﻿314: 
  Prof. 
  Lyle 
  on 
  Circular 
  Filaments 
  or 
  Circular 
  

  

  7. 
  A 
  coil 
  of 
  n 
  turns, 
  the 
  axial 
  breadth 
  £ 
  of 
  whose 
  cross 
  

   section 
  is 
  less 
  than 
  its 
  radial 
  depth 
  97, 
  of 
  mean 
  radius 
  a, 
  and 
  

   carrying 
  a 
  current 
  0, 
  can 
  be 
  replaced 
  by 
  two 
  concentric 
  and 
  

   coplanar 
  filaments 
  coaxal 
  with 
  and 
  lying 
  in 
  the 
  median 
  plane 
  

   of 
  the 
  coil, 
  each 
  carrying 
  a 
  current 
  \n&, 
  and 
  whose 
  radii 
  are 
  

   r 
  + 
  8 
  and 
  r 
  — 
  h 
  respectively, 
  where 
  

  

  _.( 
  

  

  i+ 
  £> 
  

  

  S 
  2 
  = 
  

  

  v* 
  

  

  12 
  

  

  It 
  is 
  unnecessary 
  to 
  give 
  the 
  proof 
  of 
  this, 
  as 
  it 
  follows 
  easily 
  

   on 
  the 
  same 
  lines 
  as 
  that 
  of 
  the 
  last 
  theorem. 
  A 
  coil 
  of 
  this 
  

   type 
  might 
  be 
  called 
  a 
  thin 
  double-shell 
  coil, 
  while? 
  1 
  as 
  defined 
  

   above 
  I 
  shall 
  call 
  its 
  equivalent 
  radius 
  and 
  28 
  its 
  equivalent 
  

   depth. 
  

  

  8. 
  In 
  the 
  case 
  of 
  a 
  coil 
  whose 
  cross 
  section 
  is 
  so 
  large 
  that 
  

   the 
  fourth 
  power 
  of 
  its 
  dimensions 
  divided 
  by 
  its 
  radius 
  

   cannot 
  be 
  neglected, 
  it 
  is 
  easy 
  to 
  imagine 
  it 
  divided 
  up 
  into 
  

   portions 
  whose 
  dimensions 
  are 
  small 
  enough 
  for 
  the 
  above 
  

   theorems 
  to 
  apply, 
  and 
  then 
  to 
  determine 
  the 
  system 
  of 
  fila- 
  

   ments 
  that 
  will 
  replace 
  each 
  of 
  these 
  portions. 
  A 
  few 
  extra 
  

   filaments 
  adds 
  nothing 
  to 
  the 
  difficulty 
  and 
  little 
  to 
  the 
  time 
  

   required 
  to 
  calculate 
  a 
  coefficient 
  of 
  mutual 
  induction 
  or 
  a 
  

   current-balance 
  constant, 
  seeing 
  that 
  tables 
  are 
  available 
  by 
  

   means 
  of 
  which 
  either 
  of 
  these 
  quantities 
  referring 
  to 
  any 
  

   two 
  coaxal 
  filaments 
  can 
  be 
  quickly 
  determined. 
  

  

  9. 
  The 
  principles 
  embodied 
  in 
  the 
  preceding 
  sections 
  may 
  

   be 
  established 
  in 
  a 
  totally 
  different 
  way, 
  which 
  is 
  interesting 
  

   enough 
  to 
  record 
  here. 
  

  

  Its 
  application 
  to 
  a 
  thick 
  double-shell 
  coil 
  (? 
  > 
  17) 
  will 
  be 
  

   sufficient 
  to 
  explain 
  the 
  method. 
  

  

  Jf 
  the 
  position 
  of 
  any 
  turn 
  of 
  the 
  coil 
  be 
  defined 
  by 
  x 
  

   the 
  distance 
  of 
  its 
  centre 
  from 
  a 
  fixed 
  point 
  on 
  the 
  axis, 
  and 
  

   y 
  the 
  radius 
  of 
  the 
  turn, 
  then 
  N 
  the 
  magnetic 
  flux 
  passing 
  

   through 
  this 
  turn 
  due 
  to 
  any 
  system 
  whatever 
  of 
  magnets 
  or 
  

   currents 
  will 
  be 
  a 
  function 
  of 
  x 
  and 
  y. 
  

  

  Let 
  us 
  specify 
  a 
  small 
  portion 
  of 
  the 
  coil 
  considered 
  by 
  

   rectangular 
  coordinates 
  p, 
  q 
  referred 
  to 
  axes 
  through 
  the 
  

   centre 
  of 
  the 
  section 
  parallel 
  to 
  the 
  axis 
  of 
  the 
  coil 
  and 
  its 
  

   radius 
  respectively, 
  then 
  the 
  current 
  round 
  an 
  element 
  dp, 
  

  

  do 
  of 
  the 
  coil 
  will 
  be 
  ^— 
  dp 
  . 
  da. 
  

   fa 
  

  

  | 
  and 
  rj 
  having 
  the 
  same 
  meaning 
  as 
  in 
  previous 
  sections. 
  

  

  