﻿Problem 
  of 
  the 
  Amagnetic 
  Mariner 
  s 
  Compass. 
  729 
  

  

  § 
  11, 
  To 
  obtain 
  the 
  veio3ity 
  of 
  the 
  group 
  oE 
  waves 
  ol 
  wave- 
  

   lengths 
  approximately 
  equal 
  to 
  X 
  — 
  that 
  maintaining 
  in 
  the 
  

   neighbourhood 
  o£ 
  .u 
  at 
  time 
  t 
  — 
  we 
  put 
  

  

  MT~ 
  ~iKbW~-'^''' 
  .... 
  [in 
  

  

  which 
  enables 
  us 
  to 
  write 
  down 
  the 
  group-velocity 
  thus: 
  — 
  

   cLv 
  STTKht 
  — 
  Xx 
  ,^^. 
  

  

  dt 
  = 
  ~xt 
  — 
  (^*) 
  

  

  When 
  t 
  is 
  very 
  small 
  and 
  x 
  great, 
  the 
  right-hand 
  side 
  of 
  (18) 
  

  

  is 
  approximately 
  equal 
  to 
  — 
  (~ 
  )• 
  If 
  in 
  (17) 
  we 
  take 
  the 
  

  

  wave-length 
  \ 
  as 
  small 
  compared 
  with 
  x, 
  we 
  can 
  obtain 
  the 
  

   following 
  approximate 
  expression 
  for 
  \ 
  when 
  t 
  is 
  great: 
  — 
  

  

  X 
  = 
  ^^' 
  (19) 
  

  

  With 
  this 
  value 
  for 
  X, 
  the 
  right-hand 
  side 
  of 
  (18) 
  becomes 
  

  

  (f> 
  

  

  Equations 
  (15)-(19) 
  show 
  that 
  in 
  the 
  two 
  cases, 
  when 
  t 
  is 
  

   small 
  and 
  cV 
  great, 
  and 
  when 
  t 
  is 
  great^ 
  the 
  group-velocity 
  is 
  

   twice 
  the 
  wave- 
  velocity 
  ; 
  which 
  is 
  in 
  accordance 
  with 
  the 
  

   theory 
  of 
  group-velocity 
  given 
  by 
  Osborne 
  Reynolds 
  and 
  

   extended 
  by 
  Lord 
  Rayleigh. 
  

  

  LXXV. 
  A 
  Gyrodynamical 
  Solution 
  of 
  the 
  Problem 
  of 
  the 
  

   Amagnetic 
  Mariner 
  s 
  Compass. 
  By 
  J. 
  J. 
  Taudin 
  Chabot 
  *. 
  

  

  [Plate 
  XXI.] 
  

  

  THERE 
  is 
  an 
  ever-increasing 
  desire 
  to 
  replace 
  the 
  geo- 
  

   magnetic 
  ship's 
  - 
  compass 
  by 
  an 
  instrument, 
  which 
  

   would 
  be 
  independent 
  of 
  the 
  earth's 
  magnetism, 
  as 
  it 
  

   seems 
  possible 
  to 
  do, 
  in 
  particular 
  at 
  the 
  self-rotating 
  

   planet's 
  surface, 
  by 
  the 
  use 
  of 
  rotating 
  masses, 
  whose 
  axes 
  

   of 
  rotation 
  move 
  only 
  in 
  a 
  plane, 
  unalterable 
  relatively 
  to 
  

   the 
  rotating 
  planet 
  itself. 
  

  

  The 
  precursor 
  of 
  all 
  constructions, 
  ever 
  to 
  be 
  executed 
  

   for 
  this 
  purpose, 
  is 
  Bohnenberger's 
  machine, 
  dating 
  from 
  

   1817, 
  in 
  Tubingen. 
  It 
  enabled 
  us, 
  for 
  the 
  first 
  time, 
  to 
  

   show 
  approximately 
  by 
  experiment 
  some 
  results 
  of 
  the 
  works 
  

   published 
  over 
  half 
  a 
  century 
  before 
  by 
  d'Alembert, 
  Euler, 
  

   and 
  especially 
  by 
  Lagrange, 
  on 
  rotation 
  round 
  axes 
  with 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

   Phil. 
  Mag. 
  S. 
  6. 
  Yol. 
  18. 
  No. 
  107. 
  Nov. 
  1909. 
  3 
  C 
  

  

  