﻿788 
  Mr. 
  A. 
  Eagle 
  on 
  the 
  Form 
  of 
  tlie 
  Pulses 
  

  

  quite 
  at 
  random, 
  it 
  follows 
  that 
  the 
  distribution 
  of 
  energy 
  

   in 
  each 
  pulse 
  must 
  be 
  the 
  same 
  as 
  the 
  distribution 
  of 
  

   energy 
  in 
  the 
  total 
  succession 
  of 
  pulses. 
  The 
  distribution 
  

   of 
  energy 
  in 
  the 
  spectrum 
  obviously 
  depends 
  on 
  the 
  shape 
  

   or 
  form 
  of 
  the 
  pulses 
  making 
  it. 
  Lord 
  fiayleigh 
  has 
  shown 
  

   how 
  * 
  the 
  distribution 
  of 
  energy 
  in 
  a 
  pulse 
  of 
  any 
  given 
  

   form 
  could 
  be 
  calculated, 
  and 
  calculates 
  the 
  distribution 
  of 
  

   energy 
  for 
  a 
  pulse 
  of 
  the 
  form 
  f{t) 
  = 
  e~'^^^^. 
  Other 
  sug- 
  

   gestions 
  as 
  to 
  the 
  form 
  of 
  the 
  pulse 
  have 
  been 
  made 
  by 
  

   other 
  writers, 
  and 
  the 
  distribution 
  of 
  energy 
  which 
  would 
  

   be 
  obtained 
  from 
  them 
  has 
  been 
  calculated. 
  In 
  no 
  case, 
  

   however, 
  has 
  a 
  form 
  been 
  hit 
  upon 
  which 
  gave 
  a 
  distribution 
  

   in 
  accordance 
  with 
  fact. 
  

  

  The 
  inverse 
  problem 
  — 
  viz., 
  from 
  the 
  distribution 
  of 
  energy, 
  

   to 
  find 
  the 
  form 
  of 
  pulse 
  which 
  would 
  give 
  rise 
  to 
  it 
  — 
  has 
  

   not, 
  as 
  far 
  as 
  I 
  know, 
  been 
  published, 
  and 
  its 
  solution 
  is 
  the 
  

   object 
  of 
  the 
  present 
  paper. 
  We 
  ought, 
  however, 
  to 
  state 
  

   that 
  the 
  problem 
  is 
  not 
  one 
  which 
  admits 
  of 
  a 
  definite 
  

   solution, 
  as 
  the 
  distribution 
  of 
  energy 
  in 
  the 
  spectrum 
  is 
  

   independent 
  of 
  the 
  relative 
  phases 
  of 
  the 
  infinitesimal 
  

   harmonic 
  components 
  out 
  of 
  which 
  the 
  pulse 
  may 
  be 
  con- 
  

   sidered 
  to 
  be 
  built 
  up 
  ; 
  whereas 
  its 
  form 
  must 
  obviously 
  

   depend 
  as 
  much 
  on 
  the 
  relative 
  phases 
  of 
  the 
  components 
  as 
  

   upon 
  their 
  relative 
  amplitudes. 
  

  

  Let 
  7/ 
  = 
  f(x) 
  denote 
  the 
  form 
  of 
  the 
  pulse 
  in 
  space. 
  

   Lord 
  Rayleigh 
  has 
  given 
  t 
  the 
  now 
  well-known 
  relation 
  

  

  }(.r)2 
  dx=:~ 
  f 
  1;A2 
  + 
  B2) 
  da, 
  . 
  . 
  , 
  . 
  (1) 
  

  

  where 
  

  

  A 
  = 
  1 
  f[fji) 
  cos 
  cifji 
  d/jb, 
  

  

  and 
  

  

  [ 
  

  

  B 
  = 
  I 
  /(yLt) 
  sin 
  a/jL 
  d/jb. 
  

  

  «y 
  — 
  00 
  

  

  The 
  left-hand 
  side 
  of 
  (1) 
  is 
  clearly 
  proportional 
  to 
  the 
  

   whole 
  energy 
  of 
  the 
  pulse, 
  and 
  the 
  equation 
  is 
  to 
  be 
  

   interpreted 
  as 
  implying 
  that 
  the 
  energy 
  belonging 
  to 
  the 
  

   waves 
  comprised 
  between 
  a 
  and 
  u-^-doi 
  is 
  proportional 
  to 
  

  

  (A^ 
  + 
  B^) 
  da. 
  The 
  wave-length 
  X 
  is 
  of 
  course 
  — 
  . 
  Hence, 
  

  

  * 
  Phil. 
  Mag. 
  vol. 
  xxvii. 
  p. 
  465 
  (1889), 
  or 
  Collected 
  Works, 
  vol. 
  iii. 
  

   p. 
  268. 
  

   t. 
  Op. 
  cit. 
  

  

  