﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  311 
  

  

  § 
  45. 
  Case 
  p= 
  — 
  1. 
  

  

  The 
  motion 
  in 
  an 
  arbitrary 
  ellipse 
  can 
  be 
  analysed 
  in 
  two 
  

   rectangular 
  oscillations. 
  Taking 
  a 
  new 
  system 
  o£ 
  axes 
  of 
  

   coordinates, 
  the 
  expressions 
  for 
  x 
  and 
  y 
  hold 
  the 
  same 
  form. 
  

   But 
  the 
  values 
  of 
  f 
  and 
  cp 
  for 
  a 
  certain 
  ellipse 
  will 
  be 
  altered 
  

   in 
  general. 
  Considering 
  the 
  whole 
  system 
  of 
  ellipses, 
  we 
  

   can 
  say 
  that 
  the 
  relation 
  between 
  f 
  and 
  <f> 
  will 
  also 
  be 
  altered. 
  

   The 
  new 
  values 
  of 
  f 
  and 
  cf> 
  we 
  respectively 
  call 
  £' 
  and 
  <f>'. 
  

  

  We 
  shall 
  prove 
  now 
  that 
  in 
  the 
  case 
  supposed 
  (p=— 
  1) 
  

   it 
  is 
  always 
  possible 
  to 
  turn 
  the 
  system 
  of 
  axes 
  through 
  

   an 
  angle 
  -^r 
  so 
  that 
  the 
  relation 
  between 
  f 
  ' 
  and 
  <j>' 
  takes 
  this 
  

   form 
  : 
  

  

  Or, 
  what 
  is 
  the 
  same 
  : 
  

  

  VfTT^ 
  7 
  ) 
  sin 
  <j>'= 
  ± 
  Vmf 
  + 
  n.' 
  

   The 
  new 
  coordinates 
  are 
  given 
  by 
  

  

  x' 
  = 
  x 
  cos 
  yjr 
  — 
  y 
  sin 
  -v/r, 
  y' 
  — 
  xs\x\ 
  yjr+y 
  cos 
  ty. 
  

   Therefore 
  

  

  x' 
  = 
  Vfcosi/rcos£— 
  ^1 
  — 
  f 
  sin 
  -vjr 
  cos 
  (£ 
  — 
  (/>). 
  

  

  This 
  may 
  be 
  expressed 
  in 
  a 
  single 
  trigonometrical 
  term. 
  

   Then 
  

  

  5" 
  = 
  fcos 
  2 
  ^-f 
  (1 
  — 
  f) 
  sin 
  2 
  -\|r 
  — 
  2 
  \/?(l 
  — 
  f 
  ) 
  cos 
  (/> 
  sin 
  -^ 
  cos 
  >/r. 
  

  

  From 
  (32, 
  § 
  41) 
  it 
  follows 
  that 
  ?(l-f) 
  sin 
  2 
  <£ 
  is 
  an 
  

   invariant. 
  Therefore 
  

  

  r(l-r)sin 
  2 
  <£' 
  = 
  ?(W)sin 
  2 
  <£. 
  

  

  If 
  the 
  relation 
  between 
  f 
  and 
  <£' 
  will 
  take 
  the 
  form 
  

  

  ^(l-^sin^^Tnf' 
  + 
  n, 
  

  

  then 
  it 
  must 
  be 
  possible 
  to 
  write 
  the 
  relation 
  between 
  f 
  and 
  <£ 
  

   as 
  follows 
  : 
  

  

  r(l-f)sin 
  2 
  </) 
  = 
  m{fcos 
  2 
  ^+(l-f)sin 
  2 
  f-2^f(l-^)cos(/)sin^cosf}4-w. 
  

   After 
  reduction. 
  

  

  ?(l-fjcos 
  2 
  <^-??isin2>|rA/5'(l^f)cos<^ 
  + 
  f 
  2 
  + 
  (mcos2f-l)f+msin 
  2 
  i/r-fn 
  = 
  a 
  

   Now 
  forp= 
  — 
  1 
  the 
  relation 
  between 
  £ 
  and 
  (f> 
  runs 
  

  

  r(l-Dcos 
  2 
  ^ 
  + 
  Zv/r(l-?)cos0-(-? 
  2 
  +^+r) 
  = 
  O. 
  

  

  