﻿584 
  Br. 
  H. 
  Bateman 
  on 
  

  

  It 
  should 
  first 
  of 
  all 
  be 
  noticed 
  that 
  the 
  line 
  QC 
  meets 
  the 
  

   polar 
  line 
  of 
  AA 
  with 
  regard 
  to 
  the 
  sphere, 
  for 
  if 
  QC 
  meets 
  

   the 
  sphere 
  again 
  in 
  R, 
  the 
  four 
  points 
  QRAA 
  form 
  an 
  

   harmonic 
  set 
  on 
  the 
  circle 
  Q 
  R 
  A 
  A 
  . 
  

  

  If 
  now 
  we 
  regard 
  A 
  and 
  A 
  as 
  fixed, 
  the 
  correspondence 
  

   between 
  Q 
  and 
  R 
  gives 
  a 
  conformal 
  transformation 
  of 
  the 
  

   sphere 
  into 
  itself, 
  and 
  the 
  transformation 
  is 
  of 
  such 
  a 
  nature 
  

   that 
  when 
  we 
  project 
  stereographically 
  on 
  a 
  plane 
  from 
  a 
  

   fixed 
  point 
  V 
  on 
  the 
  sphere 
  and 
  use 
  the 
  complex 
  variable 
  z 
  

   to 
  denote 
  the 
  position 
  of 
  a 
  point 
  in 
  the 
  plane, 
  the 
  affixes 
  

   q, 
  r, 
  a, 
  a 
  of 
  the 
  projections 
  of 
  Q, 
  R, 
  A, 
  A 
  are 
  connected 
  by 
  

   a 
  relation 
  of 
  the 
  form 
  * 
  

  

  qr 
  — 
  i(q 
  + 
  r)(a 
  + 
  ao) 
  + 
  aa 
  =0. 
  

  

  Again, 
  if 
  A 
  ', 
  A 
  7 
  , 
  (V 
  denote 
  the 
  displaced 
  positions 
  of 
  

   A 
  , 
  A, 
  C 
  and 
  QC' 
  meets 
  the 
  sphere 
  again 
  in 
  S, 
  the 
  affixes 
  of 
  

   the 
  stereographic 
  projections 
  of 
  Q, 
  A 
  ', 
  A', 
  S 
  are 
  connected 
  

   by 
  the 
  relation 
  

  

  qs-i(q 
  + 
  s)(a' 
  + 
  a 
  ') 
  + 
  a'a 
  ' 
  = 
  0. 
  

  

  Now 
  the 
  electric 
  vector 
  at 
  Q 
  lies 
  in 
  the 
  plane 
  QCC 
  and 
  

   so 
  is 
  in 
  the 
  direction 
  of 
  the 
  tangent 
  at 
  Q 
  to 
  the 
  circle 
  QRS 
  on 
  

   the 
  sphere. 
  Using 
  the 
  same 
  letters 
  to 
  denote 
  the 
  stereo- 
  

   graphic 
  projections 
  of 
  the 
  points 
  Q, 
  R, 
  S, 
  A, 
  A 
  &c, 
  we 
  find 
  

   that 
  the 
  stereographic 
  projections 
  of 
  the 
  lines 
  of 
  electric 
  force 
  

   are 
  such 
  that 
  the 
  line 
  through 
  Q 
  is 
  tangent 
  to 
  the 
  circle 
  

   QRS. 
  Let 
  q 
  + 
  dq 
  denote 
  the 
  complex 
  affix 
  of 
  a 
  consecutive 
  

   point 
  on 
  the 
  line 
  through 
  Q, 
  then, 
  since 
  the 
  cross 
  ratio 
  of 
  the 
  

   affixes 
  of 
  four 
  concyclic 
  points 
  is 
  real, 
  the 
  cross 
  ratio 
  of 
  

   q, 
  q 
  + 
  dq, 
  r, 
  s 
  is 
  real: 
  

  

  dq\ 
  is 
  real, 
  

  

  *Lq 
  — 
  r 
  q 
  — 
  sj 
  

  

  i 
  e 
  dq\ 
  g-*fo 
  + 
  tto) 
  _ 
  j-JrO' 
  + 
  O 
  1 
  is 
  real 
  

  

  Uq-a){q-ao) 
  {q-a 
  ! 
  ){q-<)J 
  

  

  Hence, 
  if 
  , 
  (q—a)(q 
  — 
  a 
  ) 
  , 
  , 
  . 
  , 
  

  

  *(q-a')(q-a 
  ) 
  

  

  the 
  curves 
  yjr 
  = 
  constant 
  are 
  the 
  projections 
  of 
  the 
  electric 
  

   lines 
  of 
  force, 
  and 
  it 
  follows 
  that 
  the 
  curves 
  <£ 
  = 
  constant 
  are 
  

   the 
  projections 
  of 
  the 
  magnetic 
  lines 
  of 
  force. 
  The 
  curves 
  

  

  * 
  This 
  is 
  an 
  immediate 
  consequence 
  of 
  the 
  result 
  given 
  on 
  p. 
  33 
  of 
  

   Harkness 
  and 
  Morley's 
  ' 
  Introduction 
  to 
  the 
  Theory 
  of 
  Analytic 
  

   Functions' 
  (London, 
  1898). 
  

  

  