﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  297 
  

  

  In 
  order 
  to 
  determine 
  -/- 
  we 
  have 
  to 
  differentiate 
  (20) 
  

   at 
  

   twice 
  : 
  in 
  the 
  result 
  we 
  substitute 
  sin 
  t 
  = 
  0, 
  cos 
  < 
  = 
  1, 
  sin 
  (/> 
  = 
  (), 
  

   cos 
  </>=l, 
  sm(yt 
  — 
  (j)) 
  = 
  0, 
  GOs(yt—<j>)=l. 
  Then 
  we 
  get 
  : 
  

  

  rt 
  rw(?)§(7- 
  f 
  )-/,(«/,'(«) 
  g 
  +/t(OA(o 
  (7- 
  jf)* 
  (o-o. 
  

  

  Writing 
  7 
  — 
  (7— 
  -,-) 
  instead 
  or* 
  ~, 
  where 
  it 
  appears 
  as 
  a 
  

   \ 
  dt 
  J 
  ^ 
  CM 
  „ 
  

  

  factor, 
  we 
  get 
  a 
  quadratic 
  equation 
  m 
  7— 
  -^ 
  , 
  namely: 
  

  

  +/i(?)/ 
  2 
  (?)f'(0}(7-f)-7/i(?)/ 
  2 
  '(D 
  = 
  0. 
  

  

  Therefore 
  the 
  ends 
  of 
  the 
  double 
  curves 
  are 
  double 
  points 
  

   in 
  the 
  envelope. 
  

  

  When 
  ( 
  d 
  ''\ 
  and 
  ^ 
  are 
  the 
  values 
  of 
  d 
  ;' 
  in 
  the 
  double 
  

   yu'/i 
  yu7 
  2 
  a.6* 
  

  

  point, 
  then 
  

  

  (dy\(dy\ 
  _ 
  Jf(0( 
  _d<f>\( 
  _#\ 
  Y¥ 
  s(tW(? 
  ) 
  

  

  UiiW. 
  ~ 
  7 
  /i 
  2 
  (dv 
  <W>v 
  <W* 
  _ 
  /i(o/i'(o 
  

  

  ~ 
  y 
  d[f{\01~ 
  <*L/i 
  2 
  (?>] 
  

  

  In 
  case 
  

  

  .A 
  2 
  (?)+7 
  s 
  /i 
  2 
  (?) 
  = 
  constant, 
  

  

  is 
  

  

  the 
  product 
  ( 
  ^ 
  ) 
  . 
  (-/{) 
  = 
  — 
  1, 
  &.0i 
  the 
  branches 
  are 
  in 
  th 
  

  

  case 
  rectangular. 
  

  

  As 
  J/i 
  2 
  (fj 
  is 
  the 
  value 
  of 
  the 
  energy 
  of 
  the 
  X-vibration, 
  

   iv72 
  2 
  (?) 
  the 
  value 
  of 
  the 
  energy 
  of 
  the 
  Y-vibration, 
  the 
  con- 
  

   dition 
  implies 
  the 
  conservation 
  of 
  energy. 
  

  

  AVe 
  have 
  found 
  now 
  : 
  — 
  

  

  The 
  ends 
  (meant 
  in 
  a 
  dynamical 
  sense) 
  of 
  the 
  double 
  

   curves 
  of 
  system 
  (19) 
  appear 
  as 
  double 
  points 
  in 
  the 
  envelope. 
  

   If 
  the 
  energy 
  of 
  the 
  motion 
  is 
  constant, 
  then 
  the 
  branches 
  

   in 
  the 
  double 
  points 
  are 
  rectangular. 
  

  

  In 
  case 
  of 
  the 
  motion, 
  indicated 
  by 
  (17, 
  § 
  27), 
  the 
  energy 
  

   is 
  constant 
  ; 
  so 
  for 
  the 
  different 
  systems 
  of 
  curves 
  we 
  shall 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  26. 
  No. 
  152. 
  Aug. 
  1913. 
  X 
  

  

  