﻿792 
  Mr. 
  A. 
  Eagle 
  on 
  the 
  Form 
  of 
  the 
  Pulses 
  

  

  It 
  is 
  interesting 
  to 
  observe 
  that 
  all 
  the 
  pulses 
  we 
  have 
  

  

  00 
  

  

  found 
  satisfy 
  the 
  condition 
  [ 
  f(x)d.v=0. 
  This 
  is 
  obvious 
  for 
  

  

  ' 
  —30 
  

  

  the 
  odd 
  pulses 
  ; 
  for 
  the 
  even 
  ones, 
  we 
  have 
  only 
  to 
  show 
  

   thatj7(^)6^^=0. 
  

  

  Now 
  /(.I') 
  consists 
  of 
  one 
  or 
  more 
  terms 
  of 
  the 
  form 
  

  

  I 
  

  

  cxPe 
  "^^ 
  cos 
  ux 
  dot. 
  

  

  

  

  This 
  integrated 
  vvith 
  respect 
  to 
  £0 
  gives 
  

  

  foo 
  r(p)sin|ptan-i^| 
  

  

  \ 
  u^'-' 
  e-"" 
  sin 
  U.V 
  da 
  = 
  —^ 
  . 
  (14) 
  

  

  which 
  vanishes 
  when 
  taken 
  between 
  and 
  go 
  if 
  jo 
  be 
  

   positive. 
  

  

  On 
  the 
  electron 
  theory 
  these 
  pulses 
  are 
  due 
  to 
  the 
  radiation 
  

   from 
  moving 
  electrons, 
  and 
  the 
  function 
  giving 
  the 
  form 
  of 
  

   the 
  pulse 
  as 
  a 
  function 
  of 
  t 
  is 
  the 
  same 
  as 
  the 
  function 
  giving 
  

   the 
  acceration 
  of 
  the 
  electron 
  at 
  time 
  t. 
  Hence, 
  to 
  find 
  the 
  

   displacement 
  at 
  any 
  time, 
  we 
  must 
  integrate 
  the 
  function 
  

   giving 
  the 
  form 
  of 
  the 
  pulse 
  twice. 
  By 
  integrating 
  the 
  left- 
  

   hand 
  side 
  of 
  (14) 
  twice 
  with 
  respect 
  to 
  .v, 
  we 
  observe 
  that 
  we 
  

   merely 
  change 
  jp 
  into 
  p 
  — 
  2. 
  Hence, 
  to 
  integrate 
  the 
  right- 
  

   hand 
  side 
  twice 
  we 
  have 
  only 
  to 
  change 
  p 
  into 
  /> 
  — 
  2. 
  Applying 
  

   this 
  to 
  equations 
  (12) 
  and 
  (13), 
  we 
  get 
  the 
  motion 
  of 
  an 
  

   electron 
  which 
  will 
  give 
  rise 
  to 
  these 
  pulses. 
  

  

  For 
  the 
  pulses 
  ^^2_^^^2y 
  ^nd 
  /ja::^:^' 
  obtained 
  from 
  

  

  Lord 
  Rayleigh's 
  formula, 
  the 
  motion 
  of 
  the 
  electron 
  is 
  

   given 
  by 
  

  

  y 
  = 
  A 
  log 
  (a^ 
  + 
  1^) 
  and 
  ?/ 
  = 
  B 
  tan~^ 
  t/a, 
  

  

  where 
  Ci=^^* 
  

  

  Equations 
  (10) 
  and 
  (11) 
  show 
  that 
  the 
  pulses 
  may 
  be 
  

   regarded 
  as 
  resulting 
  from 
  a 
  superposition 
  of 
  a 
  series 
  of 
  

   sine 
  curves 
  of 
  all 
  wave-lengths, 
  each 
  with 
  a 
  suitable 
  amplitude 
  

   so 
  as 
  to 
  give 
  the 
  required 
  energy 
  distribution. 
  This 
  being 
  

   so, 
  we 
  may 
  expect 
  that 
  we 
  shall 
  be 
  able 
  to 
  obtain 
  a 
  more 
  

   general 
  form 
  of 
  pulse 
  by 
  assigning 
  an 
  arbitrary 
  phase 
  to 
  the 
  

  

  