﻿the 
  Ballistic 
  Galvanometer. 
  

  

  189 
  

  

  % 
  

  

  an 
  

  

  Thu 
  

  

  (8) 
  

  

  It 
  is 
  easy 
  to 
  see 
  that 
  when 
  a 
  does 
  not 
  exceed 
  60° 
  the 
  series 
  

   for 
  <t 
  in 
  (7) 
  can 
  he 
  calculated 
  to 
  an 
  accuracy 
  of 
  one 
  per 
  cent. 
  

   by 
  retaining 
  only 
  the 
  first 
  and 
  second 
  terms: 
  and 
  this 
  will 
  

   suffice 
  for 
  most 
  cases 
  of: 
  interest. 
  For 
  largo 
  values 
  of: 
  a, 
  it 
  

   is 
  hetter 
  to 
  use 
  the 
  tables 
  for 
  K, 
  E 
  and 
  to 
  write 
  

  

  o-=(4/tt){E-(1-P)K}//c 
  2 
  . 
  . 
  . 
  . 
  (7a) 
  

   We 
  have 
  now 
  the 
  approximation 
  

  

  d 
  so 
  (4) 
  yields 
  the 
  result 
  (neglecting 
  p 
  2 
  ) 
  

  

  i) 
  2 
  = 
  4/) 
  2 
  sin 
  2 
  \x-\-47rppo- 
  sin 
  2 
  \<x. 
  

  

  &> 
  = 
  2j9sin 
  |a(l 
  + 
  l\ff)*J 
  

  

  where 
  \= 
  7r 
  pp. 
  ) 
  

  

  The 
  investigation 
  given 
  bv 
  Maxwell 
  shows 
  that 
  the 
  value 
  

   of 
  Q 
  is 
  connected 
  with 
  that 
  of 
  co 
  by 
  the 
  equation 
  

  

  L»=GQ, 
  

  

  where 
  G 
  is 
  the 
  galvanometer 
  constant 
  and 
  I 
  is 
  the 
  moment 
  

   of 
  inertia 
  of 
  the 
  needle. 
  Also, 
  if 
  H 
  is 
  the 
  horizontal 
  

   component 
  of 
  the 
  earth's 
  field 
  

  

  I(/r 
  + 
  ^) 
  = 
  H; 
  

  

  and 
  so 
  neglecting 
  p 
  2 
  , 
  in 
  agreement 
  with 
  our 
  former 
  work, 
  

   we 
  find 
  the 
  relation 
  

  

  Hence 
  from 
  (8) 
  we 
  have 
  the 
  formula 
  required 
  to 
  improve 
  

   (C). 
  namely 
  

  

  Q=^ 
  H 
  sin^(H-JXo-) 
  

  

  whore 
  o- 
  = 
  i 
  + 
  -11 
  *> 
  + 
  } 
  2 
  ' 
  ; 
  V 
  : 
  **+... 
  ; 
  

  

  = 
  ^{e-(i-^)k} 
  

  

  k 
  being 
  sin 
  \<x. 
  

  

  The 
  following 
  table 
  gives 
  the 
  values 
  of 
  the 
  factor 
  a; 
  for 
  

  

  