﻿322 
  Prof. 
  Lyle 
  on 
  Circular 
  Filaments 
  or 
  Circular 
  

  

  For 
  any 
  combination 
  of 
  two 
  coils 
  of 
  different 
  types, 
  the 
  

   factors 
  that 
  have 
  to 
  be 
  applied 
  to 
  the 
  coplanar 
  resistance- 
  

   ratio 
  to 
  get 
  the 
  second 
  and 
  third 
  ratios 
  can 
  easily 
  be 
  deduced 
  

   from 
  the 
  expressions 
  given 
  in 
  §§ 
  10, 
  11. 
  

  

  18a. 
  The 
  length 
  of 
  the 
  small 
  needle 
  in 
  the 
  magnetometer 
  

   should 
  be 
  known, 
  and 
  the 
  factors 
  for 
  getting 
  the 
  second 
  and 
  

   third 
  resistance-ratios 
  from 
  Ri/R 
  2 
  should 
  be 
  corrected 
  for 
  it. 
  

   The 
  correction 
  to 
  be 
  applied 
  to 
  each 
  factor 
  will 
  be 
  the 
  same 
  

   for 
  all 
  kinds 
  of 
  coils. 
  

  

  When 
  the 
  coils 
  and 
  needle 
  are 
  coplanar 
  there 
  is 
  equilibrium, 
  

   provided 
  

  

  27r 
  Vl 
  i 
  3 
  k 
  *\- 
  27rn 
  2{i 
  , 
  3 
  X 
  *\ 
  

  

  R,n\ 
  4^r 
  2 
  / 
  B»*i\ 
  4 
  r 
  2 
  *)> 
  

  

  where 
  2X 
  is 
  the 
  distance 
  between 
  the 
  poles 
  of 
  the 
  needle. 
  (See 
  

   Maxwell, 
  section 
  711.) 
  

  

  When 
  the 
  small 
  coil 
  is 
  displaced 
  along 
  the 
  axis 
  a 
  distance 
  

   = 
  half 
  its 
  radius, 
  there 
  will 
  be 
  equilibrium 
  provided 
  the 
  

   resistances 
  are 
  changed 
  to 
  R/ 
  and 
  R 
  2 
  ' 
  so 
  that 
  

  

  2imi/ 
  ., 
  3 
  X 
  2 
  \ 
  _ 
  27m 
  2 
  8 
  

   RT^l 
  + 
  IV}) 
  ~ 
  R 
  2 
  V 
  2 
  5V5 
  ; 
  

  

  for 
  as 
  the 
  needle 
  is 
  now 
  in 
  the 
  Gaugain 
  position 
  with 
  respect 
  

   to 
  the 
  small 
  coil, 
  its 
  length 
  disappears 
  from 
  the 
  expression 
  

   for 
  the 
  torque 
  exerted 
  by 
  the 
  small 
  coil 
  on 
  the 
  needle. 
  

   Hence 
  from 
  the 
  two 
  equations 
  above 
  we 
  get 
  

  

  Similarly 
  it 
  may 
  be 
  shown 
  for 
  the 
  other 
  factor 
  that 
  

  

  R/'-R./. 
  , 
  3X 
  2 
  \ 
  8 
  

   R 
  2 
  " 
  R^ 
  4'iV5V5 
  - 
  

  

  so 
  that 
  the 
  factors 
  in 
  § 
  18 
  would 
  become 
  

  

  5 
  

  

  and 
  

  

  5V5T 
  2 
  -a,* 
  A 
  + 
  4</' 
  

  

  and 
  similarly 
  in 
  other 
  cases. 
  

  

  19. 
  The 
  practical 
  application 
  of 
  the 
  above 
  plan 
  of 
  mea- 
  

   suring 
  the 
  equivalent 
  radii 
  of 
  coils 
  offers 
  no 
  difficulty 
  to 
  a 
  

  

  