﻿OF 
  TERRESTRIAL 
  MAGNETISM 
  AT 
  SEVERAL 
  PLACES, 
  ETC. 
  445 
  

  

  is 
  not 
  large, 
  on 
  account 
  either 
  of 
  the 
  value 
  of 
  m, 
  or 
  of 
  /' 
  — 
  /, 
  the 
  dif- 
  

   ferences 
  will 
  fall 
  much 
  below 
  the 
  errors 
  in 
  the 
  observed 
  times 
  of 
  oscil- 
  

   lation. 
  

  

  CORRECTION 
  FOR 
  ARC. 
  

  

  In 
  many 
  observations 
  in 
  the 
  First 
  Series, 
  the 
  horizontal 
  oscilla- 
  

   tions 
  were 
  performed 
  through 
  arcs 
  of 
  very 
  different 
  extent. 
  With 
  

   needle 
  ~No. 
  1, 
  the 
  semi-arc 
  of 
  vibration 
  at 
  the 
  commencement 
  of 
  the 
  

   experiment 
  was, 
  in 
  some 
  cases, 
  30°, 
  and 
  in 
  others 
  not 
  more 
  than 
  20°, 
  

   the 
  arc 
  at 
  the 
  conclusion 
  of 
  the 
  experiment 
  depending, 
  of 
  course, 
  

   upon 
  its 
  duration. 
  Similar 
  variations 
  occur 
  with 
  the 
  other 
  needles. 
  

   The 
  most 
  simple 
  method, 
  therefore, 
  of 
  rendering 
  the 
  results 
  compara- 
  

   ble 
  is 
  to 
  reduce 
  the 
  times 
  of 
  oscillation 
  to 
  what 
  they 
  would 
  have 
  been 
  

   in 
  indefinitely 
  small 
  arcs. 
  The 
  formula 
  for 
  this 
  purpose 
  is 
  the 
  same 
  

   with 
  that 
  investigated 
  by 
  Borda 
  for 
  the 
  pendulum.* 
  By 
  applying 
  the 
  

   known 
  values 
  of 
  the 
  arcs 
  observed 
  at 
  beginning 
  and 
  ending 
  the 
  experi- 
  

   ment, 
  the 
  times 
  are 
  reduced 
  in 
  the 
  tables 
  which 
  follow. 
  Some 
  error 
  

   is, 
  no 
  doubt, 
  introduced, 
  particularly 
  when 
  these 
  arcs 
  are 
  large, 
  by 
  

   the 
  difficulty 
  of 
  observing 
  accurately 
  the 
  extent 
  of 
  the 
  arc 
  of 
  vibration, 
  

   To 
  diminish 
  these, 
  as 
  far 
  as 
  possible, 
  the 
  arcs 
  of 
  vibration 
  should 
  be 
  

   reduced 
  to 
  the 
  smallest 
  practicable 
  limit, 
  in 
  order 
  that 
  the 
  times 
  in 
  the 
  

   different 
  arcs 
  may 
  not 
  vary 
  too 
  rapidly. 
  The 
  practice 
  of 
  oscillating 
  

   in 
  different 
  arcs 
  leading 
  to 
  a 
  troublesome 
  correction, 
  is 
  to 
  be 
  avoided. 
  

   We 
  find 
  as 
  the 
  greatest 
  semi-arc 
  of 
  observation 
  suitable 
  to 
  be 
  employed 
  

   in 
  such 
  observations 
  about 
  fifteen 
  degrees. 
  The 
  oscillation 
  in 
  a 
  rare- 
  

   fied 
  medium 
  permits 
  this 
  to 
  be 
  much 
  reduced. 
  

  

  In 
  the 
  Second 
  Series 
  all 
  the 
  observations 
  were 
  made 
  within 
  the 
  

   same 
  arcs, 
  and 
  are 
  directly 
  comparable. 
  Needle 
  A 
  made 
  300 
  oscilla- 
  

   tions 
  between 
  the 
  semi-arcs 
  of 
  4° 
  and 
  2° 
  when 
  the 
  mercury 
  gauge 
  

   stood 
  at 
  three 
  inches. 
  At 
  the 
  same 
  pressure 
  C 
  and 
  B 
  made 
  300 
  oscil- 
  

   lations, 
  between 
  6° 
  and 
  2°. 
  In 
  such 
  small 
  arcs, 
  the 
  difference 
  in 
  the 
  

  

  * 
  T' 
  = 
  T 
  i 
  1 
  — 
  Sin 
  - 
  ( 
  A 
  + 
  a) 
  • 
  Sin. 
  (A 
  — 
  a) 
  v 
  

  

  V 
  . 
  32 
  M 
  (Log. 
  Sin. 
  A 
  — 
  Log. 
  Sin. 
  a}) 
  

   in 
  which 
  T' 
  is 
  the 
  reduced 
  time 
  of 
  a 
  given 
  number 
  of 
  vibrations 
  ; 
  T, 
  the 
  observed 
  time 
  ; 
  A 
  

   and 
  a 
  the 
  arcs 
  at 
  beginning 
  and 
  ending 
  ; 
  M, 
  the 
  modulus 
  of 
  the 
  common 
  logarithms. 
  

  

  