﻿(10) 
  

  

  about 
  a 
  Position 
  of 
  Equilibrium. 
  285 
  

  

  As 
  BR 
  BR 
  

  

  B& 
  = 
  Wy 
  = 
  

  

  we 
  have, 
  according 
  to 
  (1) 
  (p. 
  283) 
  : 
  

  

  <r/« 
  x 
  - 
  da 
  y 
  

   dt 
  _ 
  ~dt 
  

   pn 
  x 
  qn 
  y 
  ~ 
  

  

  This 
  enables 
  us 
  to 
  express 
  the 
  m 
  a's 
  by 
  the 
  help 
  of 
  one 
  new 
  

   variable 
  f 
  only 
  , 
  this 
  new 
  variable 
  f 
  we 
  will 
  choose 
  in 
  such 
  

  

  7 
  f* 
  

  

  a 
  way 
  that 
  each 
  of 
  the 
  members 
  o£ 
  (10) 
  is 
  equal 
  to 
  R 
  2 
  /t 
  2 
  — 
  . 
  

   where 
  R 
  is 
  a 
  constant 
  of 
  integration 
  of 
  moderate 
  value. 
  

   Now 
  we 
  have 
  

  

  a 
  x 
  =pn 
  x 
  ^li% 
  a, 
  = 
  qn 
  I/ 
  -R 
  Vi%C 
  2 
  +^, 
  (11) 
  

  

  where 
  C 
  2 
  , 
  are 
  (hi 
  — 
  1) 
  constants 
  of 
  integration. 
  

  

  As 
  in 
  (10) 
  the 
  sum 
  of 
  the 
  denominators 
  is 
  zero 
  (§ 
  13, 
  p. 
  282), 
  

   the 
  sum 
  of 
  the 
  numerators 
  is 
  also 
  zero. 
  Therefore 
  

  

  dotx 
  doty 
  _ 
  

  

  dt 
  + 
  dt 
  + 
  ~ 
  

  

  Or 
  

  

  «x+«z/+ 
  = 
  constant 
  (12) 
  

  

  In 
  case 
  m=2, 
  then 
  qn 
  y 
  ' 
  = 
  —pn 
  x 
  . 
  In 
  this 
  case 
  we 
  have 
  

  

  Introducing 
  f 
  instead 
  of 
  f 
  and 
  R 
  ' 
  instead 
  of 
  R 
  in 
  such 
  

   a 
  way 
  that 
  

  

  ?= 
  ~~ 
  C 
  2 
  ? 
  , 
  Ro 
  2 
  — 
  — 
  "TT^-^ 
  ' 
  

  

  we 
  get, 
  after 
  omitting 
  the 
  accents 
  again 
  : 
  

  

  ax= 
  B 
  Wh% 
  «,-R 
  WA 
  2 
  (l-r). 
  • 
  . 
  (13) 
  

  

  § 
  17. 
  When 
  we 
  differentiate 
  R 
  according 
  to 
  the 
  time, 
  

   we 
  get 
  

  

  dR_BR 
  da 
  x 
  BR 
  d&x 
  BR 
  rf^ 
  BR 
  d/3 
  y 
  BR 
  

  

  rf« 
  = 
  ~"B«x* 
  rf* 
  BA' 
  ^ 
  3"« 
  y 
  ' 
  dt 
  1/3/ 
  dt 
  + 
  ar 
  

  

  From 
  (7) 
  (p. 
  283) 
  we 
  deduce 
  

  

  dR 
  _ 
  BR 
  

   dt 
  " 
  "dt 
  ' 
  

  

  