﻿734 
  Mr. 
  Taudin 
  Chabot 
  : 
  Gyrodynamical 
  Solution 
  of 
  the 
  

  

  Of 
  these 
  cases 
  Sire 
  specially 
  dealt 
  with 
  No. 
  6 
  h, 
  Gilbert 
  

   with 
  Nos. 
  6 
  a 
  and 
  6 
  h, 
  without, 
  however, 
  the 
  gyrostat 
  in 
  the 
  

   state 
  of 
  rest 
  being 
  an 
  astatic 
  system. 
  Sire 
  arrived 
  at 
  the 
  

   stability 
  of 
  his 
  gyroscopic 
  pendulum 
  * 
  in 
  the 
  state 
  of 
  rest 
  

   merely 
  in 
  that 
  he 
  laid 
  the 
  axis 
  7 
  a 
  good 
  deal 
  lower, 
  than 
  the 
  

   axis 
  y, 
  Gilbert, 
  in 
  his 
  barogyroscope 
  t? 
  in 
  that 
  he 
  kept 
  both 
  

   axes 
  in 
  one 
  plane, 
  balanced 
  the 
  system 
  most 
  carefully 
  about 
  

   the 
  axis 
  y 
  and 
  then 
  added 
  an 
  adjustable 
  weight. 
  Gilbert 
  

   studied 
  the 
  special 
  case 
  of 
  identity 
  of 
  the 
  axis 
  V 
  with 
  the 
  axis 
  

   of 
  rotation 
  of 
  his 
  planet 
  (Earth) 
  — 
  getting, 
  therefore, 
  wj. 
  as 
  

   the 
  angular 
  velocity 
  of 
  the 
  planet's 
  rotation, 
  — 
  with 
  the 
  con- 
  

   sequence, 
  that 
  his 
  model 
  scarcely 
  showed 
  more 
  than 
  a 
  gyrostat 
  

   with 
  a 
  permanently 
  horizontal 
  axis 
  y 
  in 
  a 
  forked-bearing, 
  

   rotative 
  in 
  a 
  foot 
  ; 
  whereas 
  Sire 
  kept 
  the 
  axis 
  V 
  as 
  the 
  

   principal 
  axis 
  of 
  his 
  model, 
  which, 
  according 
  to 
  the 
  relative 
  

   direction 
  of 
  rotation 
  round 
  the 
  axis 
  y, 
  was 
  approximated 
  

   (even 
  in 
  opposition 
  to 
  the 
  so-called 
  centrifugal 
  force) 
  or 
  

   fled 
  by 
  the 
  gyrostatic 
  pendulum. 
  

  

  Instead 
  of 
  giving 
  the 
  unbalanced 
  gyrostat 
  a 
  position 
  of 
  

   rest 
  by 
  means 
  of 
  a 
  gravitational 
  moment, 
  we 
  can 
  make 
  the 
  

   balanced 
  gyrostat 
  stable 
  in 
  any 
  chosen 
  position 
  by 
  means 
  of 
  

   an 
  elastic 
  force 
  (acting 
  about 
  the 
  axis 
  ?/), 
  and 
  then 
  observe 
  

   the 
  deflexion, 
  brought 
  about 
  as 
  a 
  consequence 
  of 
  rotation 
  at 
  

   the 
  same 
  time 
  round 
  F. 
  The 
  cases 
  Nos. 
  6 
  and 
  7 
  are 
  identical, 
  

   as 
  is 
  evident, 
  if 
  <^ 
  = 
  90°, 
  viz. 
  the 
  axes 
  x 
  and 
  V 
  coincide. 
  

  

  In 
  case 
  No. 
  8, 
  being 
  here 
  of 
  special 
  interest, 
  the 
  pendulum- 
  

   motion 
  of 
  the 
  gyrostat 
  round 
  the 
  axis 
  ?/ 
  results, 
  as 
  ordinarily, 
  

   from 
  the 
  intermittent 
  action 
  of 
  a 
  deflecting 
  force 
  against 
  a 
  

   directing 
  one, 
  which 
  every 
  time 
  brings 
  back 
  the 
  pendulum 
  

   system 
  into 
  its 
  position 
  of 
  stable 
  equilibrium. 
  The 
  directing 
  

   force 
  arises 
  from 
  the 
  simultaneous 
  rotation 
  (which 
  also 
  might 
  

   be 
  considered 
  as 
  a 
  rotary 
  oscillation) 
  round 
  the 
  axis 
  x^ 
  or 
  by 
  

   means 
  of 
  a 
  control-spring 
  ; 
  whilst, 
  in 
  an 
  altogether 
  similar 
  

   manner, 
  a 
  force 
  acting 
  around 
  the 
  axis 
  y 
  arises 
  from 
  the 
  

   rotation 
  round 
  the 
  axis 
  F, 
  which, 
  purely 
  a 
  directive 
  force, 
  

   in 
  this 
  case 
  means 
  a 
  deflecting 
  one, 
  every 
  time 
  then 
  (and 
  

   thus 
  intermittently) 
  growing 
  to 
  a 
  maximum, 
  when 
  the 
  

   ^y-axis 
  passes 
  the 
  azimuth 
  normal 
  to 
  the 
  Vx 
  plane. 
  The 
  

   y-axis 
  describes 
  in 
  consequence 
  a 
  conoid-surface 
  round 
  x 
  

   with 
  about 
  the 
  spiral-ellipse 
  (similar 
  to 
  the 
  Lissajou-figure 
  

   for 
  oscillation-ratios 
  of 
  1 
  to 
  2 
  for 
  0^ 
  and 
  180'' 
  or 
  90° 
  and 
  

   270° 
  phase- 
  difference) 
  as 
  genera 
  ting-line 
  and 
  its 
  apex 
  in 
  

   the 
  intersection 
  of 
  the 
  axes 
  x 
  and 
  y. 
  Moreover, 
  the 
  chief 
  

  

  * 
  Sire, 
  Pendule 
  gyroscopique, 
  Archives 
  des 
  Sciences 
  physiques 
  et 
  

   naturelles, 
  vol, 
  i. 
  1858. 
  

  

  t 
  Gilbert, 
  Barogyroscope, 
  Comptes 
  rendus, 
  vol. 
  xciv., 
  1882. 
  

  

  