﻿Magnetic 
  Shells 
  equivalent 
  to 
  Circular 
  Coils. 
  317 
  

  

  and 
  that 
  of 
  a 
  thin 
  double-shell 
  coil 
  is 
  

  

  2nirr 
  2n7r 
  /„ 
  . 
  B 
  2s 
  

  

  Znirr 
  ^ 
  zutt 
  /-, 
  , 
  o 
  \ 
  

   r 
  2 
  — 
  8' 
  2 
  r 
  \ 
  r 
  2 
  J 
  

  

  1 
  2.E 
  4 
  -llflV 
  t 
  + 
  2y* 
  g2 
  \ 
  for 
  a 
  thin 
  double 
  

   p* 
  ■ 
  2' 
  p 
  7 
  J 
  shell, 
  

  

  where 
  r 
  is 
  its 
  equivalent 
  radius 
  and 
  28 
  its 
  equivalent 
  depth. 
  

  

  11. 
  The 
  magnetic 
  force 
  H 
  at 
  any 
  point 
  on 
  the 
  axis 
  of 
  the 
  

   three 
  types 
  of 
  coils 
  distant 
  x 
  from 
  their 
  centres 
  is 
  given 
  by 
  

  

  r 
  2 
  

   H 
  = 
  27mC-3 
  for 
  a 
  single 
  shell, 
  

   P 
  

  

  H 
  = 
  2t™C^ 
  |l+|.- 
  4 
  * 
  8 
  T 
  r 
  V 
  } 
  for 
  a 
  thick 
  double 
  shell, 
  

  

  -and 
  

  

  H=2™C 
  ( 
  ? 
  + 
  - 
  2 
  . 
  -, 
  * 
  | 
  

  

  where 
  p 
  2 
  = 
  r 
  2 
  + 
  # 
  2 
  . 
  

  

  It 
  will 
  be 
  noticed 
  that 
  in 
  the 
  expression 
  for 
  the 
  axial 
  H 
  of 
  

   a 
  thick 
  double 
  -shell 
  coil 
  the 
  second 
  term, 
  depending 
  on 
  the 
  

   equivalent 
  breadth, 
  will 
  disappear 
  when 
  x 
  = 
  rj2: 
  hence 
  a 
  

   Helmholtz 
  galvanometer 
  made 
  of 
  equal 
  coils 
  of 
  either 
  the 
  

   first 
  or 
  second 
  type 
  which 
  are 
  placed 
  at 
  half 
  their 
  equivalent 
  

   radii 
  apart 
  is 
  an 
  ideal 
  instrument 
  of 
  its 
  kind, 
  as 
  the 
  cor- 
  

   rections 
  both 
  for 
  the 
  cross 
  sections 
  of 
  its 
  coils 
  and 
  for 
  the 
  

   length 
  of 
  its 
  needle 
  disappear 
  from 
  its 
  constant. 
  

  

  .12. 
  In 
  practice 
  the 
  equivalent 
  radius 
  of 
  a 
  coil 
  may 
  be 
  

   determined 
  in 
  one 
  or 
  other 
  of 
  three 
  ways. 
  

  

  a. 
  By 
  measurement 
  of 
  the 
  mean 
  radius 
  and 
  cross 
  section 
  

   and 
  substituting 
  the 
  values 
  so 
  obtained 
  in 
  the 
  expressions 
  

   given 
  for 
  the 
  equivalent 
  radii 
  of 
  the 
  three 
  types 
  of 
  coils 
  in 
  

   §§ 
  4, 
  6, 
  and 
  7. 
  

  

  I. 
  By 
  comparison 
  by 
  Bosscha's 
  method 
  with 
  a 
  standard 
  coil, 
  

   preferably 
  a 
  single-shell 
  one, 
  whose 
  equivalent 
  radius 
  has 
  

   been 
  carefully 
  determined 
  by 
  method 
  a. 
  

  

  c. 
  By 
  a 
  second 
  electrical 
  method 
  which 
  I 
  will 
  presently 
  

   describe. 
  

  

  13. 
  It 
  is 
  important 
  to 
  note 
  that 
  the 
  comparison 
  of 
  two 
  

   single-shell 
  coils 
  by 
  Bosscha's 
  method 
  gives 
  directly 
  the 
  

   ratio 
  of 
  their 
  equivalent 
  radii, 
  no 
  corrections 
  having 
  to 
  be 
  

   added 
  if 
  the 
  length 
  of 
  the 
  small 
  needle 
  at 
  the 
  common 
  centre 
  

   of 
  the 
  two 
  coils 
  be 
  neglected. 
  (This 
  latter 
  correction 
  is, 
  

   however, 
  larger 
  than, 
  I 
  think, 
  many 
  people 
  imagine 
  and 
  

   should 
  be 
  applied 
  in 
  most 
  cases, 
  taking 
  rive- 
  sixths 
  of 
  the 
  

   actual 
  length 
  of 
  the 
  small 
  magnet 
  for 
  the 
  distance 
  between 
  

   its 
  poles 
  *".) 
  

  

  * 
  See 
  W. 
  Hallock 
  and 
  F. 
  Kphlrausch, 
  Wied. 
  Ann. 
  xxii. 
  p. 
  411, 
  or 
  

   abstract 
  in 
  Phil. 
  Mag. 
  [5] 
  vol. 
  xviii. 
  p. 
  390, 
  " 
  On 
  the 
  Distance 
  apart 
  of 
  

   the 
  Poles 
  of 
  a 
  Magnet." 
  

  

  