﻿Relativity 
  and 
  Electrodynamics. 
  333 
  

  

  m 
  

   in 
  order 
  by 
  — 
  l 
  (x 
  + 
  u), 
  etc., 
  and 
  add 
  and 
  use 
  (1). 
  Then 
  we 
  

  

  find 
  

  

  m 
  2 
  m 
  2 
  

  

  or 
  

  

  *= 
  (m2 
  ~ 
  mi) 
  S/V 
  2 
  where 
  V 
  = 
  (x 
  + 
  uf 
  + 
  (y 
  + 
  v 
  y 
  + 
  {k 
  + 
  w)K 
  

   Hence 
  the 
  Cartesian 
  equations 
  of 
  motion 
  are 
  

  

  m 
  iy 
  = 
  Y+ 
  {m 
  *- 
  m 
  > 
  ] 
  ( 
  y 
  + 
  v) 
  {X(* 
  + 
  u) 
  + 
  Y(y 
  + 
  v) 
  + 
  Z(h 
  + 
  W 
  )}, 
  

  

  m 
  2 
  i 
  = 
  Z+ 
  (m2 
  ~ 
  mi){z 
  + 
  ™' 
  ) 
  {X( 
  ! 
  h 
  + 
  u-) 
  + 
  Y(y 
  + 
  v) 
  + 
  Z(i+w)}. 
  

  

  Also 
  for 
  a 
  central 
  attraction 
  

  

  X=— 
  fixjr 
  1 
  , 
  Y=—/jLy/r*, 
  Z=—/jbz/r 
  z 
  . 
  

  

  From 
  these 
  it 
  appears 
  that 
  a 
  planet's 
  orbit 
  cannot 
  remain 
  

   a 
  plane 
  orbit 
  during 
  transference 
  through 
  the 
  aether, 
  except 
  

   in 
  the 
  special 
  case 
  when 
  the 
  direction 
  of 
  the 
  sun's 
  motion 
  is 
  

   in 
  and 
  remains 
  in 
  the 
  plane 
  of 
  the 
  orbit. 
  

  

  We 
  can 
  readily 
  transform 
  the 
  equations 
  to 
  polars 
  since 
  

   the 
  disturbance 
  is 
  along 
  the 
  direction 
  of 
  the 
  resultant 
  

   velocity 
  in 
  space. 
  

  

  For 
  the 
  example 
  under 
  recent 
  discussion 
  by 
  Sir 
  Oliver 
  

   Lodge 
  and 
  as 
  a 
  first 
  approximation, 
  we 
  might 
  take 
  

  

  mi 
  = 
  m 
  (l 
  + 
  aY 
  2 
  /<?), 
  m 
  2 
  = 
  m 
  (l 
  + 
  6V 
  2 
  /c 
  2 
  ), 
  

  

  and 
  neglecting 
  squares 
  of 
  V 
  2 
  /c 
  2 
  , 
  treat 
  the 
  problem 
  as 
  one 
  of 
  

   a 
  disturbed 
  central 
  orbit. 
  

  

  Astronomers 
  doubtless 
  know 
  the 
  best 
  mode 
  of 
  dealing 
  

   with 
  this, 
  and 
  could 
  obtain 
  a 
  solution 
  without 
  difficulty. 
  

  

  It 
  is 
  important 
  to 
  remember 
  that 
  these 
  equations 
  are 
  first 
  

   approximations 
  only 
  to 
  the 
  general 
  equations 
  connecting 
  

   moving 
  material 
  systems 
  with 
  the 
  state 
  of 
  the 
  aether. 
  

  

  A 
  special 
  case 
  amenable 
  to 
  elementary 
  treatment 
  is 
  that 
  

  

  