﻿314 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  (2) 
  The 
  system 
  of 
  ellipses 
  does 
  not 
  contain 
  double 
  curves. 
  

   The 
  enveloping 
  conic 
  sections 
  are 
  ellipses, 
  one 
  o£ 
  them 
  being 
  

   the 
  outward 
  envelope, 
  the 
  other 
  the 
  inward 
  envelope. 
  

  

  When 
  the 
  two 
  values 
  of 
  C 
  are 
  equal, 
  then 
  the 
  two 
  en- 
  

   veloping 
  ellipses 
  coincide 
  ; 
  we 
  are 
  in 
  the 
  case 
  of 
  the 
  periodic 
  

   motion 
  in 
  an 
  ellipse. 
  

  

  § 
  48. 
  f(g) 
  is 
  a 
  linear 
  function 
  of 
  f. 
  According 
  to 
  § 
  44 
  

   this 
  occurs 
  when 
  

  

  p{4r+P)=q\ 
  

  

  In 
  this 
  case 
  

  

  The 
  relation 
  between 
  f 
  and 
  <£ 
  degenerates 
  into 
  

  

  and 
  v 
  /^T=S)ooB#=-v!>f-(gJn+ll). 
  

  

  These 
  relations 
  are 
  of 
  the 
  form 
  

  

  */?(l 
  — 
  £) 
  cos 
  (f> 
  = 
  m£+ 
  n. 
  

   We 
  have 
  now 
  

  

  According 
  to 
  (34) 
  of 
  p. 
  309 
  for 
  this 
  case 
  -j- 
  = 
  const. 
  

  

  Therefore 
  : 
  The 
  envelope 
  is 
  degenerated 
  into 
  four 
  straight 
  

   lines, 
  which 
  are 
  the 
  sides 
  of 
  a 
  rectangle. 
  

  

  The 
  direction 
  of 
  these 
  sides 
  may 
  be 
  found 
  by 
  means 
  of 
  (34 
  ). 
  

   This 
  equation 
  may 
  be 
  written 
  as 
  follows 
  : 
  

  

  2^ 
  

   das 
  1 
  

  

  dy\> 
  

  

  -0 
  

  

  m 
  

  

  If 
  ,- 
  = 
  tan 
  «, 
  then 
  

  

  dx 
  cot2ft) 
  = 
  m. 
  ...... 
  (39) 
  

  

  The 
  motion 
  of 
  the 
  point 
  takes 
  place 
  exactly 
  as 
  in 
  the 
  

   pereral 
  case 
  where 
  no 
  relation 
  exists. 
  

  

  § 
  49. 
  I 
  — 
  and 
  p^—q. 
  If 
  we 
  suppose 
  n 
  y 
  to 
  be 
  nearly 
  

   equal 
  to 
  n 
  Xi 
  then 
  the 
  lowest 
  point 
  of 
  the 
  surface 
  over 
  which 
  

   the 
  point 
  moves 
  is 
  supposed 
  to 
  be 
  nearly 
  an 
  ombilic. 
  Now 
  

   we 
  will 
  make, 
  moreover, 
  the 
  supposition 
  that 
  the 
  surface 
  has 
  

  

  