﻿Dispersion 
  of 
  Light 
  in 
  Space. 
  875 
  

  

  below), 
  the 
  velocity 
  o£ 
  propagation 
  is 
  a 
  function 
  of 
  the 
  

   frequency 
  p/27r, 
  or 
  in 
  other 
  words, 
  it 
  is 
  a 
  function 
  of 
  the 
  

   wave-length. 
  

  

  10. 
  If 
  w 
  is 
  the 
  distance 
  from 
  the 
  rth 
  mass 
  to 
  the 
  (7' 
  + 
  ^)th, 
  

   then 
  q 
  = 
  d'/b 
  and 
  (10) 
  may 
  be 
  written 
  

  

  7/=Acos(pt±0/b,.v)', 
  .... 
  (11) 
  

  

  the 
  suffixes 
  being 
  suppressed. 
  The 
  velocity 
  of 
  propagation 
  

   is 
  thus 
  

  

  y=:pbH3 
  (12) 
  

  

  = 
  pb/2 
  sin-i 
  {irp^{bm/S)} 
  

   =^{hSlm)(l-^p^bm/8...). 
  . 
  . 
  (13) 
  

  

  11. 
  For 
  infinitely 
  great 
  wave-lengths 
  (when 
  p 
  = 
  0) 
  (13) 
  

   reduces 
  to 
  

  

  V 
  =^{bB/m); 
  (14) 
  

  

  and 
  for 
  wave-lengths 
  (X) 
  which 
  are 
  great 
  enough 
  to 
  make 
  

   -^^p^bmjS 
  a 
  small 
  fraction, 
  p 
  is 
  equal 
  to 
  27rY^/X 
  to 
  a 
  first 
  

   approximation. 
  Thus 
  (13) 
  becomes 
  

  

  v=v. 
  (l-7^^^,V6x^.. 
  ) 
  (15) 
  

  

  12. 
  Now 
  let 
  U 
  be 
  the 
  group-velocity* 
  for 
  disturbances 
  of 
  

   wave-length 
  \. 
  Then 
  

  

  V=Y-\dYldX; 
  

   or 
  from 
  (15) 
  

  

  U 
  = 
  Y^(l-7r2672X2...). 
  . 
  . 
  (16) 
  

  

  13. 
  Before 
  any 
  numerical 
  values 
  are 
  inserted 
  in 
  (15) 
  or 
  

   (16) 
  it 
  may 
  be 
  remarked 
  that 
  the 
  law 
  of 
  dispersion 
  thus 
  

   established 
  for 
  a 
  loaded 
  string 
  is 
  readily 
  seen 
  to 
  be 
  applicable 
  

   to 
  a 
  suitably 
  constituted 
  medium 
  having 
  extension 
  in 
  three 
  

   dimensions. 
  We 
  have 
  only 
  to 
  imagine 
  a 
  multitude 
  of 
  equal 
  

   masses 
  arranged 
  in 
  cubical 
  order, 
  and 
  connected 
  by 
  uniformly 
  

   tense 
  massless 
  strings 
  running 
  in 
  three 
  mutually 
  perpen- 
  

   dicular 
  directions. 
  For 
  laminar 
  disturbances 
  propagated 
  in 
  

   any 
  one 
  of 
  these 
  three 
  directions, 
  the 
  mode 
  of 
  propagation 
  

   will 
  be 
  precisely 
  the 
  same 
  as 
  for 
  the 
  single 
  loaded 
  string 
  

   already 
  considered, 
  and 
  there 
  will 
  be 
  the 
  same 
  law 
  of 
  dis- 
  

   persion. 
  In 
  certain 
  other 
  cases 
  — 
  longitudinal 
  vibrations 
  of 
  

   an 
  extensible 
  loaded 
  string, 
  or 
  transverse 
  vibrations 
  of 
  a 
  

   tense 
  chain 
  with 
  elongated, 
  smoothly 
  articulated 
  links 
  — 
  there 
  

  

  * 
  Cf. 
  Rayleigh, 
  'Theory 
  of 
  Sound,' 
  2nd 
  ed. 
  vol. 
  i. 
  Appendix. 
  I 
  

   believe 
  it 
  was 
  first 
  pointed 
  out 
  by 
  Lord 
  Rayleio-h 
  that, 
  if 
  there 
  is 
  any 
  

   dispersion 
  of 
  light 
  in 
  space 
  — 
  that 
  is 
  if 
  there 
  is 
  any 
  difference 
  between 
  

   the 
  waye-velocity 
  and 
  the 
  group-velocity, 
  it 
  is 
  with 
  the 
  group-velocity 
  

   that 
  the 
  tidings 
  of 
  any 
  recognizable 
  astronomical 
  event 
  are 
  transmitted 
  

   to 
  us. 
  

  

  