﻿Magnetic 
  Shells 
  equivalent 
  to 
  Circular 
  Coils. 
  311 
  

  

  2. 
  The 
  potential 
  V 
  ; 
  of 
  a 
  circular 
  filament 
  concentric 
  and 
  

   coaxal 
  with 
  the 
  coil 
  of 
  radius 
  r 
  and 
  with 
  current 
  nO 
  flowing 
  

   is 
  given 
  by 
  

  

  v 
  '= 
  2 
  - 
  c 
  { 
  1 
  -vfe}' 
  

  

  which 
  expanded 
  in 
  ascending 
  powers 
  of 
  x/r 
  gives 
  

  

  \ 
  r 
  2r 
  d 
  2 
  . 
  4 
  . 
  r° 
  2 
  . 
  4 
  . 
  6 
  . 
  r 
  1 
  ) 
  

  

  3. 
  The 
  two 
  potentials 
  V 
  in 
  (1) 
  and 
  V 
  in 
  (2) 
  will 
  be 
  

   identical 
  provided 
  

  

  1 
  _lf 
  3^-2^ 
  1 
  

   r 
  ~a\ 
  L 
  24a 
  2 
  J 
  ' 
  

  

  IJ/i 
  3(5P-V 
  ) 
  1 
  

   r 
  3 
  a 
  3 
  t 
  24a 
  2 
  J' 
  

  

  1-1 
  fi 
  5(7P- 
  6^ 
  2 
  )^ 
  

   r 
  5 
  a 
  5 
  | 
  24a 
  2 
  J 
  ' 
  

  

  which 
  equations 
  can 
  all 
  be 
  satisfied 
  to 
  the 
  order 
  of 
  approxima- 
  

   tion 
  adopted 
  if 
  

  

  3f 
  2 
  -27 
  ? 
  2 
  = 
  5f-47/ 
  2 
  = 
  7? 
  2 
  -67 
  ? 
  2 
  =&c, 
  

  

  that 
  is 
  if 
  | 
  2 
  = 
  V 
  2 
  

  

  and 
  1 
  __ 
  1 
  / 
  ^ 
  y 
  2 
  \ 
  

  

  r 
  a 
  \ 
  

  

  or 
  

  

  24a 
  2 
  J 
  

   = 
  *( 
  1+ 
  8£?) 
  

  

  4. 
  If 
  we 
  expand 
  the 
  potentials 
  in 
  ascending 
  powers 
  of 
  ajx 
  

   the 
  same 
  result 
  is 
  arrived 
  at, 
  hence 
  the 
  potentials 
  of 
  a 
  circular 
  

   ooil 
  of 
  square 
  cross 
  section 
  (77, 
  77), 
  mean 
  radius 
  a, 
  with 
  n 
  turns 
  

   and 
  carrying 
  a 
  current 
  C, 
  and 
  of 
  a 
  circular 
  filament 
  of 
  radius 
  

  

  a 
  ( 
  1 
  + 
  ~-^ 
  J 
  carrying 
  a 
  current 
  nC, 
  lying 
  in 
  the 
  median 
  

  

  plane 
  of, 
  and 
  coaxal 
  with 
  the 
  coil, 
  are 
  identical 
  at 
  all 
  points 
  

   on 
  the 
  common 
  axis, 
  and 
  hence 
  by 
  Legendre's 
  theorem 
  

   identical 
  at 
  all 
  points 
  of 
  space 
  without 
  the 
  coil. 
  

  

  This 
  particular 
  filament, 
  therefore, 
  is 
  equivalent 
  to 
  the 
  coil 
  

   .and 
  can 
  replace 
  it, 
  and 
  the 
  radius 
  of 
  this 
  filament 
  I 
  shall 
  call 
  

  

  