﻿Magnetic 
  Shells 
  equivalent 
  to 
  Circular 
  Coils. 
  325 
  

  

  For, 
  the 
  mutual 
  inductance 
  of 
  two 
  coils 
  being 
  their 
  mutual 
  

   energy 
  when 
  unit 
  current 
  is 
  circulating 
  in 
  each, 
  and 
  as 
  unit 
  

   current 
  in 
  a 
  double-shell 
  coil 
  is 
  replaced 
  by 
  a 
  current 
  of 
  ^n 
  

  

  in 
  each 
  of 
  its 
  equivalent 
  filaments, 
  then, 
  if 
  ab 
  be 
  Maxwell's 
  M 
  

   for 
  two 
  coaxal 
  filaments 
  a 
  and 
  b, 
  and 
  if 
  a 
  a/ 
  be 
  the 
  pair 
  

   equivalent 
  to 
  one 
  coil 
  and 
  b 
  V 
  be 
  the 
  pair 
  equivalent 
  to 
  the 
  

   other, 
  the 
  mutual 
  energy 
  of 
  currents 
  of 
  \n 
  circulating 
  in 
  a 
  

   and 
  a' 
  with 
  respect 
  to 
  currents 
  \n 
  l 
  circulating 
  in 
  b 
  and 
  b 
  1 
  will 
  

   be 
  

  

  i/yj/vi' 
  

  

  

  {ab 
  + 
  ab' 
  + 
  a'b 
  + 
  a'b 
  1 
  } 
  

  

  = 
  nn 
  1 
  X 
  mean 
  of 
  the 
  four 
  M's. 
  

  

  Thus 
  four 
  determinations 
  of 
  an 
  M 
  are 
  required 
  in 
  the 
  

   case 
  of 
  double-shell 
  coils, 
  but 
  this 
  does 
  not 
  involve 
  four 
  times 
  

   the 
  labour 
  of 
  a 
  single 
  determination 
  if 
  the 
  four 
  calculations 
  

   are 
  worked 
  concurrently, 
  as 
  corresponding 
  numbers 
  in 
  the 
  

   four 
  will 
  be 
  found 
  on 
  the 
  same 
  or 
  consecutive 
  pages 
  of 
  a 
  

   logarithm 
  book. 
  

  

  23. 
  As 
  an 
  example 
  I 
  will 
  give 
  the 
  leading 
  figures 
  in 
  the 
  

   calculation 
  of 
  the 
  mutual 
  inductance 
  of 
  Rowland's 
  coils 
  A 
  and 
  

   B 
  used 
  in 
  his 
  determination 
  of 
  the 
  ohm. 
  

  

  Coil 
  A 
  %=i3-710, 
  £i 
  = 
  -84, 
  ^=-90,^=154. 
  

  

  CoilB 
  a 
  2 
  = 
  13-690, 
  £ 
  2 
  = 
  '84, 
  ?7 
  2 
  = 
  *90, 
  rc 
  2 
  = 
  154. 
  

  

  Distance 
  apart 
  = 
  6*534. 
  

  

  So 
  both 
  coils 
  are 
  thin 
  double-shell 
  ones 
  and 
  are 
  specified 
  in 
  

   the 
  notation 
  used 
  by 
  

  

  "' 
  = 
  ai 
  ( 
  1+ 
  2&) 
  = 
  13 
  ' 
  7121 
  > 
  

   8 
  1 
  2 
  = 
  ^llL 
  = 
  -0087, 
  8 
  1= 
  -0933. 
  

  

  r, 
  = 
  a*(l+ 
  2 
  -fy 
  =13-6921, 
  

  

  (^ 
  = 
  8, 
  = 
  -0933. 
  

  

  Hence 
  the 
  radii 
  of 
  the 
  filaments 
  equivalent 
  to 
  A 
  being 
  

  

  /*! 
  + 
  $! 
  and 
  ^ 
  — 
  §1 
  are 
  

  

  13-8054 
  and 
  13*6188 
  

   and 
  to 
  B 
  are 
  

  

  13*7854 
  and 
  13*5988, 
  

  

  B 
  

  

  