﻿Magnetic 
  Shells 
  equivalent 
  to 
  Circular 
  Coils. 
  313 
  

  

  to 
  be 
  satisfied 
  : 
  — 
  

  

  1 
  /i_g_ffl_ 
  Mi 
  3 
  £ 
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  &c. 
  &c. 
  

  

  It 
  is 
  easily 
  seen 
  that 
  this 
  series 
  of 
  equations 
  will 
  be 
  satisfied 
  

   to 
  the 
  order 
  of 
  approximation 
  adopted 
  if 
  

  

  or 
  

  

  and 
  

  

  r 
  a\ 
  24a 
  2 
  / 
  

  

  p 
  12 
  

  

  I£ 
  we 
  expand 
  the 
  potentials 
  in 
  ascending 
  powers 
  of 
  a/as 
  for 
  

   the 
  portion 
  of 
  the 
  axis 
  where 
  x 
  is 
  > 
  a 
  the 
  same 
  result 
  will 
  

   be 
  arrived 
  at 
  ; 
  hence 
  the 
  potentials 
  of 
  the 
  coil 
  and 
  of 
  the 
  

   double 
  filament 
  are 
  equal 
  at 
  all 
  points 
  in 
  space 
  external 
  to 
  

   the 
  coil, 
  and 
  in 
  all 
  calculations 
  (except 
  those 
  bearing 
  on 
  self- 
  

   induction) 
  the 
  coil 
  can 
  be 
  replaced 
  by 
  this 
  double-shell 
  or 
  

   filament. 
  

  

  A 
  coil 
  of 
  this 
  type 
  (i.e. 
  in 
  which 
  £>??) 
  I 
  shall 
  call 
  a 
  thick 
  

   double-shell 
  coil 
  ; 
  the 
  radius 
  of 
  either 
  of 
  the 
  two 
  replacing 
  

   filaments 
  I 
  shall 
  call 
  the 
  equivalent 
  radius 
  of 
  this 
  coil 
  ; 
  and 
  the 
  

   distance 
  2/3 
  between 
  the 
  two 
  filaments 
  I 
  shall 
  call 
  the 
  equiva- 
  

   lent 
  breadth 
  of 
  the 
  coil. 
  

  

  It 
  is 
  obvious 
  now 
  that 
  the 
  coil 
  will 
  be 
  completely 
  specified 
  

   when 
  its 
  type, 
  its 
  equivalent 
  radius, 
  and 
  its 
  equivalent 
  breadth 
  

   are 
  given 
  ; 
  that 
  all 
  magnetic 
  actions 
  of 
  the 
  coil 
  on 
  systems 
  

   external 
  to 
  itself 
  will 
  be 
  identical 
  with 
  those 
  of 
  the 
  equivalent 
  

   filaments; 
  and 
  that 
  all 
  magnetic 
  action 
  on 
  the 
  coil 
  from 
  

   external 
  systems 
  will 
  be 
  identical 
  with 
  the 
  action 
  of 
  these 
  

   systems 
  on 
  the 
  filaments, 
  the 
  order 
  of 
  approximation 
  being 
  

   up 
  to 
  the 
  fourth 
  pow 
  r 
  er 
  of 
  the 
  dimensions 
  of 
  the 
  section 
  of 
  

   the 
  coil 
  divided 
  by 
  its 
  radius. 
  

  

  