﻿the 
  Ballistic 
  Galvanometer. 
  187" 
  

  

  then, 
  allowing 
  for 
  damping, 
  a 
  formula 
  is 
  obtained 
  which 
  in 
  

   the 
  case 
  of 
  slight 
  damping 
  reduces 
  to 
  

  

  Q=(H/Gp)(l+£V)«, 
  .... 
  (B) 
  

  

  X 
  beino- 
  the 
  logarithmic 
  decrement 
  of 
  the 
  oscillating 
  needle. 
  

  

  Now 
  if 
  the 
  discharge 
  Q 
  is 
  large 
  enough 
  to 
  necessitate 
  the 
  

   use 
  of 
  sin 
  \ 
  ex. 
  in 
  place 
  of 
  i«, 
  and 
  if 
  at 
  the 
  same 
  time 
  the 
  

   effect 
  of 
  damping 
  is 
  needed, 
  it 
  is 
  natural 
  to 
  conjecture 
  that 
  

   the 
  formula 
  might 
  be 
  taken 
  as 
  

  

  Q=(2H/G^)(l 
  + 
  iX)sinJ 
  a 
  , 
  . 
  . 
  . 
  (C) 
  

  

  derived 
  by 
  a 
  sort 
  of 
  superposition 
  of 
  (A), 
  (B). 
  Actually 
  the 
  

   formula 
  (C) 
  is 
  sometimes 
  quoted 
  in 
  books 
  on 
  Physics 
  as 
  

   given 
  by 
  Maxwell 
  ; 
  but 
  the 
  formula 
  is 
  not 
  to 
  be 
  found 
  in 
  the 
  

   articles 
  mentioned 
  above. 
  

  

  The 
  question 
  now 
  suggests 
  itself 
  : 
  — 
  To 
  what 
  extent 
  is 
  (C) 
  

   correct, 
  and 
  how 
  can 
  the 
  formula 
  be 
  improved 
  ? 
  

  

  This 
  question 
  is 
  answered 
  below 
  : 
  as 
  might 
  be 
  expected,. 
  

   (C) 
  is 
  not 
  exact, 
  but 
  is 
  a 
  good 
  approximation 
  so 
  long 
  as 
  a 
  is 
  

   not 
  too 
  large. 
  The 
  more 
  exact 
  formula 
  will 
  be 
  found 
  in 
  

   (9); 
  and 
  a 
  glance 
  at 
  the 
  table 
  will 
  indicate 
  the 
  necessary 
  

   correction 
  to 
  (0). 
  

  

  Approximate 
  Theory 
  of 
  the 
  Ballistic 
  Galvanometer. 
  

  

  Suppose 
  that 
  in 
  a 
  small 
  oscillation 
  of 
  the 
  needle 
  it 
  is 
  

   found 
  that 
  the 
  period 
  is 
  T 
  and 
  that 
  the 
  logarithmic 
  decrement 
  

   is 
  X 
  ; 
  then 
  in 
  the 
  small 
  oscillations 
  the 
  equation 
  of 
  motion 
  is 
  

  

  W 
  + 
  ^tt 
  + 
  (r+P 
  2 
  )tf=o, 
  • 
  • 
  • 
  (i) 
  

  

  where 
  T=2w/p 
  and 
  \=^pT=7rp/p. 
  

  

  If 
  now 
  the 
  oscillation 
  is 
  of 
  larger 
  amplitude, 
  the 
  equation 
  

   of 
  motion 
  will 
  be 
  

  

  ^+2p^ 
  + 
  (p*+p>) 
  ain0=O, 
  . 
  . 
  , 
  (2) 
  

  

  instead 
  of 
  (1) 
  : 
  multiply 
  this 
  equation 
  (2) 
  by 
  dd/dt 
  and 
  

   integrate 
  from 
  the 
  position 
  of 
  equilibrium 
  (0 
  = 
  0) 
  over 
  part 
  

   of 
  the 
  first 
  swing 
  of 
  the 
  needle. 
  We 
  then 
  find 
  

  

  if 
  (o 
  is 
  the 
  initial 
  angular 
  velocity 
  of 
  the 
  needle, 
  produced 
  

   by 
  passing 
  the 
  amount 
  Q 
  of 
  electricity 
  round 
  the 
  coil. 
  

  

  