﻿about 
  a 
  Position 
  of 
  Equilibrium, 
  321 
  

  

  osculating 
  "curves 
  themselves 
  we 
  can 
  call 
  Lissajous 
  twisted 
  

   curves. 
  

  

  Such 
  a 
  twisted 
  curve 
  remains 
  enclosed 
  inside 
  a 
  rectangular 
  

   parallelopiped 
  bounded 
  by 
  the 
  planes 
  

  

  ,*=±^S, 
  y 
  =±^ 
  z=±^E. 
  

  

  n 
  x 
  % 
  n 
  z 
  

  

  In 
  consequence 
  o£ 
  the 
  variability 
  of 
  the 
  a's 
  this 
  parallelo- 
  

   piped 
  varies 
  continually.. 
  According 
  to 
  (10) 
  (p. 
  285) 
  the 
  

   following 
  relations 
  exist 
  between 
  the 
  a's 
  : 
  

  

  — 
  — 
  = 
  constant, 
  

  

  P 
  n 
  x 
  <l 
  n 
  y 
  

  

  * 
  x 
  + 
  *y 
  + 
  * 
  z 
  — 
  constant. 
  

  

  Therefore 
  the 
  vertices 
  of 
  the 
  circumscribed 
  parallelopiped 
  

   move 
  along 
  a 
  twisted 
  curve, 
  situated 
  on 
  an 
  ellipsoid, 
  whose 
  

   axes 
  lying 
  on 
  the 
  axes 
  of 
  coordinates 
  are 
  in 
  the 
  ratio 
  

  

  1.1.1 
  

  

  n 
  x 
  ' 
  n 
  y 
  ' 
  n 
  z 
  ' 
  

  

  This 
  curve 
  projects 
  itself 
  on 
  the 
  planes 
  of 
  coordinates 
  as 
  an 
  

   ellipse 
  or 
  as 
  an 
  hyperbola 
  ; 
  e. 
  g. 
  the 
  XY 
  projection 
  is 
  an 
  

   ellipse 
  if 
  p 
  and 
  ^are 
  of 
  different 
  sign, 
  an 
  hyperbola 
  if 
  p 
  and 
  q 
  

   are 
  of 
  the 
  same 
  sign 
  ; 
  it 
  is 
  clear 
  that 
  two 
  of 
  the 
  projections 
  

   are 
  ellipses, 
  the 
  third 
  is 
  an 
  hyperbola. 
  

  

  As 
  the 
  a's 
  change 
  periodically 
  between 
  definite 
  limits, 
  the 
  

   vertices 
  will 
  move 
  to 
  and 
  fro 
  between 
  two 
  extreme 
  positions. 
  

  

  § 
  55. 
  Besides 
  on 
  the 
  a's 
  the 
  form 
  of 
  an 
  osculating 
  curve 
  

   depends 
  on 
  the 
  /3 
  ; 
  s. 
  However, 
  for 
  an 
  osculating 
  curve 
  

   described 
  in 
  a 
  definite 
  parallelopiped 
  it 
  depends 
  not 
  on 
  3, 
  

   but 
  only 
  on 
  2 
  quantities, 
  as 
  is 
  evident 
  when 
  we 
  change 
  the 
  

   origin 
  of 
  time. 
  We 
  can 
  get 
  : 
  

  

  ^ 
  

  

  cos 
  

  

  y= 
  

  

  l 
  y 
  

  

  {n 
  x 
  t 
  + 
  2n 
  v 
  {l3 
  x 
  -t3 
  z 
  Y 
  

   n/ 
  'a. 
  

  

  cos 
  nj. 
  

  

  The 
  form 
  of 
  the 
  osculating 
  curve 
  evidently 
  depends 
  on 
  the 
  

  

  