﻿about 
  a 
  Position 
  of 
  Equilibrium., 
  289 
  

  

  the 
  circle 
  f=0. 
  By 
  the 
  maximal 
  positive 
  and 
  negative 
  value 
  

  

  of 
  it 
  li=±Q^/3] 
  the 
  curve 
  has 
  contracted 
  into 
  an 
  isolated 
  

  

  point. 
  So 
  we 
  have 
  in 
  general 
  the 
  case 
  of 
  libration, 
  for 
  &=0 
  

  

  the 
  asymptotic, 
  and 
  for 
  k=±~\/o 
  the 
  periodic 
  form 
  of 
  

  

  motion. 
  The 
  general 
  form 
  of 
  motion 
  does 
  not 
  occur. 
  

   (Fig. 
  1. 
  PL 
  VII.) 
  

  

  Now 
  we 
  suppose 
  p 
  ^ 
  0. 
  Then 
  

  

  fv'W 
  cos 
  £=*+//£ 
  

  

  AVe 
  investigate 
  the 
  site 
  and 
  shape 
  of 
  the 
  £-</> 
  curves 
  for 
  

   positive 
  values 
  of 
  p 
  1 
  and 
  for 
  all 
  possible 
  values 
  of 
  k. 
  

  

  For 
  /; 
  = 
  there 
  is 
  degeneration 
  into 
  the 
  circle 
  £V0, 
  and 
  a 
  

   straight 
  line 
  normal 
  to 
  the 
  orioin 
  of 
  the 
  anodes 
  at 
  a 
  distance 
  

   p 
  from 
  pole 
  (the 
  degeneration 
  points 
  to 
  an 
  asymptotic 
  

   form 
  of 
  motion). 
  

  

  We 
  have 
  two 
  cases 
  now 
  : 
  p' 
  < 
  1 
  and 
  p' 
  > 
  1. 
  

  

  p 
  <1. 
  Let 
  us 
  now 
  investigate 
  the 
  shape 
  of 
  the 
  curves 
  for 
  

   different 
  values 
  of 
  k. 
  For 
  Z:>0 
  they 
  lie 
  to 
  the 
  right 
  of 
  the 
  

   straight 
  line 
  just 
  mentioned 
  (case 
  of 
  libration) 
  ; 
  for 
  increasing- 
  

   value 
  of 
  k 
  they 
  contract 
  more 
  and 
  more 
  until 
  for 
  the 
  maximal 
  

   value 
  of 
  /.-, 
  belonging 
  to 
  a 
  certain 
  value 
  of 
  p' 
  \ 
  we 
  get 
  an 
  

   isolated 
  point 
  (periodic 
  form 
  of 
  motion). 
  If 
  — 
  p 
  f 
  <k<0 
  the 
  

   curves 
  surround 
  point 
  (general 
  form 
  of 
  motion) 
  ; 
  if 
  k= 
  — 
  p' 
  

   we 
  have 
  a 
  curve 
  through 
  0, 
  for 
  k< 
  —p' 
  they 
  lie 
  to 
  the 
  left 
  

   of 
  (case 
  of 
  libration) 
  ; 
  for 
  the 
  minimal 
  value 
  of 
  k 
  we 
  again 
  

   get 
  an 
  isolated 
  point 
  (periodic 
  form 
  of 
  motion). 
  (Fig. 
  2.) 
  

  

  For 
  increasing 
  values 
  of 
  p 
  the 
  straight 
  line 
  separating 
  the 
  

   domains 
  k>0 
  and 
  k<0 
  moves 
  to 
  the 
  right. 
  The 
  domain 
  

   k>0 
  (the 
  domain 
  of 
  the 
  former 
  case 
  of 
  libration) 
  becomes 
  

   smaller 
  and 
  vanishes 
  for 
  p'—l. 
  For 
  p 
  f 
  >l 
  we 
  therefore 
  

   have 
  curves 
  surrounding 
  and 
  curves 
  to 
  the 
  left 
  of 
  only. 
  

   When 
  p' 
  increases 
  still 
  more 
  the 
  remaining 
  isolated 
  point 
  

   approaches 
  to 
  (the 
  domain 
  of 
  the 
  latter 
  case 
  of 
  libration 
  

   becomes 
  smaller) 
  and 
  the 
  curves 
  farther 
  from 
  approach 
  to 
  

   circles. 
  The 
  general 
  form 
  of 
  motion 
  becomes 
  preponderant 
  

   more 
  and 
  more. 
  

  

  § 
  23. 
  //■ 
  [ 
  + 
  n. 
  2 
  — 
  n 
  ? 
  , 
  = 
  p. 
  We 
  take 
  f 
  as 
  radius 
  vector. 
  We 
  

   take 
  first 
  the 
  case 
  of 
  the 
  strict 
  relation. 
  Therefore 
  : 
  

  

  A/r(r-e 
  2 
  )(c 
  3 
  -r)cos<£=*. 
  

  

  The 
  £~<t> 
  curves 
  remain 
  to 
  the 
  right 
  or 
  to 
  the 
  left 
  of 
  

   according 
  as 
  k 
  is 
  positive 
  or 
  negative 
  (in 
  general 
  we 
  have 
  

   the 
  case 
  of 
  libration). 
  For 
  the 
  maximal 
  value 
  of 
  k' 
  2 
  belonging 
  

  

  