﻿322 
  Mr, 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  quantities 
  fi 
  x 
  —py 
  and 
  8 
  y 
  — 
  iS 
  z 
  . 
  So 
  if 
  we 
  put 
  

  

  n, 
  Wy 
  n, 
  

  

  then 
  we 
  find 
  : 
  

  

  # 
  = 
  A 
  cos 
  (w,^ 
  + 
  a) 
  , 
  "^ 
  

  

  ^=Bcos(w 
  y 
  < 
  + 
  6), 
  f 
  ...... 
  (41) 
  

  

  z 
  = 
  G 
  cos 
  n 
  z 
  t. 
  J 
  

  

  In 
  the 
  extreme 
  parallelopipeds 
  curves 
  are 
  described 
  for 
  

   which 
  <£ 
  = 
  or 
  it 
  (conf. 
  § 
  24). 
  Therefore 
  

  

  %(p 
  n 
  x 
  Px 
  + 
  Q 
  n 
  yfit/ 
  + 
  rn 
  zfiz) 
  = 
  or 
  tt. 
  

   Since 
  

  

  we 
  may 
  write 
  

  

  Or- 
  

  

  pa 
  + 
  qb 
  = 
  or 
  7r*. 
  

  

  § 
  56. 
  Let 
  us 
  put 
  in 
  (41) 
  t— 
  + 
  Tand 
  t= 
  — 
  t, 
  and 
  let 
  us 
  call 
  

   the 
  values 
  of 
  #, 
  y, 
  and 
  £ 
  belonging 
  to 
  these 
  values 
  of 
  t, 
  

   respectively 
  ^ 
  1? 
  y 
  1? 
  ^ 
  and 
  # 
  2 
  > 
  ^/2 
  5 
  ^2? 
  then 
  we 
  find 
  : 
  

  

  ■!(#! 
  + 
  #2) 
  = 
  A 
  cos 
  a 
  cos 
  rijT, 
  \{y 
  x 
  -f 
  ?/ 
  2 
  ) 
  = 
  B 
  cos 
  6 
  cos 
  n 
  y 
  T, 
  z 
  x 
  = 
  # 
  s 
  . 
  

  

  The 
  curve 
  represented 
  by 
  (41) 
  has 
  therefore 
  with 
  respect 
  to 
  

   directions 
  of 
  chords 
  parallel 
  to 
  the 
  XY-plane 
  as 
  diameter 
  a 
  

   curve 
  represented 
  by 
  the 
  equations 
  : 
  

  

  # 
  3 
  = 
  A.' 
  cos 
  n 
  x 
  t, 
  ^ 
  

  

  y 
  3 
  =B'cos%£, 
  I 
  (42) 
  

  

  z 
  =Ccos?i 
  z 
  £, 
  > 
  

   where 
  

  

  A' 
  = 
  Acosa, 
  B' 
  = 
  Bcos&. 
  

  

  To 
  investigate 
  the 
  curves 
  represented 
  by 
  (41) 
  we 
  can 
  start 
  

   from 
  the 
  curves 
  represented 
  by 
  (42). 
  In 
  fig. 
  54 
  such 
  a 
  

   curve 
  is 
  given 
  perspectively, 
  in 
  fig. 
  55 
  (continuous 
  lines) 
  

  

  * 
  The 
  Lissajous 
  twisted 
  curves 
  have 
  been 
  discussed 
  by 
  A.. 
  Eighi 
  

   (II 
  Nuovo 
  Cimento, 
  vols. 
  ix. 
  & 
  x., 
  1873). 
  

  

  