﻿188 
  Dr. 
  T. 
  J. 
  I'a. 
  Bromwich 
  on 
  

  

  The 
  actual 
  observation 
  which 
  can 
  be 
  made 
  on 
  the 
  needle 
  

   will 
  give 
  the 
  value 
  — 
  a. 
  at 
  which 
  the 
  needle 
  first 
  comes 
  to 
  

   rest 
  ; 
  so 
  that 
  we 
  can 
  find 
  a) 
  by 
  putting 
  = 
  oc 
  in 
  (3), 
  which 
  

   gives 
  

  

  co*=i(p°-+p*)sm>U 
  + 
  ip^ydt, 
  . 
  . 
  (4) 
  

  

  the 
  integral 
  extending 
  over 
  the 
  time 
  of 
  swing 
  from 
  — 
  to 
  

  

  It 
  appears 
  to 
  be 
  impossible 
  to 
  evaluate 
  the 
  integral 
  in 
  (4) 
  

   exactly 
  ; 
  but 
  an 
  approximate 
  solution 
  is 
  easily 
  found 
  when 
  

   p 
  is 
  small 
  enough 
  to 
  allow 
  p 
  2 
  to 
  be 
  neglected. 
  On 
  this 
  

   hypothesis, 
  we 
  may 
  neglect 
  p 
  entirely 
  when 
  calculating 
  the 
  

   integral, 
  because 
  in 
  (4) 
  the 
  integral 
  is 
  multiplied 
  by 
  p; 
  thus, 
  

   in 
  finding 
  the 
  integral, 
  we 
  replace 
  (3) 
  by 
  the 
  approximation 
  

  

  d0\ 
  2 
  

  

  -j- 
  I 
  =4/r(sinHa 
  — 
  sin 
  2 
  } 
  2 
  0) 
  

  

  or 
  — 
  = 
  2p 
  sin 
  \ol 
  cos 
  d> 
  

  

  dt 
  p 
  2 
  9 
  \, 
  

  

  where 
  sin 
  \Q 
  = 
  sin 
  \ol 
  sin 
  <j> 
  

  

  and 
  then 
  the 
  swing 
  from 
  6 
  — 
  to 
  a 
  corresponds 
  to 
  a 
  range 
  

   in 
  (/> 
  from 
  to 
  -Jvr. 
  

  

  Thus 
  we 
  may 
  write 
  in 
  (4) 
  

  

  ■*)" 
  

  

  J\dt/ 
  J 
  dt 
  r 
  " 
  J 
  </(!— 
  sm 
  2 
  -Jasin 
  tt 
  i 
  

  

  It 
  is 
  easy 
  to 
  express 
  the 
  last 
  integral 
  in 
  terms 
  of 
  the 
  

   complete 
  elliptic 
  integrals 
  K, 
  E 
  to 
  modulus 
  /j 
  = 
  sin^a: 
  in 
  

   fact 
  we 
  have 
  

  

  *W<fr^ 
  _ 
  ^ 
  } 
  

  

  V 
  (1 
  — 
  k 
  sm-<£) 
  

  

  I 
  

  

  But 
  if 
  a 
  does 
  not 
  exceed 
  60°, 
  the 
  value 
  of 
  k 
  2 
  will 
  not 
  exceed 
  

   :^, 
  and 
  then 
  a 
  series 
  will 
  be 
  usually 
  simpler 
  to 
  work 
  with 
  ; 
  

   thus 
  expanding 
  the 
  denominator 
  by 
  the 
  binomial 
  theorem 
  

   and 
  integrating 
  term-by-term 
  we 
  find 
  

  

  cos 
  2 
  j> 
  dcj> 
  _ 
  it 
  ~\ 
  

  

  where 
  \ 
  (7) 
  

  

  J. 
  

  

  I 
  2 
  72 
  1 
  2 
  .3 
  2 
  Z4 
  1 
  2 
  .3 
  2 
  .5 
  2 
  | 
  

  

  a_i+ 
  2.4 
  /L 
  ' 
  + 
  2.4 
  2 
  .6 
  A; 
  + 
  2~.4:K()KS 
  k 
  + 
  '" 
  J 
  

  

  