﻿312 
  Prof. 
  Lyle 
  on 
  Circular 
  Filaments 
  or 
  Circular 
  

  

  the 
  equivalent 
  radius 
  of 
  the 
  coil, 
  and 
  a 
  coil 
  of 
  square 
  cross- 
  

   section 
  I 
  shall 
  call 
  a 
  single-shell 
  coil. 
  

  

  5. 
  In 
  the 
  construction 
  of 
  coils 
  there 
  is 
  no 
  reason 
  why 
  

   with 
  ordinary 
  care 
  a 
  very 
  approximately 
  square 
  section 
  conld 
  

   not 
  be 
  attained, 
  and 
  then 
  the 
  above 
  theorem 
  would, 
  as 
  1 
  will 
  

   explain, 
  simplify 
  the 
  theory 
  of 
  galvanometers 
  and 
  electro- 
  

   dynamometers, 
  as 
  well 
  as 
  the 
  determination 
  of 
  coil-constants, 
  

   coefficients 
  of 
  mutual 
  induction, 
  and 
  current-balance 
  con- 
  

   stants. 
  

  

  6. 
  A 
  coil 
  of 
  n 
  turns 
  the 
  axial 
  breadth 
  f 
  of 
  whose 
  section 
  

   is 
  greater 
  than 
  its 
  radial 
  depth 
  ??, 
  of 
  mean 
  radius 
  a 
  and 
  

   carrying 
  a 
  current 
  C, 
  can 
  be 
  replaced 
  by 
  two 
  equal 
  filaments 
  

   coaxal 
  with 
  the 
  coil, 
  each 
  carrying 
  a 
  current 
  -JnC, 
  whose 
  

  

  radii 
  are 
  a 
  1 
  1 
  + 
  ^~ 
  2 
  p 
  an 
  ^ 
  wn 
  i° 
  n 
  are 
  placed 
  at 
  equal 
  distances 
  

   /3 
  on 
  either 
  side 
  of 
  the 
  median 
  plane 
  of 
  the 
  coil, 
  where 
  

  

  P 
  12 
  ' 
  

  

  For: 
  

  

  The 
  potential 
  of 
  the 
  two 
  filaments 
  as 
  specified 
  above, 
  at 
  

   any 
  point 
  on 
  the 
  common 
  axis 
  distant 
  x 
  from 
  their 
  median 
  

   plane, 
  is 
  

  

  v,-»«5/. 
  1 
  _.-=^ 
  + 
  i--±i!l 
  

  

  2 
  ( 
  pi 
  p-2 
  ) 
  

  

  where 
  

  

  Pi* 
  = 
  r" 
  + 
  -Pf, 
  pi 
  = 
  r* 
  + 
  (• 
  + 
  /3) 
  2 
  , 
  

  

  r 
  being 
  the 
  radius 
  of 
  either 
  filament. 
  Expanding 
  by 
  Taylor's 
  

   theorem 
  

  

  (where 
  p 
  2 
  = 
  x 
  2 
  + 
  r 
  2 
  ) 
  

  

  and 
  in 
  order 
  that 
  this 
  may 
  be 
  identical 
  with 
  the 
  axial 
  potential 
  

   of 
  the 
  coil 
  given 
  in 
  section 
  1, 
  the 
  following 
  equations 
  have 
  

  

  