﻿(32) 
  

  

  about 
  a 
  Position 
  of 
  Equilibrium. 
  309 
  

  

  axis, 
  and 
  if 
  9 
  is 
  the 
  angle 
  we 
  have 
  in 
  view, 
  then 
  these 
  

   relations 
  become 
  

  

  1_ 
  1 
  1 
  ] 
  

  

  A 
  2 
  + 
  B 
  2 
  -f(W)sin 
  2 
  c£' 
  

  

  1 
  1 
  

  

  A 
  2 
  B 
  2 
  ~£(l-£)sin 
  2 
  </>' 
  f* 
  • 
  

  

  tan2 
  , 
  = 
  ^(wT 
  cos 
  ^| 
  

  

  1 
  — 
  iL, 
  

  

  From 
  the 
  first 
  and 
  the 
  second 
  of 
  these 
  relations 
  we 
  deduce 
  

   A 
  2 
  + 
  B 
  2 
  =l. 
  The 
  vertices 
  of 
  an 
  arbitrary 
  rectangle 
  circum- 
  

   scribed 
  to 
  any 
  one 
  of 
  the 
  ellipses 
  lie 
  on 
  the 
  circumference 
  of 
  

   a 
  circle 
  with 
  unity 
  as 
  radius. 
  

  

  In 
  a 
  system 
  of 
  ellipses 
  there 
  are 
  two 
  double 
  curves, 
  or 
  

   there 
  are 
  no 
  double 
  curves. 
  

  

  § 
  42. 
  Envelope. 
  (22) 
  of 
  p. 
  298 
  becomes 
  for 
  this 
  case 
  

  

  (1-f) 
  sin 
  2 
  (£-<£) 
  -f 
  sin 
  2 
  £ 
  = 
  2 
  •£(! 
  -?;/'(£) 
  sin 
  (*-<£) 
  sin 
  f. 
  (33) 
  

   We 
  may 
  write 
  it 
  in 
  this 
  form 
  : 
  

  

  fsin 
  2 
  * 
  - 
  - 
  /ig 
  Vfsin^ 
  

  

  According 
  to 
  (21, 
  § 
  31) 
  we 
  have 
  

  

  dy 
  __ 
  Vl-£sin(£ 
  — 
  0) 
  

  

  <&e~ 
  Vfsini 
  

  

  Therefore 
  

  

  (l) 
  2 
  - 
  2 
  /'®i-l=0. 
  . 
  . 
  . 
  (34) 
  

  

  This 
  quadratic 
  equation 
  in 
  -p 
  gives 
  the 
  direction 
  of 
  the 
  

   envelope 
  in 
  those 
  points 
  in 
  which 
  it 
  envelopes 
  a 
  same 
  

   ellipse, 
  defined 
  by 
  the 
  value 
  of 
  f. 
  To 
  each 
  value 
  of 
  ~~ 
  

  

  (XX 
  

  

  two 
  points 
  of 
  the 
  ellipse 
  belong. 
  The 
  product 
  of 
  the 
  roots 
  of 
  

   the 
  equation 
  being 
  —1, 
  we 
  can 
  say 
  that 
  the 
  tangential 
  lines 
  

   to 
  the 
  ellipse 
  in 
  the 
  points 
  in 
  which 
  it 
  is 
  enveloped 
  form 
  a 
  

   rectangle. 
  

  

  § 
  43. 
  We 
  have 
  found 
  already 
  (§ 
  20) 
  that 
  in 
  the 
  case 
  7 
  = 
  1 
  

  

  -j 
  may 
  become 
  zero 
  not 
  only 
  for 
  sin 
  = 
  0, 
  but 
  also 
  for 
  

  

  cos 
  <f> 
  = 
  , 
  

  

  9 
  2^(1- 
  J) 
  • 
  

  

  