﻿304 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillat 
  

  

  ions 
  

  

  enveloping 
  parabolas 
  have 
  coincided 
  (fig. 
  14). 
  The 
  X- 
  and 
  

   Y-amplitudes 
  being 
  resp. 
  Vf 
  and 
  J 
  i/-^, 
  they 
  have 
  a 
  ratio 
  

   of 
  2 
  \/2 
  : 
  1. 
  This 
  result 
  was 
  also 
  found 
  in 
  another 
  way 
  by 
  

   Lord 
  Rayleigh 
  *. 
  

  

  The 
  asymptotic 
  form 
  of 
  motion 
  we 
  have 
  for 
  k 
  = 
  0. 
  The 
  

   relation 
  between 
  f 
  and 
  <£ 
  degenerates 
  into 
  f=0, 
  f=l, 
  

   cos 
  (f> 
  = 
  0. 
  In 
  case 
  of 
  the 
  asymptotic 
  form 
  of 
  motion 
  cos<£ 
  — 
  

   invariably 
  ; 
  the 
  motion 
  approaches 
  asymptotically 
  to 
  a 
  

   motion 
  in 
  the 
  YZ-plane 
  (fig. 
  16). 
  The 
  f 
  2 
  enveloping 
  para- 
  

   bola 
  has 
  degenerated 
  into 
  the 
  Y-axis. 
  

  

  § 
  37. 
  Approximate 
  relation. 
  — 
  The 
  focus 
  of 
  the 
  enveloping 
  

   parabolse 
  is 
  at 
  a 
  distance 
  of 
  — 
  ^p' 
  from 
  0. 
  According 
  to 
  

   §22 
  we 
  may 
  for. 
  a 
  certain 
  value 
  of 
  p 
  < 
  1 
  distinguish 
  the 
  

   following 
  forms 
  of 
  motion 
  : 
  — 
  

  

  k>0. 
  All 
  is 
  in 
  main 
  points 
  the 
  same 
  as 
  in 
  the 
  case 
  p' 
  = 
  0. 
  

  

  k 
  has 
  its. 
  maximal 
  value. 
  The 
  periodic 
  form 
  of 
  motion 
  

   corresponding 
  to 
  this 
  value 
  of 
  Jc 
  differs 
  from 
  the 
  periodic 
  

   form 
  of 
  motion 
  we 
  had 
  for 
  p' 
  = 
  only 
  by 
  the 
  value 
  of 
  £. 
  

  

  k 
  = 
  0. 
  The 
  f— 
  cf> 
  relation 
  degenerates 
  into 
  f=0 
  and 
  

   VI— 
  X 
  cos 
  4>=p'. 
  Now 
  according 
  to 
  § 
  34 
  (p. 
  300) 
  

  

  —J 
  \/l 
  — 
  f 
  cos</> 
  

  

  is 
  the 
  distance 
  of 
  the 
  node 
  of 
  an 
  osculating 
  curve 
  from 
  

   the 
  X-axis. 
  Therefore 
  in 
  the 
  asymptotic 
  case 
  all 
  osculating 
  

   curves 
  have 
  their 
  nodes 
  at 
  the 
  same 
  point 
  of 
  the 
  Y-axis, 
  

   at 
  a 
  distance 
  of 
  — 
  |- 
  Vl 
  — 
  fcos 
  (j>— 
  — 
  \p' 
  from 
  the 
  X-axis; 
  

   i. 
  o. 
  w. 
  the 
  nodes 
  are 
  at 
  B. 
  The 
  f 
  2 
  enveloping 
  parabola 
  has 
  

   degenerated 
  into 
  the 
  Y-axis. 
  For 
  k 
  = 
  the 
  Xi 
  an 
  d 
  A, 
  

   parabolse 
  exchange 
  their 
  place 
  ; 
  for 
  a 
  positive 
  value 
  of 
  k 
  the 
  

   X 
  parabola 
  has 
  its 
  opening 
  turned 
  downwards, 
  the 
  £\ 
  para- 
  

   bola 
  upwards 
  ; 
  for 
  a 
  negative 
  value 
  of 
  k 
  the 
  tables 
  are 
  

   turned 
  (iig. 
  17). 
  

  

  —p'<k< 
  0. 
  The 
  nodes 
  of 
  the 
  osculating 
  curves 
  lie 
  on 
  

   both 
  sides 
  of 
  O 
  (fig. 
  18). 
  

  

  k= 
  — 
  p 
  . 
  One 
  of 
  the 
  Lissajous 
  double 
  curves 
  has 
  degene- 
  

   rated 
  into 
  the 
  X-axis 
  ; 
  the 
  f 
  2 
  enveloping 
  parabola 
  passes 
  

   through 
  O 
  ; 
  the 
  nodes 
  of 
  the 
  osculating 
  curves 
  lie 
  above 
  O. 
  

  

  h 
  < 
  — 
  p' 
  . 
  We 
  may 
  get 
  the 
  system 
  of 
  curves 
  and 
  their 
  

   envelope 
  by 
  turning 
  the 
  system 
  we 
  have 
  for 
  k>0 
  over 
  1£0°. 
  

  

  k 
  has 
  its 
  minimal 
  value. 
  Periodic 
  form 
  of 
  motion 
  ; 
  

   <f) 
  — 
  7r 
  invariably. 
  

  

  For 
  p 
  >1 
  the 
  focus 
  of 
  the 
  enveloping 
  parabolaa 
  lies 
  out- 
  

   side 
  the 
  ellipse 
  # 
  2 
  -f-4?/ 
  2 
  =l. 
  According 
  to 
  §22 
  we 
  have 
  

  

  * 
  Phil. 
  Mag. 
  [6] 
  vol. 
  xx. 
  (1010). 
  

  

  