﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  295 
  

  

  necessary 
  Eor 
  such 
  a 
  curve 
  may 
  be 
  proved 
  as 
  follows 
  : 
  In 
  a 
  

   double 
  curve 
  cos 
  £ 
  and 
  cos 
  (yt 
  — 
  <j>) 
  take 
  the 
  same 
  value 
  for 
  

   two 
  different 
  values 
  of 
  t 
  ; 
  for 
  this 
  value 
  of 
  t 
  sin 
  t 
  and 
  

   sin 
  (7 
  t 
  — 
  (f>) 
  have 
  a 
  different 
  sign. 
  Let 
  ^ 
  be 
  one 
  of 
  the 
  values 
  

   for 
  t, 
  then 
  the 
  other 
  must 
  be 
  t'=^2r 
  1 
  ir 
  — 
  t 
  , 
  and 
  in 
  the 
  same 
  

   time 
  yt' 
  — 
  (f> 
  = 
  2): 
  2 
  7r 
  — 
  yt 
  + 
  <j>, 
  where 
  r 
  x 
  and 
  r. 
  2 
  are 
  integers. 
  

   From 
  this 
  it 
  follows 
  that 
  cp= 
  (^7— 
  r 
  2 
  )7r 
  ; 
  therefore 
  sin 
  (p 
  = 
  0. 
  

   That 
  the 
  condition 
  sin^ 
  = 
  is 
  sufficient 
  for 
  such 
  a 
  curve 
  

   follows 
  directly 
  by 
  substituting 
  ^> 
  — 
  yir 
  in 
  the 
  expression 
  for 
  y. 
  

  

  For 
  cos<£ 
  = 
  the 
  X-axis 
  and 
  Y-axis 
  are 
  lines 
  of 
  symmetry. 
  

  

  Now 
  in 
  § 
  20 
  we 
  have 
  found 
  that 
  in 
  general 
  f 
  varies 
  

   periodically 
  between 
  two 
  extremes 
  ; 
  the 
  vertices 
  of 
  the 
  rect- 
  

   angles, 
  in 
  which 
  the 
  osculating 
  curves 
  are 
  described, 
  move 
  

   to 
  and 
  fro 
  along 
  the 
  ellipse 
  between 
  two 
  limits 
  ; 
  at 
  the 
  same 
  

  

  time 
  <b 
  varies. 
  In 
  the 
  cases 
  Si 
  = 
  3 
  and 
  Si 
  = 
  4 
  . 
  =0 
  for 
  

  

  at 
  

  

  sin 
  <f> 
  = 
  Q 
  (§ 
  20, 
  p. 
  287) 
  ; 
  so 
  in 
  the 
  extreme 
  rectangles 
  Lissajous 
  

   double 
  curves 
  are 
  described. 
  However, 
  in 
  the 
  case 
  Sj 
  = 
  2 
  

  

  ly 
  

  

  = 
  for 
  sin 
  c/> 
  = 
  or 
  (in 
  case 
  / 
  = 
  in 
  § 
  17) 
  for 
  cos<£ 
  = 
  

  

  (§ 
  20, 
  p. 
  287) 
  ; 
  so 
  here 
  it 
  may 
  occur 
  that 
  in 
  one 
  of 
  the 
  

   extreme 
  rectangles 
  or 
  in 
  both 
  a 
  symmetric 
  Lissajous 
  curve 
  is 
  

   described. 
  

  

  We 
  have 
  seen 
  (§ 
  21) 
  that 
  there 
  are 
  two 
  different 
  special 
  

   cases 
  : 
  

  

  1°. 
  The 
  periodic 
  case, 
  f 
  remains 
  constant. 
  In 
  the 
  cases 
  

   7 
  = 
  2 
  and 
  7 
  = 
  3 
  sinc£ 
  = 
  ; 
  the 
  same 
  curve, 
  a 
  Lissajous 
  double 
  

   curve, 
  is 
  continually 
  described. 
  In 
  the 
  case 
  7 
  = 
  1 
  sin 
  $ 
  = 
  () 
  

   or 
  cos 
  (f) 
  = 
  Q 
  ; 
  a 
  double 
  curve 
  or 
  a 
  symmetric 
  curve 
  is 
  con- 
  

   tinually 
  described. 
  

  

  2°. 
  The 
  asymptotic 
  case. 
  To 
  the 
  form 
  of 
  motion, 
  just 
  

   described, 
  the 
  motion 
  approaches 
  asymptotically. 
  

  

  Envelope 
  of 
  the 
  system 
  of 
  Lissajous 
  curves*. 
  

  

  § 
  29. 
  In 
  order 
  to 
  find 
  the 
  envelope 
  of 
  the 
  system 
  of 
  curves 
  

  

  x=Fi(S,<l>,t), 
  y=F 
  2 
  (£;<M), 
  <I>=X(0> 
  

  

  we 
  may 
  regard 
  that 
  envelope 
  as 
  the 
  locus 
  of 
  the 
  intersecting 
  

   points 
  of 
  a 
  certain 
  curve, 
  given 
  by 
  a 
  certain 
  value 
  of 
  £ 
  with 
  

  

  * 
  In 
  my 
  prize 
  essay 
  was 
  only 
  discussed 
  the 
  envelope 
  for 
  system 
  (17, 
  

   27). 
  I'rof. 
  Korteweg, 
  being 
  one 
  of 
  the 
  examiners, 
  proved 
  that 
  some 
  

  

  of 
  my 
  results 
  remained 
  true 
  for 
  the 
  more 
  general 
  system 
  (19). 
  Of 
  his 
  

  

  • 
  >b.-ervations 
  I 
  have 
  made 
  use 
  in 
  the 
  following- 
  pages. 
  

  

  