﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  301 
  

  

  § 
  35. 
  Envelope. 
  For 
  this 
  case 
  (22, 
  § 
  31) 
  takes 
  the 
  form 
  : 
  

   2(1- 
  J) 
  sin 
  (2«-0) 
  sin 
  (*-£)— 
  ? 
  sin 
  $ 
  sin* 
  

  

  = 
  V^lHTsin 
  (2*— 
  £) 
  sin 
  *. 
  

   When 
  multiplying 
  with 
  sin 
  <£ 
  we 
  get 
  : 
  

  

  (1-J)sin 
  (*—£){ 
  sin?*— 
  sin 
  2 
  (t 
  — 
  <j>)}— 
  £ 
  sin 
  2 
  < 
  cos 
  * 
  sine/) 
  

   = 
  /°Vl-?sin 
  £{ 
  sin 
  2 
  t- 
  sin 
  2 
  ft— 
  (/>)}. 
  

  

  When 
  multiplying 
  with 
  ^ 
  and 
  making 
  use 
  of 
  (21) 
  

  

  m° 
  t 
  

  

  v'l-fsinft-^V 
  / 
  Vl-?sin(«-^ 
  

  

  _ 
  ( 
  v.-^c-^ 
  V( 
  

  

  sin 
  £ 
  

  

  * 
  + 
  *>'-*'( 
  

  

  Vl-gsin 
  (*—<£) 
  

  

  , 
  sin 
  t 
  

  

  Pntfcing 
  y'l 
  — 
  £sin(f-<ft) 
  _ 
  £ 
  + 
  p'u 
  

  

  sin 
  t 
  v 
  ' 
  

  

  then 
  we 
  get 
  : 
  

  

  v\i-v)-(k+ 
  P 
  'vy=o. 
  

  

  Now 
  this 
  cubic 
  equation 
  has 
  the 
  same 
  coefficients 
  as 
  the 
  

   equation 
  which 
  serves 
  to 
  determine 
  the 
  values 
  of 
  f 
  corre- 
  

   sponding 
  to 
  sin 
  </>=0. 
  The 
  equation 
  has 
  two 
  positive 
  roots 
  

   which 
  we 
  shall 
  call 
  £\ 
  and 
  f 
  a 
  (we 
  suppose 
  f 
  a 
  >?i) 
  all( 
  i 
  one 
  

   negative 
  root 
  — 
  \. 
  

  

  >So 
  the 
  relation 
  between 
  f 
  and 
  £ 
  degenerates 
  into 
  : 
  

  

  Vl'- 
  r 
  sin 
  (7-0) 
  _ 
  ft-jV^ 
  V 
  I3f 
  gin 
  ft 
  -eft) 
  _ 
  k 
  + 
  p% 
  

   sin 
  i 
  f 
  x 
  ' 
  sin 
  t 
  fa 
  

  

  Vl^sin 
  ft— 
  0) 
  _lc-p\ 
  

   sin 
  £ 
  —A, 
  

  

  Now 
  we 
  reduce 
  the 
  first 
  member 
  of 
  these 
  equations 
  as 
  

   follows 
  : 
  — 
  

  

  \j\— 
  £sin 
  ft 
  — 
  <M 
  V 
  7 
  1 
  — 
  fsin 
  (2t 
  — 
  (f>) 
  cos 
  £— 
  a/1 
  — 
  f 
  cos 
  (2t 
  — 
  <f>) 
  sin 
  £ 
  

  

  Si 
  J 
  ^ 
  in 
  ' 
  

  

  VfsinJ 
  b 
  b 
  Y; 
  ^ 
  •' 
  

  

  