﻿582 
  Dr. 
  H. 
  Bateman 
  on 
  

  

  position 
  of 
  the 
  gun 
  and 
  lie 
  on 
  a 
  radius 
  through 
  this 
  position. 
  

   The 
  magnitudes 
  of 
  the 
  vectors, 
  however, 
  vary 
  inversely 
  as 
  

   the 
  distance 
  from 
  the 
  gun, 
  and 
  so 
  the 
  energy 
  per 
  unit 
  volume 
  

   obeys 
  the 
  inverse-square 
  law 
  as 
  far 
  as 
  points 
  on 
  the 
  same 
  

   radius 
  are 
  concerned. 
  It 
  may 
  be 
  verified 
  after 
  some 
  laborious 
  

   algebra 
  that 
  the 
  electric 
  and 
  magnetic 
  vectors 
  are 
  equal 
  in 
  

   magnitude 
  and 
  at 
  right 
  angles 
  to 
  one 
  another 
  *. 
  

  

  § 
  4. 
  To 
  obtain 
  more 
  convenient 
  expressions 
  for 
  the 
  com- 
  

   ponents 
  of 
  the 
  electric 
  and 
  magnetic 
  vectors, 
  we 
  write 
  

  

  #— 
  f=«(*— 
  t), 
  y— 
  f 
  n- 
  , 
  K 
  t 
  - 
  T 
  )\ 
  z-?=7(*-Ti)i> 
  

   and 
  regard 
  «, 
  /3, 
  7 
  as 
  constants. 
  We 
  may 
  then 
  write 
  

  

  TT 
  --R- 
  1 
  d 
  l 
  ~ 
  U 
  

  

  x 
  ~ 
  Mdrla 
  + 
  mfi 
  + 
  ny-l 
  

  

  H 
  =R 
  1 
  d 
  n 
  $~ 
  m 
  l 
  

  

  M.drloc+m/3 
  + 
  ny—l 
  

  

  Now 
  let 
  (A,, 
  fi, 
  v), 
  (\ 
  , 
  a^oj 
  ^0) 
  be 
  the 
  direction-cosines 
  of 
  

   the 
  barrels 
  of 
  the 
  gun 
  at 
  time 
  r. 
  We 
  then 
  have 
  

  

  l\ 
  + 
  mfjb 
  + 
  nv 
  = 
  l, 
  l\ 
  +m/n 
  + 
  nv 
  = 
  l, 
  

  

  E 
  = 
  - 
  1 
  & 
  r 
  V^g 
  , 
  \~ 
  a 
  1 
  

  

  * 
  2M^rUa 
  + 
  /x^+j/7-l 
  i 
  "X 
  a 
  + 
  Mo/3 
  + 
  vo7-lJ 
  

   XT 
  _ 
  1 
  dr 
  v/3—w 
  ?off— 
  w 
  1 
  

  

  To 
  prove 
  this 
  we 
  take 
  the 
  plane 
  containing 
  the 
  two 
  barrels 
  

   of 
  the 
  gun 
  at 
  time 
  r 
  as 
  the 
  plane 
  ?/ 
  = 
  and 
  the 
  tangent 
  to 
  

   G's 
  path 
  as 
  axis 
  of 
  x. 
  Then 
  

  

  ? 
  = 
  v, 
  V 
  = 
  0, 
  ? 
  = 
  0, 
  1=1, 
  m 
  = 
  i\/Q 
  2 
  -iy 
  n 
  = 
  0; 
  

  

  \z=v, 
  //,= 
  0, 
  y 
  = 
  N 
  /(l 
  — 
  v 
  2 
  ) 
  ; 
  

   X 
  = 
  v, 
  ^o 
  = 
  0; 
  V 
  =— 
  x/Cl 
  — 
  V 
  2 
  ) 
  ; 
  

  

  Z 
  — 
  a 
  1 
  — 
  V* 
  , 
  1— 
  va 
  

  

  2Rt— 
  : 
  ^^— 
  — 
  1= 
  7 
  — 
  „. 
  , 
  ;q-77TZZ^\ 
  ~ 
  t 
  ~ 
  

  

  I^H^H^T^T 
  ct-v 
  + 
  ipy/(l-v*) 
  a-t>-t£i/(l-w 
  s 
  ) 
  

  

  V 
  — 
  a 
  

  

  "Xcc 
  + 
  fi/S'+vy—l 
  X 
  a 
  + 
  ^ 
  /3 
  + 
  Vo7— 
  1# 
  

  

  The 
  other 
  identities 
  can 
  be 
  established 
  in 
  the 
  same 
  way. 
  

   * 
  A 
  geometrical 
  proof 
  of 
  this 
  result 
  is 
  given 
  below. 
  

  

  