﻿580 
  Dr. 
  H. 
  Bateman 
  on 
  

  

  the 
  velocity 
  of 
  G, 
  which 
  is 
  supposed 
  to 
  be 
  less 
  than 
  the 
  

   velocity 
  of 
  light 
  (fig. 
  1). 
  

  

  Eiir. 
  1 
  

  

  Cos 
  ex. 
  — 
  V 
  

  

  The 
  electric 
  and 
  magnetic 
  vectors 
  at 
  an 
  arbitrary 
  point 
  

   (as, 
  y, 
  z, 
  t) 
  can 
  be 
  calculated 
  from 
  the 
  state 
  of 
  motion 
  of 
  the 
  

   gun 
  at 
  one 
  particular 
  time 
  r 
  (depending 
  on 
  te 
  9 
  y, 
  z, 
  t), 
  the 
  

   corresponding 
  position 
  of 
  the 
  gun 
  may 
  be 
  called 
  the 
  effective 
  

   position 
  for 
  the 
  point 
  (x, 
  y, 
  z, 
  t). 
  When 
  the 
  barrels 
  of 
  the 
  

   gun 
  do 
  not 
  alter 
  their 
  direction 
  the 
  electric 
  and 
  magnetic 
  

   vectors 
  vanish 
  ; 
  hence 
  the 
  electromagnetic 
  field 
  only 
  comes 
  

   into 
  existence 
  when 
  the 
  directions 
  of 
  the 
  barrels 
  change. 
  

  

  § 
  3. 
  Let 
  the 
  motion 
  of 
  the 
  gun 
  be 
  represented 
  by 
  the 
  

   equations 
  

  

  #=1:0), 
  y=fl/W, 
  2=?(t), 
  

  

  where 
  

  

  F*(T) 
  +!j'»(t)-+ 
  J*(T)<1. 
  

  

  For 
  each 
  point 
  (%, 
  y, 
  z, 
  i) 
  in 
  space 
  there 
  is 
  then 
  just 
  one 
  

   value 
  of 
  t 
  for 
  which 
  

  

  [*-f(T)] 
  2 
  + 
  U,- 
  v 
  ( 
  T) 
  y 
  +[z-^)Y= 
  [t-Tf, 
  er. 
  

  

  (The 
  unit 
  of 
  time 
  is 
  chosen 
  so 
  that 
  the 
  velocity 
  of 
  light 
  is 
  

   represented 
  by 
  unity.) 
  

  

  Now 
  let 
  functions 
  /(r), 
  m{r), 
  n(r) 
  be 
  chosen 
  so 
  that 
  

  

  l 
  2 
  + 
  m 
  2 
  + 
  n 
  2 
  = 
  l, 
  

  

  then 
  if 
  

  

  w^l{x-i)+m(y- 
  v 
  )+n{z-X)-{t-T), 
  

  

  it 
  is 
  easy 
  to 
  verify 
  that 
  when 
  f 
  (t) 
  is 
  an 
  arbitrary 
  function, 
  

  

  to 
  

   is 
  a 
  solution 
  of 
  the 
  wave 
  equation 
  

  

  'dm 
  d 
  2 
  n 
  yg 
  yn 
  

  

  "da? 
  + 
  dy 
  2 
  + 
  d- 
  2 
  ~ 
  dt 
  2 
  ' 
  

  

  