﻿about 
  a 
  Position 
  of 
  Equilibrium* 
  305 
  

  

  now 
  only 
  to 
  consider 
  k<0; 
  so 
  only 
  the 
  four 
  last 
  forms 
  of 
  

   motion 
  are 
  possible. 
  For 
  increasing 
  values 
  of 
  p' 
  the 
  general 
  

   form 
  of 
  motion 
  (—p'<k 
  <0) 
  becomes 
  preponderant 
  more 
  

   and 
  more 
  ; 
  the 
  focus 
  of 
  the 
  enveloping 
  parabolas 
  moves 
  

   farther 
  ; 
  tig. 
  19 
  shows 
  how 
  the 
  limitation 
  approaches 
  more 
  

   and 
  more 
  to 
  the 
  rectangle 
  of 
  the 
  general 
  case 
  where 
  no 
  

   relation 
  exists. 
  

  

  § 
  38. 
  The 
  osculating 
  curves 
  are 
  given 
  by 
  : 
  

   a»= 
  \7fcos 
  t, 
  

  

  -6v} 
  * 
  ' 
  ■ 
  < 
  27 
  > 
  

  

  y=iVl-?cos(3e-^ 
  

  

  whilst 
  

  

  The 
  vertices 
  of 
  the 
  circumscribed 
  rectangles 
  lie 
  in 
  the 
  

   circumference 
  of 
  the 
  ellipse 
  : 
  

  

  & 
  +9^=1. 
  

  

  The 
  Lissajous 
  curve 
  corresponding 
  to 
  an 
  arbitrary 
  value 
  

   of 
  cf> 
  has 
  two 
  nodes. 
  For 
  cos<£ 
  = 
  the 
  nodes 
  lie 
  in 
  the 
  

  

  /F 
  

  

  X-axis 
  on 
  either 
  side 
  of 
  at 
  distances 
  -^— 
  from 
  ; 
  for 
  an 
  

   arbitrary 
  value 
  of 
  cj) 
  the 
  distance 
  of 
  the 
  nodes 
  from 
  

   the 
  Y-axis 
  is 
  also—, 
  the 
  distance 
  from 
  the 
  X-axis 
  is 
  

   }. 
  \/l 
  — 
  £ 
  cos 
  (/>. 
  One 
  of 
  the 
  nodes 
  is 
  reached 
  at 
  the 
  moments 
  

  

  IT 
  7T 
  

  

  t= 
  - 
  and 
  t= 
  -ry 
  , 
  the 
  other 
  at 
  the 
  moments 
  t 
  — 
  \ir 
  and 
  

  

  t 
  = 
  ?.7T. 
  

  

  The 
  Lissajous 
  double 
  curves 
  pass 
  through 
  0. 
  

  

  In 
  the 
  case 
  of 
  the 
  libration 
  as 
  well 
  as 
  in 
  the 
  case 
  of 
  the 
  

   general 
  form 
  of 
  motion 
  in 
  the 
  extreme 
  rectangles, 
  double 
  

   curves 
  aro 
  described. 
  In 
  the 
  former 
  case 
  the 
  ends 
  of 
  the 
  

   double 
  curves 
  lie 
  two 
  by 
  two 
  in 
  opposite 
  quadrants 
  ; 
  in 
  the 
  

   latter 
  case 
  in 
  each 
  of 
  the 
  four 
  quadrants 
  one 
  end 
  lies. 
  

  

  § 
  39. 
  Enveloping 
  curve 
  s. 
  — 
  "We 
  shall 
  restrict 
  ourselves 
  to 
  

   the 
  case 
  that 
  p=0 
  and 
  q 
  = 
  in 
  the 
  relation 
  between 
  f 
  and 
  

  

  </>. 
  Then 
  

  

  rVr(l-?)cos$=r 
  (28) 
  

  

  We 
  suppose 
  /• 
  to 
  be 
  positive. 
  

  

  