﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  277 
  

  

  equation 
  for 
  x 
  fco 
  consider 
  as 
  disturbing 
  the 
  terms 
  with 
  yz 
  

   and 
  yz. 
  We 
  may 
  replace 
  yz 
  by 
  nnj/z, 
  and 
  in 
  the 
  terms 
  of 
  

   the 
  2nd 
  order 
  we 
  may 
  pur 
  n 
  . 
  + 
  n 
  7 
  — 
  »„ 
  = 
  0. 
  In 
  this 
  way 
  we 
  

   arrive 
  at 
  the 
  following 
  form 
  for 
  the 
  first 
  equation 
  : 
  

  

  (l> 
  being 
  the 
  coefficient 
  of 
  the 
  term 
  xyz 
  in 
  U 
  3 
  ). 
  

  

  In 
  the 
  same 
  way 
  we 
  can 
  also 
  simplify 
  the 
  two 
  remaining 
  

   equations 
  ; 
  we 
  must 
  hear 
  in 
  mind 
  that 
  in 
  the 
  second 
  equation 
  

   xz 
  must 
  be 
  replaced 
  by 
  n 
  x 
  n 
  z 
  xz,in 
  the 
  third 
  equation 
  however 
  

  

  The 
  result 
  is 
  that 
  the 
  equations 
  may 
  be 
  written 
  in 
  this 
  

   form 
  : 
  

  

  x+n 
  x 
  *x— 
  ~- 
  =0, 
  

  

  where, 
  if 
  y/ 
  is 
  written 
  instead 
  oi 
  p— 
  (a 
  ljz 
  ihjn 
  z 
  + 
  b 
  xz 
  n 
  x 
  iu—c 
  ry 
  ii 
  x 
  n 
  :/ 
  ) 
  

   11= 
  —p'xyz. 
  

   Si=4. 
  

   s 
  8. 
  There 
  are 
  five 
  relations 
  to 
  be 
  considered, 
  namely 
  : 
  

  

  3n 
  x 
  — 
  Wy=/3, 
  

  

  w 
  x 
  +zw 
  y 
  — 
  n 
  z 
  =p 
  

  

  n 
  J 
  . 
  + 
  n 
  I/ 
  ±n 
  z 
  — 
  n 
  u 
  = 
  p. 
  

  

  In 
  the 
  equations 
  of 
  motion 
  no 
  term 
  of 
  the 
  second 
  order 
  is 
  

   disturbing. 
  So 
  the 
  terms 
  of 
  the 
  second 
  order 
  may 
  in 
  the 
  

   equations 
  be 
  left 
  out 
  ; 
  in 
  T 
  and 
  U 
  the 
  terms 
  of 
  the 
  third 
  

   order 
  may 
  be 
  left 
  out. 
  Among 
  the 
  terms 
  of 
  the 
  third 
  order 
  

   in 
  tin* 
  equations 
  are 
  disturbing 
  terms 
  as 
  well 
  of 
  the 
  first 
  kind 
  

   as 
  of 
  the 
  second 
  kind. 
  Terms 
  of 
  higher 
  order 
  than 
  h 
  3 
  we 
  

   may 
  omit. 
  The 
  disturbing 
  terms 
  of 
  the 
  second 
  kind 
  are 
  

   different 
  in 
  the 
  different 
  cases 
  mentioned 
  above; 
  those 
  of 
  

   the 
  first 
  kind 
  are 
  always 
  the 
  same. 
  We 
  shall 
  first 
  discuss 
  

   these 
  terms. 
  

  

  It 
  is 
  clear 
  that 
  disturbing 
  terms 
  of 
  the 
  first 
  kind 
  may 
  

   appear 
  in 
  all 
  equations 
  and 
  that 
  they 
  will 
  contain 
  in 
  general 
  

   all 
  coordinate-. 
  In 
  the 
  equation 
  for 
  x 
  disturbing 
  terms 
  of 
  

   the 
  first 
  kind 
  of 
  order 
  h* 
  are 
  those 
  with 
  : 
  x\ 
  x*". 
  rx 
  2 
  

  

  '--- 
  - 
  ;v 
  < 
  «", 
  in 
  z' 
  z 
  * 
  , 
  .'v 
  

  

  , 
  _ 
  , 
  wyy 
  

  

  