﻿Magnetic 
  Shells 
  equivalent 
  to 
  Circular 
  Coils. 
  315 
  

  

  "J 
  The 
  mutual 
  energy 
  o£ 
  this 
  filament 
  and 
  the 
  external 
  mag- 
  

   netic 
  field 
  will 
  be 
  

  

  nQ 
  r- 
  T 
  /rfN\ 
  /dN\ 
  1 
  f 
  2 
  ^ 
  2 
  N 
  , 
  d 
  2 
  N 
  

  

  where 
  N 
  is 
  the 
  value 
  of 
  N 
  at 
  the 
  centre 
  of 
  the 
  section. 
  

  

  Integrating 
  between 
  the 
  limits 
  ( 
  + 
  ■£?, 
  — 
  if) 
  ( 
  + 
  i*?? 
  •— 
  i 
  1 
  ?) 
  we 
  

   get 
  the 
  total 
  mutual 
  energy 
  

  

  Again, 
  if 
  we 
  have 
  two 
  equal 
  filaments 
  whose 
  jo 
  and 
  q 
  coordi- 
  

   nates 
  are 
  (a, 
  /3), 
  and 
  ( 
  — 
  a, 
  /3) 
  respectively, 
  and 
  if 
  a 
  current 
  

   jftC 
  circulates 
  in 
  each, 
  the 
  mutual 
  energy 
  of 
  the 
  external 
  

   system 
  and 
  the 
  filaments 
  will 
  be 
  

  

  2 
  flb 
  </?/ 
  1 
  . 
  2 
  L 
  d.V 
  dxdy 
  dy 
  2 
  J 
  J 
  

  

  ~ 
  [\ 
  T 
  ^dN 
  , 
  1 
  f 
  ,d 
  2 
  N 
  no^N 
  ~l 
  . 
  

  

  but 
  

  

  ?/ 
  rf# 
  dx 
  2 
  dy 
  1 
  

  

  JNo 
  + 
  C/Syo 
  + 
  i^)^? 
  + 
  {fy 
  + 
  ip) 
  a 
  - 
  

  

  (See 
  Maxwell, 
  section 
  703) 
  ; 
  

   then 
  the 
  energy 
  of 
  the 
  two 
  filaments 
  reduces 
  to 
  

  

  dy 
  

  

  and 
  in 
  order 
  that 
  this 
  may 
  be 
  identical 
  with 
  the 
  expression 
  

   for 
  the 
  energy 
  of 
  the 
  coil 
  

  

  /32/o 
  + 
  i« 
  2 
  =£ 
  and 
  #/ 
  + 
  i/S 
  2 
  =J. 
  

  

  Solving 
  the 
  second 
  equation 
  we 
  have 
  to 
  the 
  order 
  of 
  approxi- 
  

   mation 
  adopted 
  

  

  