﻿Directive 
  System 
  of 
  Wireless 
  Telegraphy. 
  645 
  

  

  It 
  is 
  evident 
  from 
  the 
  figure, 
  that 
  the 
  resultant 
  maximum 
  

   electromagnetic 
  field 
  is 
  directed 
  to 
  one 
  side 
  only 
  ; 
  that 
  it 
  has 
  

   a 
  value 
  double 
  that 
  of 
  the 
  component 
  electromagnetic 
  fields 
  ; 
  

   and 
  that 
  the 
  field 
  on 
  the 
  other 
  side 
  is 
  zero. 
  

  

  The 
  intensity 
  of 
  the 
  electromagnetic 
  field 
  set 
  up 
  by 
  the 
  

   directive 
  system 
  at 
  a 
  point 
  whose 
  direction 
  makes 
  an 
  angle 
  a 
  

   with 
  the 
  maximum 
  radiation, 
  is 
  expressed 
  by 
  C 
  cos 
  a. 
  Let 
  

   M 
  be 
  the 
  intensity 
  of 
  the 
  electromagnetic 
  field 
  produced 
  by 
  

   the 
  circularly 
  radiating 
  system 
  at 
  the 
  same 
  point, 
  and 
  let 
  cf> 
  

   be 
  the 
  phase-difference 
  between 
  the 
  electromagnetic 
  field 
  of 
  

   the 
  circular 
  system 
  and 
  that 
  of 
  one 
  side 
  of 
  the 
  directive 
  

   system. 
  

  

  The 
  intensity 
  of 
  the 
  resultant 
  electromagnetic 
  field 
  at 
  the 
  

   point 
  considered 
  will 
  be 
  

  

  1= 
  \/(M 
  -t- 
  O 
  cos 
  a 
  cos 
  <p) 
  2 
  + 
  C 
  2 
  cos 
  2 
  a 
  sin 
  2 
  (/> 
  

  

  = 
  y/ 
  M 
  2 
  + 
  C 
  2 
  cos 
  2 
  a 
  + 
  2MC 
  cos 
  * 
  cos 
  <j>. 
  

  

  The 
  minimum 
  of 
  I 
  with 
  reference 
  to 
  a 
  is 
  obtained 
  when 
  

  

  M 
  a 
  

   cosa= 
  — 
  -p 
  COS 
  9. 
  

  

  This 
  value 
  of 
  cos 
  a 
  is 
  imaginary 
  when 
  M 
  cos 
  <f> 
  >C. 
  

  

  In 
  the 
  case 
  when 
  M 
  cos 
  0<C, 
  one 
  has 
  I 
  m 
  i 
  n 
  = 
  M 
  sin 
  $ 
  ; 
  when 
  

   M 
  cos 
  (j> 
  > 
  C 
  one 
  has 
  

  

  I 
  min=x 
  /M 
  2 
  + 
  C 
  2 
  cos 
  2 
  «-2M(Jcos 
  cc 
  cos 
  0. 
  

  

  In 
  the 
  special 
  case 
  when 
  (/> 
  = 
  the 
  equation 
  of 
  the 
  resultant 
  

   electromagnetic 
  field 
  transforms 
  itself 
  into 
  

  

  I 
  = 
  M 
  + 
  Ccosa, 
  

  

  which 
  is 
  the 
  equation 
  of 
  a 
  curve 
  that 
  can 
  have 
  three 
  

   different 
  forms 
  according 
  to 
  the 
  value 
  of 
  the 
  ratio 
  M/C. 
  The 
  

   curve 
  represented 
  by 
  the 
  condition 
  M=(J 
  is 
  the 
  cardioid 
  

   above 
  mentioned. 
  

  

  But 
  since 
  in 
  wireless 
  telegraphy 
  the 
  action 
  depends 
  chiefly 
  

   upon 
  the 
  energy, 
  it 
  will 
  be 
  useful 
  to 
  consider 
  this 
  in 
  prefer- 
  

   ence 
  to 
  the 
  intensity 
  of 
  the 
  electromagnetic 
  field. 
  

  

  In 
  the 
  general 
  case 
  the 
  energy 
  radiated 
  in 
  the 
  different 
  

   directions 
  is 
  expressed 
  by 
  the 
  equation 
  — 
  

  

  W 
  = 
  M 
  2 
  + 
  C 
  2 
  COS 
  2 
  a 
  + 
  2MC 
  COS 
  a 
  cos 
  <£, 
  

  

  and 
  in 
  the 
  case 
  of 
  the 
  cardioid 
  by 
  the 
  equation 
  — 
  

  

  W=M 
  2 
  (l 
  + 
  cosa) 
  2 
  , 
  

  

  