﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  291 
  

  

  curves 
  surrounding 
  (general 
  form 
  of 
  motion) 
  is 
  very 
  small 
  

   for 
  small 
  values 
  of 
  p' 
  and 
  it 
  extends 
  according 
  as 
  p 
  increases, 
  

   so 
  that 
  those 
  curves 
  are 
  most 
  important 
  for 
  great 
  values 
  of 
  p~ 
  

   Furthermore, 
  we 
  notice 
  that 
  according 
  as 
  p 
  increases 
  the 
  

   curves 
  surrounding 
  approach 
  to 
  circles 
  with 
  as 
  centre. 
  

  

  For 
  each 
  value 
  of 
  p' 
  we 
  got 
  for 
  the 
  maximal 
  and 
  minimal 
  

   value 
  of 
  k 
  an 
  isolated 
  point 
  on 
  the 
  axis 
  of 
  the 
  angles 
  (periodic 
  

   form 
  of 
  motion). 
  

  

  Fig. 
  4 
  gives 
  some 
  curves 
  for 
  a 
  rather 
  small 
  value 
  of 
  p\ 
  

   tig. 
  5 
  for 
  a 
  rather 
  great 
  value 
  of 
  p' 
  ; 
  the 
  • 
  — 
  lines 
  indi- 
  

   cate 
  the 
  degenerated 
  curves. 
  

  

  S 
  1= 
  4. 
  

  

  § 
  24. 
  ',)n 
  v 
  —n, 
  / 
  = 
  p. 
  The 
  relation 
  between 
  f 
  and 
  </> 
  runs 
  : 
  

  

  In 
  fig. 
  0' 
  the 
  different 
  possibilities 
  have 
  been 
  represented, 
  

   f 
  is 
  taken 
  as 
  radius 
  vector. 
  The 
  general 
  case 
  is 
  indicated 
  

   by 
  a 
  continuous 
  line, 
  the 
  case 
  of 
  libration 
  too. 
  In 
  the 
  

   periodic 
  case 
  sin 
  $ 
  = 
  0, 
  and 
  in 
  the 
  asymptotic 
  case 
  the 
  form 
  

   of 
  motion 
  approaches 
  to 
  sin 
  $ 
  = 
  (). 
  

  

  n 
  r 
  + 
  2n 
  lf 
  — 
  n 
  z 
  = 
  p, 
  —n 
  x 
  + 
  2nt,—n 
  s 
  = 
  p, 
  n 
  T 
  -\-n 
  J/ 
  + 
  v 
  z 
  -n 
  u 
  = 
  p. 
  

  

  n 
  .r 
  + 
  ^]/ 
  — 
  n 
  ~ 
  — 
  n 
  u 
  = 
  P- 
  

  

  The 
  relations 
  between 
  f 
  and 
  </> 
  are 
  to 
  be 
  found 
  in 
  § 
  17. 
  

   In 
  case 
  of 
  the 
  first 
  relation 
  f 
  remains 
  between 
  and 
  the 
  

   smaller 
  of 
  C\ 
  and 
  C 
  2 
  , 
  in 
  case 
  of 
  the 
  second 
  relation 
  between 
  

   and 
  the 
  smaller 
  of 
  (\ 
  and 
  C 
  3 
  , 
  in 
  case 
  of 
  the 
  third 
  relation 
  

   between 
  and 
  the 
  smallest 
  of 
  (\, 
  C 
  2 
  , 
  and 
  C 
  3 
  , 
  in 
  case 
  of 
  the 
  

   fourth 
  between 
  and 
  the 
  smaller 
  of 
  Ui 
  and 
  C 
  2 
  . 
  For 
  the 
  rest 
  

   the 
  £*-(/> 
  curves 
  are 
  of 
  quite 
  the 
  same 
  kind 
  as 
  in 
  the 
  case 
  of 
  

   the 
  relation 
  'dn 
  T 
  — 
  ihj 
  = 
  p. 
  

  

  S!=2. 
  

  

  § 
  2.5. 
  A\ 
  e 
  shall 
  restrict 
  ourselves 
  to 
  the 
  case 
  1=0. 
  p"= 
  —7. 
  

   Then 
  the 
  relation 
  between 
  f 
  and 
  </> 
  (§ 
  17) 
  runs 
  : 
  

  

  J(l-$)cos 
  s 
  0= 
  ? 
  g(l-f) 
  + 
  r. 
  

  

  In 
  figs. 
  7-13 
  (PI. 
  VII.) 
  f 
  is 
  taken 
  as 
  radius 
  vector. 
  Every 
  

   one 
  of 
  these 
  figures 
  corresponds 
  to 
  a 
  certain 
  value 
  of 
  q 
  ; 
  in 
  each 
  

   of 
  th«' 
  figures 
  tli<- 
  curves 
  correspond 
  to 
  different 
  values 
  of 
  r. 
  

  

  