
  for 
  a 
  

   constant 
  value 
  of 
  q, 
  if 
  r 
  approaches 
  to 
  zero. 
  For 
  a 
  positive 
  

   value 
  of 
  r 
  we 
  have 
  two 
  systems 
  of 
  ellipses, 
  each 
  of 
  them 
  having 
  

   its 
  own 
  envelope. 
  For 
  r 
  = 
  the 
  two 
  systems 
  are 
  no 
  longer 
  

   separated 
  ; 
  they 
  have 
  in 
  common 
  the 
  double 
  curves 
  along 
  the 
  

   axes 
  of 
  coordinates. 
  We 
  have 
  to 
  do 
  with 
  an 
  asymptotic 
  form 
  

   of 
  motion. 
  

  

  For 
  a 
  negative 
  value 
  of 
  r 
  we 
  have 
  but 
  one 
  system 
  of 
  

   ellipses 
  ; 
  if 
  this 
  value 
  is 
  not 
  too 
  little, 
  then 
  the 
  envelope 
  

   consists 
  of 
  two 
  concentric 
  curves, 
  as 
  is 
  represented 
  in 
  fig. 
  39. 
  

  

  