﻿of 
  Atoms 
  and 
  Molecules, 
  865 
  

  

  corresponding 
  to 
  vibrations 
  perpendicular 
  to 
  the 
  plane 
  of 
  

   the 
  ring 
  is 
  more 
  than 
  three 
  times 
  as 
  great 
  as 
  the 
  frequency 
  

   in 
  question, 
  and 
  consequently 
  of 
  negligible 
  influence 
  on 
  the 
  

   dispersion. 
  

  

  In 
  order 
  to 
  determine 
  the 
  frequency 
  of 
  vibration 
  of 
  the 
  

   system 
  corresponding 
  to 
  displacement 
  of 
  the 
  nuclei 
  relative 
  

   to 
  each 
  other, 
  let 
  us 
  consider 
  a 
  configuration 
  in 
  which 
  the 
  

   radius 
  of 
  the 
  ring 
  is 
  equal 
  to 
  y, 
  and 
  the 
  distance 
  apart 
  of 
  

   the 
  nuclei 
  2x. 
  The 
  radial 
  force 
  acting 
  on 
  one 
  of 
  the 
  

   electrons 
  and 
  due 
  to 
  the 
  attraction 
  from 
  the 
  nuclei 
  and 
  the 
  

   repulsion 
  from 
  the 
  other 
  electron 
  is 
  

  

  R 
  = 
  2 
  ' 
  2// 
  

  

  Let 
  us 
  now 
  consider 
  a 
  slow 
  displacement 
  of 
  the 
  system 
  

  

  during 
  which 
  the 
  radial 
  force 
  balances 
  the 
  centrifugal 
  force 
  

  

  due 
  to 
  the 
  rotation 
  of 
  the 
  electrons, 
  and 
  the 
  angular 
  momen- 
  

  

  e 
  2 
  

   turn 
  of 
  the 
  latter 
  remains 
  constant. 
  Putting 
  R 
  = 
  — 
  F, 
  we 
  

  

  y 
  

  

  have 
  seen 
  on 
  p. 
  859 
  that 
  the 
  radius 
  of 
  the 
  ring 
  is 
  inversely 
  

   proportional 
  to 
  F. 
  Therefore, 
  during 
  the 
  displacement 
  con- 
  

   sidered, 
  Ry 
  3 
  remains 
  constant. 
  This 
  gives 
  by 
  differentiation 
  

  

  ^if^rZ2ifx 
  2 
  -{x 
  2 
  + 
  y 
  2 
  f)dy-2^xyhJx 
  = 
  0. 
  

   Introducing 
  x 
  = 
  b 
  and 
  y 
  = 
  a, 
  we 
  get 
  

  

  $ 
  = 
  -¥-* 
  0-834. 
  

  

  dx 
  21V3-4 
  

  

  The 
  force 
  acting 
  on 
  one 
  of 
  the 
  nuclei 
  due 
  to 
  the 
  attraction 
  

   from 
  the 
  ring 
  and 
  the 
  repulsion 
  from 
  the 
  other 
  nucleus 
  is 
  

  

  2e 
  2 
  x 
  e 
  2 
  

  

  Q 
  = 
  

  

  (x 
  2 
  +y 
  2 
  )% 
  4*T 
  

  

  For 
  x 
  = 
  b, 
  y=a 
  this 
  force 
  is 
  equal 
  to 
  0. 
  

  

  Corresponding 
  to 
  a 
  small 
  displacement 
  of 
  the 
  system 
  for 
  

  

  which 
  x 
  = 
  a+Sx 
  we 
  get, 
  using 
  the 
  above 
  value 
  for 
  — 
  and 
  

  

  dx 
  

  

  putting 
  Q 
  = 
  — 
  g 
  H&r, 
  

  

  f.V 
  

  

  H 
  = 
  S(V3-2) 
  ~ 
  1-515. 
  

  

  For 
  the 
  frequency 
  of 
  vibration 
  corresponding 
  to 
  the 
  dis- 
  

   placement 
  in 
  question 
  we 
  get, 
  denoting 
  the 
  mass 
  of 
  one 
  of 
  

   Phil 
  Mag. 
  S. 
  6. 
  Vol. 
  26. 
  No. 
  155. 
  Nov. 
  1913. 
  3 
  Bf 
  

  

  