﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  323 
  

  

  by 
  projections 
  for 
  the 
  simple 
  case 
  that 
  we 
  have 
  n 
  x 
  : 
  n 
  y 
  :n 
  z 
  = 
  

   1:2:3 
  (so 
  we 
  are 
  in 
  the 
  case 
  S 
  = 
  3); 
  the 
  Lissajous 
  twisted 
  

   curve 
  begins 
  and 
  ends 
  in 
  two 
  vertices 
  of 
  the 
  circumscribed 
  

   parallelopiped 
  and 
  is 
  described 
  backwards 
  and 
  forwards. 
  

  

  When 
  a 
  curve 
  (42) 
  is 
  constructed 
  we 
  must 
  bear 
  in 
  mind 
  

   that 
  

  

  x 
  = 
  x 
  d 
  — 
  Bi 
  s'mnj, 
  ?/=y 
  3 
  — 
  S 
  2 
  sinw/, 
  z=z 
  d 
  , 
  

   where 
  

  

  6\ 
  = 
  Asina, 
  S 
  2 
  =Bsin5. 
  

  

  So 
  we 
  can 
  think 
  of 
  the 
  curve 
  (41) 
  as 
  described 
  by 
  a 
  point 
  

   moving 
  along 
  the 
  curve 
  (42) 
  and 
  vibrating 
  at 
  the 
  same 
  time 
  

   according 
  to 
  the 
  X 
  and 
  Y 
  direction. 
  From 
  this 
  we 
  can 
  see 
  

   how 
  the 
  osculating 
  curve 
  changes 
  for 
  increasing 
  values 
  of 
  a 
  

   and 
  b. 
  In 
  fig. 
  55 
  the 
  projections 
  are 
  represented 
  (dotted 
  

   lines) 
  of 
  an 
  osculating 
  curve 
  for 
  n 
  x 
  : 
  n 
  y 
  : 
  n 
  z 
  = 
  l 
  : 
  2 
  : 
  3 
  and 
  

   small 
  values 
  of 
  a 
  and 
  b. 
  

  

  Special 
  cases. 
  — 
  According 
  to 
  § 
  24 
  the 
  periodic 
  case 
  occurs 
  

   only 
  for 
  sin 
  = 
  0. 
  The 
  circumscribed 
  parallelopiped 
  does 
  

   not 
  change. 
  The 
  /3's 
  increase 
  uniformly 
  with 
  the 
  time 
  ; 
  the 
  

   osculating 
  curve 
  changes 
  its 
  form 
  ; 
  however, 
  invariably 
  

   sin 
  = 
  0. 
  The 
  osculating 
  curve 
  is 
  thus 
  represented 
  by 
  (41) 
  ; 
  

   a 
  increases 
  uniformly 
  with 
  the 
  time, 
  and 
  sin 
  (pa 
  + 
  qb) 
  = 
  0. 
  

  

  For 
  n 
  x 
  + 
  ny—n 
  z 
  =p 
  in 
  the 
  asymptotic 
  case 
  the 
  motion 
  

  

  approaches 
  to 
  C 
  2 
  — 
  ?=0 
  or 
  C 
  3 
  -£ 
  = 
  0, 
  <£=^or-^- 
  (§23). 
  

  

  For 
  the 
  other 
  relations 
  in 
  the 
  asymptotic 
  case 
  the 
  motion 
  

   approaches 
  to 
  sin 
  = 
  (§ 
  24). 
  

  

  Mechanism 
  with 
  four 
  degrees 
  of 
  freedom. 
  

  

  § 
  57. 
  The 
  coordinates 
  of 
  the 
  representative 
  point 
  are 
  

   given 
  by 
  : 
  

  

  \A.r 
  

  

  cos(n 
  it 
  * 
  + 
  2n 
  a 
  & 
  c 
  ), 
  

  

  y=^A 
  008(^ 
  + 
  2*^), 
  

  

  n 
  y_ 
  

   z= 
  v^?cos(nJ+2nB), 
  

  

  u= 
  VLf»cos(M 
  + 
  2nA). 
  

  

  . 
  (43) 
  

  

  The 
  point 
  moves 
  in 
  a 
  space 
  R 
  4 
  . 
  An 
  osculating 
  curve 
  may 
  

   be 
  called 
  a 
  Lissajous 
  wrung 
  curve. 
  Such 
  a 
  curve 
  remains 
  

  

  