﻿318 
  Mr, 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  We 
  may 
  ask 
  in 
  what 
  manner 
  the 
  transition 
  takes 
  place 
  from 
  

   fig. 
  39 
  to 
  fig. 
  36. 
  In 
  fig. 
  37 
  the 
  envelope 
  is 
  drawn 
  for 
  a 
  

   small 
  (negative) 
  value 
  of 
  r; 
  there 
  are 
  in 
  the 
  envelope 
  4 
  double 
  

   points 
  and 
  8 
  cusps. 
  In 
  fig. 
  38 
  r 
  has 
  a 
  greater 
  (negative) 
  

   value 
  ; 
  the 
  cusps 
  have 
  moved 
  towards 
  the 
  double 
  points, 
  whilst 
  

   the 
  double 
  points 
  move 
  from 
  0. 
  There 
  will 
  belong 
  to 
  a 
  given 
  

   value 
  of 
  q 
  a 
  certain 
  value 
  of 
  r 
  for 
  which 
  the 
  cusps 
  have 
  

   coincided 
  two 
  by 
  two; 
  then 
  we 
  are 
  in 
  the 
  case 
  of 
  fig. 
  39. 
  In 
  

   order 
  to 
  determine 
  this 
  value 
  of 
  r 
  for 
  which 
  the 
  cusps 
  vanish 
  

   (that 
  is 
  to 
  say, 
  coincide 
  two 
  by 
  two) 
  we 
  turn 
  the 
  axes 
  of 
  

   coordinates 
  through 
  an 
  angle 
  of 
  45° 
  and 
  determine 
  the 
  inter- 
  

   secting 
  points 
  of 
  the 
  envelope 
  with 
  the 
  Y-axis. 
  Then 
  we 
  get 
  

   the 
  points 
  corresponding 
  to 
  cos<£'=0 
  and 
  the 
  double 
  points 
  

   in 
  the 
  Y'-axis. 
  If 
  now 
  the 
  cusps 
  disappear, 
  then 
  for 
  such 
  a 
  

   double 
  point 
  eos<£' 
  = 
  0. 
  af 
  = 
  for 
  ^ 
  = 
  and 
  cos£' 
  = 
  0. 
  By 
  

   substituting 
  cos 
  t' 
  in 
  (33) 
  (p. 
  309) 
  we 
  get 
  

  

  (W) 
  co 
  S 
  v-?=2^nww) 
  cosf 
  =2/(o/'(r). 
  

  

  Now 
  we 
  have 
  : 
  

  

  / 
  2 
  (r)= 
  ? 
  r(i-r 
  , 
  )+^ 
  

   swcrwa-sr)- 
  

  

  Therefore 
  

  

  (i-n 
  cosv-r=s'(i-2n- 
  

  

  Eliminating 
  <£/ 
  from 
  this 
  relation, 
  and 
  

  

  r(i-r)cos 
  2 
  f= 
  ? 
  T(i-r)+/, 
  

  

  we 
  get 
  

  

  If 
  for 
  this 
  value 
  of 
  f 
  cos 
  <£' 
  = 
  (), 
  then 
  

  

  Making 
  use 
  of 
  40 
  (§ 
  49), 
  then 
  we 
  find 
  this 
  value 
  of 
  r 
  : 
  

  

  '•=(<?- 
  1 
  ){(^h?-i}- 
  

  

  § 
  53. 
  Now 
  that 
  we 
  have 
  examined 
  the 
  special 
  cases 
  we 
  shall 
  

   give 
  a 
  review 
  of 
  the 
  cases 
  which 
  may 
  occur 
  for 
  jo= 
  — 
  q 
  and 
  

   Z 
  = 
  0, 
  for 
  different 
  values 
  of 
  q 
  and 
  r. 
  We 
  may 
  for 
  that 
  

   purpose 
  follow 
  the 
  scheme 
  of 
  p. 
  292. 
  

  

  