﻿738 
  Problem 
  of 
  the 
  Amagnetic 
  Mar'mer's 
  Compass. 
  

  

  require 
  ; 
  in 
  Fa;, 
  on 
  the 
  other 
  hand, 
  with 
  a 
  dependent 
  variable, 
  

   directly 
  proportional 
  to 
  wi-, 
  wy, 
  ly, 
  and 
  cos-^/r. 
  Whilst 
  ^ 
  = 
  90° 
  

   or 
  270°, 
  i|r 
  results 
  from 
  the 
  combined 
  action 
  of 
  F^ 
  or 
  Fr 
  and 
  

   that 
  deflecting 
  force, 
  Fr, 
  which 
  arises 
  from 
  the 
  simultaneous 
  

   rotation 
  round 
  the 
  planet's 
  axis 
  F, 
  exactly 
  as 
  ¥x 
  arises 
  from 
  

   that 
  round 
  .r, 
  and 
  hence 
  is 
  proportional 
  to 
  a>r» 
  Wy, 
  and 
  ly, 
  

   with 
  a 
  maximum 
  value 
  for 
  '*/r 
  = 
  90° 
  — 
  <^, 
  and 
  a 
  minimum 
  value 
  

   for 
  yjr=z(j). 
  

  

  It 
  is 
  further 
  necessary 
  to 
  choose 
  (o^ 
  >«», 
  lest 
  cox) 
  in 
  general, 
  

   in 
  relation 
  to 
  the 
  surface 
  of 
  the 
  planet, 
  might 
  become 
  ^ 
  zero. 
  

   The 
  rotating-velocity 
  (Ox, 
  and 
  the 
  oscillating-one 
  t^^, 
  must 
  be 
  

   such 
  as 
  to 
  suit 
  one 
  another, 
  that 
  is, 
  such, 
  that 
  they 
  synchronize 
  

   (a 
  to-and-fro 
  motion 
  round 
  y 
  corresponding 
  to 
  a 
  complete 
  

   rotation 
  round 
  c^'), 
  whence 
  the 
  amplitude 
  of 
  this 
  oscillation 
  

   may 
  be 
  small. 
  One 
  recalls 
  further 
  the 
  fact, 
  that 
  the 
  time- 
  

   period 
  of 
  an 
  oscillation, 
  27rPF~% 
  will 
  be 
  determined 
  by 
  

   the 
  magnitude 
  of 
  the 
  moment 
  of 
  inertia, 
  I, 
  and 
  the 
  directing 
  

   force 
  F, 
  this 
  one 
  as 
  shown 
  previously, 
  the 
  other 
  built 
  up 
  by 
  

   the 
  factor 
  ly, 
  and 
  a 
  gyrostatic 
  term, 
  which 
  is 
  a 
  quadratic 
  

   function 
  essentially 
  of 
  ly 
  and 
  a)y. 
  

  

  The 
  described 
  arrangement 
  with 
  horizontal 
  bearings 
  for 
  

   the 
  gyrostat 
  introduces 
  the 
  important 
  advantage 
  over 
  all 
  those 
  

   with 
  vertical 
  gyrostatic 
  axes, 
  that 
  a 
  construction 
  is 
  possible, 
  

   which 
  permits 
  this 
  axis 
  to 
  be 
  regarded 
  as 
  very 
  nearly 
  indeed 
  

   rigidly 
  fixed, 
  and 
  at 
  the 
  same 
  time 
  to 
  content 
  with 
  a 
  

   very 
  small 
  directing 
  force, 
  viz. 
  inertia 
  ; 
  the 
  small 
  angular 
  

   motion, 
  instead 
  of 
  complete 
  rotation, 
  allows 
  one 
  the 
  use 
  of 
  

   the 
  finest 
  cut 
  bearings 
  obtainable, 
  or 
  possibly 
  of 
  magnetic 
  

   setting 
  or 
  adjustment 
  of 
  the 
  pressure 
  on 
  the 
  bearings, 
  so 
  that 
  

   in 
  consequence 
  the 
  directing 
  forces 
  to 
  be 
  chosen 
  (or 
  the 
  

   impulses 
  ly 
  Q)y) 
  can 
  be 
  smaller 
  than 
  is 
  usual 
  with 
  gyrostats, 
  

   this 
  meaning 
  right 
  away 
  a 
  shortening 
  of 
  the 
  time-period 
  of 
  

   the 
  oscillations. 
  And 
  for 
  a 
  completely 
  rigid 
  axis 
  the 
  period 
  

   of 
  oscillation 
  would 
  be 
  simply 
  inversely 
  proportional 
  to 
  the 
  

   square-root 
  of 
  the 
  restoring 
  force 
  in 
  the 
  actual 
  model, 
  subject 
  

   then 
  to 
  a 
  small 
  correction 
  for 
  the 
  non-absolute 
  rigidity 
  of 
  the 
  

   material 
  of 
  which 
  it 
  is 
  made. 
  

  

  Valuable 
  is 
  a 
  comparison 
  of 
  the 
  formulae 
  given 
  formerly 
  at 
  

   Montreal, 
  however 
  in 
  another 
  connexion, 
  by 
  Lord 
  Kelvin 
  *, 
  

   who 
  saw 
  still 
  the 
  principal 
  contents 
  of 
  the 
  present 
  communi- 
  

   cation. 
  

  

  The 
  synchronism 
  being 
  obtained 
  between 
  the 
  rotation 
  

  

  round 
  a; 
  and 
  the 
  oscillations 
  about 
  ?/, 
  we 
  generally 
  get 
  forced 
  

  

  oscillations, 
  that 
  is, 
  oscillations, 
  which 
  are 
  differently 
  possible 
  

  

  between 
  the 
  limits, 
  determined 
  by 
  the 
  selected 
  values 
  of 
  

  

  * 
  lleports 
  of 
  the 
  British 
  Associalion, 
  liv., 
  1884. 
  

  

  