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  XL 
  VII. 
  Corpuscular 
  Radiation. 
  By 
  H. 
  Bateman, 
  M.A., 
  

   Ph.D., 
  Teaching 
  and 
  Research 
  Scholar, 
  Johns 
  Hopkins 
  

   University 
  *. 
  

  

  § 
  1. 
  TN 
  the 
  mathematical 
  theory 
  of 
  electromagnetism 
  it 
  is 
  

   JL 
  customary 
  to 
  adopt 
  the 
  hypothesis 
  that 
  an 
  electro- 
  

   magnetic 
  field 
  which 
  occurs 
  in 
  nature 
  can 
  be 
  obtained 
  by 
  

   superposing 
  a 
  number 
  of 
  elementary 
  electromagnetic 
  fields, 
  

   each 
  of 
  which 
  has 
  a 
  simple 
  real 
  point 
  singularity 
  moving 
  

   through 
  the 
  aether 
  with 
  a 
  velocity 
  less 
  than 
  that 
  of 
  light. 
  

   The 
  solution 
  of 
  Maxwell's 
  equations 
  which 
  corresponds 
  to 
  

   this 
  elementary 
  type 
  of 
  field 
  has 
  been 
  given 
  by 
  Lienard 
  t 
  

   and 
  Wiechert 
  J 
  ; 
  it 
  is 
  found 
  that 
  a 
  constant 
  (positive 
  or 
  

   negative) 
  charge 
  is 
  associated 
  with 
  the 
  singularity, 
  and 
  on 
  

   this 
  account 
  the 
  elementary 
  field 
  is 
  sometimes 
  regarded 
  as 
  a 
  

   model 
  of 
  that 
  due 
  to 
  a 
  moving 
  electron 
  : 
  better 
  results, 
  how- 
  

   ever, 
  have 
  been 
  obtained 
  by 
  regarding 
  the 
  electron 
  as 
  an 
  

   aggregate 
  of 
  such 
  point 
  charges 
  distributed 
  throughout 
  a 
  

   small 
  volume. 
  

  

  It 
  is 
  implicitly 
  assumed 
  that 
  electromagnetic 
  fields 
  with 
  

   other 
  types 
  of 
  singularities 
  are 
  non-existent. 
  This 
  hypothesis, 
  

   however, 
  is 
  of 
  such 
  a 
  sweeping 
  nature 
  that 
  it 
  ought 
  only 
  to 
  

   be 
  adopted 
  after 
  a 
  careful 
  study 
  of 
  the 
  different 
  tvpes 
  of 
  

   singularities 
  which 
  solutions 
  of 
  Maxwell's 
  equations 
  can 
  

   possess. 
  After 
  examining 
  a 
  few 
  of 
  these 
  I 
  have 
  come 
  across 
  

   one 
  type 
  of 
  electromagnetic 
  field 
  which 
  may 
  perhaps 
  be 
  of 
  

   some 
  physical 
  interest 
  as 
  its 
  chief 
  characteristic 
  is 
  a 
  cor- 
  

   puscular 
  type 
  of 
  radiation. 
  The 
  brief 
  description 
  which 
  has 
  

   been 
  given 
  elsewhere 
  § 
  will 
  now 
  be 
  amplified 
  by 
  a 
  geometrical 
  

   construction 
  for 
  the 
  electric 
  and 
  magnetic 
  vectors 
  at 
  an 
  

   arbitrary 
  point 
  of 
  space. 
  

  

  § 
  2. 
  The 
  primary 
  singularity 
  may 
  be 
  described 
  as 
  a 
  

   double-barreled 
  gun 
  G 
  which 
  moves 
  through 
  the 
  aether 
  in 
  

   an 
  arbitrary 
  manner 
  and 
  keeps 
  up 
  a 
  perpetual 
  fire 
  of 
  

   secondary 
  singularities 
  (bullets) 
  || 
  which 
  move 
  with 
  the 
  speed 
  

   of 
  light 
  along 
  straight 
  lines 
  in 
  the 
  direction 
  of 
  the 
  barrels 
  of 
  

   the 
  gun. 
  These 
  directions 
  vary 
  with 
  the 
  velocity 
  of 
  Gr 
  in 
  

   such 
  a 
  way 
  that 
  the 
  angle 
  between 
  them 
  is 
  always 
  bisected 
  

   by 
  the 
  direction 
  of 
  G's 
  motion 
  and 
  its 
  magnitude 
  depends 
  on 
  

  

  * 
  Communicated 
  by 
  the 
  Author, 
  

   t 
  Leclairage 
  electrique, 
  pp. 
  5, 
  53, 
  106 
  (1898). 
  

   £ 
  Archives 
  neerlandaises 
  (2) 
  v. 
  p. 
  54 
  (1900). 
  

   § 
  The 
  Annals 
  of 
  Mathematics, 
  Dec. 
  1913. 
  

  

  || 
  As 
  far 
  as 
  I 
  can 
  make 
  out 
  there 
  is 
  no 
  total 
  charge 
  associated 
  with 
  

   either 
  the 
  gun 
  or 
  the 
  bullets. 
  

  

  