﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  313 
  

  

  Dividing 
  by 
  Vf 
  sin 
  t, 
  and 
  making 
  use 
  of 
  

  

  d 
  V 
  — 
  s/l—% 
  sin 
  (t 
  — 
  <f>) 
  

  

  o 
  dy 
  , 
  ., 
  , 
  n 
  dy 
  n 
  . 
  

  

  we 
  get 
  x 
  2 
  y^- 
  +x—xy 
  z 
  -\ 
  — 
  y-¥--\ 
  — 
  # 
  = 
  0. 
  

  

  to 
  ° 
  dx 
  m^ 
  ax 
  m 
  

  

  The 
  integral 
  of 
  this 
  differential 
  equation 
  runs 
  

  

  ^+y>-i=0(«»+^\ 
  .... 
  (37) 
  

  

  where 
  C 
  is 
  a 
  constant 
  of 
  integration. 
  

  

  The 
  envelope 
  is 
  degenerated 
  into 
  central 
  conic 
  sections, 
  

   having 
  their 
  axes 
  along 
  the 
  axes 
  of 
  coordinates. 
  

  

  § 
  47. 
  In 
  order 
  to 
  determine 
  the 
  value 
  of 
  we 
  make 
  use 
  of 
  

   the 
  fact 
  that 
  the 
  envelope 
  contains 
  the 
  points 
  of 
  intersection 
  

   of 
  an 
  ellipse 
  corresponding 
  to 
  cos^> 
  = 
  with 
  the 
  axes 
  of 
  co- 
  

   ordinates 
  ((36) 
  of 
  § 
  46 
  is 
  satisfied 
  by 
  cos$ 
  = 
  0, 
  sin£ 
  = 
  

   or 
  +1). 
  The 
  intersecting 
  points 
  of 
  such 
  an 
  ellipse 
  with 
  the 
  

   Y-axis 
  are 
  given 
  by 
  

  

  <*=0, 
  y=±v/W- 
  

  

  cos 
  <j> 
  = 
  for 
  the 
  values 
  o£ 
  f 
  satisfying 
  the 
  equation 
  

  

  Kl-fl-(mff+n)=0. 
  

  

  From 
  these 
  three 
  equations 
  and 
  the 
  equation 
  of 
  the 
  

   envelope 
  

  

  * 
  2 
  +y 
  2 
  -i=c(^+^) 
  

  

  we 
  must 
  eliminate 
  x, 
  y 
  } 
  and 
  f. 
  

   The 
  result 
  is 
  : 
  

  

  C 
  2 
  +-(l-m)C+ 
  — 
  =0. 
  . 
  . 
  . 
  (,38) 
  

  

  This 
  equation 
  being 
  of 
  the 
  second 
  degree, 
  we 
  may 
  say 
  

   that 
  the 
  envelope 
  is 
  degenerated 
  into 
  two 
  conic 
  sections. 
  

  

  Now 
  there 
  are 
  two 
  cases 
  : 
  — 
  

  

  (1) 
  The 
  system 
  of 
  ellipses 
  contains 
  double 
  curves. 
  This 
  

   will 
  occur 
  when 
  sin 
  <f> 
  is 
  zero 
  for 
  values 
  of 
  f 
  between 
  and 
  1. 
  

   Therefore 
  ?nf+n 
  must 
  be 
  zero 
  for 
  a 
  value 
  of 
  f 
  between 
  

   and 
  1. 
  In 
  this 
  case 
  there 
  are 
  two 
  double 
  curves, 
  for 
  the 
  

   system 
  of 
  ellipses 
  is 
  symmetrical 
  in 
  respect 
  to 
  the 
  axes 
  of 
  

   coordinates. 
  In 
  the 
  ends 
  of 
  these 
  double 
  curves 
  the 
  en- 
  

   veloping 
  conic 
  sections 
  intersect 
  under 
  right 
  angles. 
  When 
  

   we 
  examine 
  the 
  values 
  of 
  C 
  it 
  will 
  be 
  clear 
  that 
  one 
  of 
  the 
  

   conic 
  sections 
  is 
  an 
  ellipse, 
  the 
  other 
  an 
  hyperbola. 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  2Q. 
  No. 
  152. 
  Aug. 
  1913. 
  Y 
  

  

  