﻿316 
  Prof. 
  Lyle 
  on 
  Circular 
  Filaments 
  or 
  Circular 
  

   and 
  then 
  from 
  the 
  first 
  

  

  2 
  _f 
  2 
  -V; 
  

  

  but 
  the 
  radius 
  of 
  the 
  replacing 
  filaments 
  being 
  ya 
  + 
  fi 
  is 
  

  

  " 
  -'< 
  1+ 
  i£?) 
  

  

  and 
  their 
  distance 
  apart 
  is 
  2a 
  where 
  a 
  2 
  = 
  10 
  - 
  as 
  before. 
  

  

  As 
  an 
  illustration 
  of 
  this 
  method 
  of 
  treating 
  the 
  problem, 
  

   the 
  following 
  case 
  may 
  be 
  considered 
  : 
  — 
  

  

  Let 
  the 
  external 
  field 
  be 
  uniform 
  and 
  the 
  axis 
  of 
  the 
  coil 
  

   parallel 
  to 
  the 
  lines 
  of 
  force 
  of 
  the 
  field. 
  

  

  The 
  mutual 
  energy 
  is 
  CSN, 
  where 
  N 
  is 
  the 
  magnetic 
  flux 
  

   through 
  any 
  one 
  turn. 
  

  

  But 
  C2N= 
  (n 
  . 
  C 
  . 
  H) 
  x 
  mean 
  area 
  of 
  the 
  coil 
  

  

  sGHx^+J) 
  

  

  as 
  is 
  easily 
  shown 
  by 
  integration 
  : 
  a 
  being 
  the 
  mean 
  radius 
  

   as 
  before. 
  

  

  In 
  this 
  case 
  the 
  equivalence 
  of 
  the 
  filaments 
  replacing 
  a 
  

   thick 
  double 
  shell 
  or 
  a 
  single 
  shell 
  to 
  their 
  corresponding 
  

   coils 
  is 
  obvious. 
  

  

  For 
  a 
  thin 
  double 
  shell 
  the 
  mean 
  area 
  of 
  the 
  two 
  fila- 
  

   ments 
  is 
  

  

  7r(r 
  2 
  -f8 
  2 
  ) 
  

  

  --{*+&+*#} 
  

   —{«•+§}. 
  

  

  and 
  this 
  is 
  again 
  equal 
  to 
  the 
  mean 
  area 
  of 
  the 
  corresponding 
  

   coil. 
  

  

  Applications 
  of 
  the 
  foregoing 
  Principles. 
  

  

  10. 
  The 
  constant 
  of 
  a 
  single-shell 
  galvanometer-coil 
  cor- 
  

   rected 
  for 
  cross 
  section 
  is 
  2nir/r 
  where 
  r 
  is 
  its 
  equivalent 
  

   radius 
  ; 
  while 
  that 
  of 
  a 
  thick 
  double-shell 
  coil 
  is 
  

  

  where 
  r 
  is 
  its 
  equivalent 
  radius 
  and 
  2/3 
  its 
  equivalent 
  breadth, 
  

  

  