﻿302 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  tJie 
  Oscillations 
  

  

  So 
  the 
  first 
  equation 
  becomes 
  : 
  

  

  Or, 
  after 
  integration 
  : 
  

  

  The 
  envelope 
  has 
  degenerated 
  into 
  three 
  parabolse, 
  having 
  

   the 
  same 
  axis 
  ; 
  according 
  to 
  § 
  31 
  (p. 
  296) 
  they 
  intersect 
  (in 
  

   the 
  ends 
  of 
  the 
  double 
  curves) 
  under 
  right 
  angles, 
  so 
  they 
  

   are 
  confocal. 
  

  

  In 
  order 
  to 
  determine 
  the 
  value 
  of 
  the 
  constants 
  C 
  we 
  

   have 
  but 
  to 
  recollect 
  that 
  one 
  of 
  the 
  parabolse 
  passes 
  through 
  

  

  the 
  points 
  ( 
  V£i» 
  \ 
  Vl 
  — 
  £i) 
  and 
  ( 
  Vfa* 
  \ 
  Vl 
  — 
  £2)* 
  Let 
  it 
  

   be 
  the 
  parabola 
  

  

  

  ^(V-CV 
  

  

  Then 
  we 
  have 
  : 
  

  

  

  

  ivwx-cji-Cb 
  

  

  

  i^J=S=06-C 
  1 
  . 
  

  

  Therefore 
  

  

  

  

  c 
  _ 
  Vl-Si— 
  Vl— 
  & 
  

  

  2(fe-&) 
  

  

  The 
  equation 
  from 
  which 
  fj 
  and 
  f 
  2 
  have 
  been 
  found 
  runs 
  

  

  {*(l-S)-(*Vtf=° 
  (25) 
  

  

  From 
  this 
  we 
  may 
  deduce 
  : 
  

  

  

  Vi- 
  

  

  -a- 
  

  

  

  , 
  Vi- 
  

  

  ■6= 
  

  

  = 
  5 
  + 
  ' 
  ' 
  

  

  

  Then 
  

  

  we 
  get 
  

  

  * 
  

  

  k 
  

  

  

  

  

  

  

  C- 
  

  

  . 
  6" 
  

  

  -6)~ 
  

  

  1 
  h 
  

   »6& 
  

  

  = 
  - 
  

  

  1 
  £\ 
  

  

  2 
  $i&X" 
  

  

  

  The 
  product 
  

  

  of 
  th< 
  

  

  3 
  roots 
  of 
  the 
  cubic 
  

  

  equation 
  

  

  being 
  — 
  P, 
  

  

  

  

  

  c= 
  

  

  X 
  

  

  2k' 
  

  

  

  

  

  Now 
  

  

  we 
  mus 
  

  

  t 
  decide, 
  whether 
  

  

  

  

  

  

  o, 
  

  

  

  

  or 
  

  

  —X* 
  

  

  

  