﻿about 
  a 
  Position 
  of 
  Equilibrium, 
  27 
  ( 
  .) 
  

  

  wo 
  find 
  that 
  they 
  arc 
  the 
  derivatives, 
  resp. 
  according 
  to 
  

   ar, 
  //. 
  of 
  

  

  (I 
  is 
  the 
  coeffieient 
  of 
  the 
  term 
  .r 
  : 
  V/ 
  in 
  U 
  t 
  .) 
  

  

  So 
  the 
  equations 
  of 
  motion 
  may 
  be 
  written 
  in 
  this 
  form 
  : 
  

  

  f-x 
  i 
  9R 
  -0 
  

   x 
  + 
  n 
  x 
  *x 
  — 
  ^— 
  — 
  U, 
  

  

  1 
  "J. 
  • 
  S1! 
  ft. 
  

  

  where, 
  if 
  — 
  / 
  is 
  written 
  instead 
  of 
  ( 
  — 
  ^A, 
  x 
  + 
  |a 
  a 
  y 
  ) 
  ;?/ 
  — 
  / 
  : 
  

  

  R 
  = 
  K*# 
  4 
  + 
  K^ 
  4 
  + 
  + 
  M^AV 
  + 
  MyJB'AV 
  + 
  -/afy. 
  

  

  § 
  10. 
  ±yij-f2% 
  — 
  n 
  z 
  — 
  p 
  . 
  — 
  In 
  this 
  case 
  : 
  

  

  T=i.v 
  2 
  +Af+ii 
  2 
  +iP.„.;' 
  2 
  +p., 
  y 
  .^ 
  + 
  , 
  

  

  where 
  

  

  Pzz=i«2z# 
  2 
  + 
  Jftt# 
  s 
  + 
  £c«* 
  a 
  4- 
  e^xy 
  +f 
  xx 
  xz 
  -f 
  // 
  x 
  ,//~, 
  

   P^ 
  = 
  Ja^a? 
  2 
  + 
  J&^y 
  2 
  + 
  JflayS 
  8 
  + 
  e 
  xy 
  xy 
  -YfxyXz 
  + 
  fc^ys. 
  

  

  Further 
  

  

  where 
  U 
  4 
  represents 
  a 
  homogeneous 
  function 
  of 
  degree 
  4 
  in 
  

   .<•, 
  y, 
  and 
  z. 
  The 
  result 
  is 
  that 
  the 
  disturbing 
  terms 
  of 
  the 
  

   second 
  kind 
  in 
  the 
  different 
  equations 
  after 
  reduction 
  are 
  

   the 
  derivatives, 
  resp. 
  according 
  to 
  x, 
  y, 
  and 
  z 
  of 
  : 
  

  

  ( 
  =F 
  njriylixy 
  ± 
  V 
  1 
  A* 
  — 
  1 
  » 
  y 
  %y 
  + 
  %**fye 
  -p)xifz. 
  

   ( 
  p 
  is 
  the 
  coefficient 
  of 
  the 
  term 
  xy 
  2 
  z 
  in 
  U 
  4 
  .) 
  

  

  Therefore 
  the 
  equations 
  of 
  motion 
  may 
  be 
  written 
  as 
  

   follows 
  : 
  

  

  •• 
  2 
  3R 
  n 
  

  

  a? 
  + 
  71x0?— 
  s 
  =0, 
  

  

  //+"/7/- 
  g- 
  =0, 
  

  

  ; 
  + 
  „/*-— 
  =0, 
  

  

  where, 
  if 
  — 
  / 
  is 
  written 
  instead 
  of 
  +n 
  x 
  n 
  y 
  h 
  X 
  y+n 
  x 
  nJb 
  xz 
  

  

  R= 
  K„r 
  4 
  + 
  K 
  y 
  / 
  + 
  + 
  M«A 
  2 
  AV 
  + 
  M 
  y 
  , 
  B 
  2 
  AV 
  + 
  -/>/; 
  . 
  

  

  