﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  303 
  

  

  The 
  parabola 
  we 
  have 
  in 
  view 
  lias 
  its 
  top 
  on 
  the 
  outside 
  

   of 
  the 
  ellipse 
  that 
  contains 
  the 
  vertices 
  of 
  the 
  circumscribed 
  

   rectangles. 
  Now 
  according 
  to 
  (25) 
  : 
  

  

  Only 
  the 
  last 
  of 
  these 
  expressions 
  surpasses 
  unity. 
  So 
  

   one 
  of 
  the 
  enveloping 
  parabolas 
  has 
  the 
  equation 
  : 
  

  

  *_ 
  ^ 
  .,*+ 
  * 
  v= 
  o. 
  

  

  The 
  focus 
  of 
  this 
  parabola 
  is 
  at 
  a 
  distance 
  of 
  — 
  } 
  2 
  p 
  from 
  

   the 
  X-axis 
  ; 
  this 
  focus 
  is 
  at 
  the 
  same 
  time 
  the 
  focus 
  of 
  the 
  

   remaining 
  enveloping 
  parabolas. 
  Therefore 
  the 
  equations 
  

  

  become 
  : 
  

  

  2y— 
  ^+ 
  F 
  +p' 
  =0 
  ^ 
  parabola 
  

  

  2y- 
  | 
  *»+ 
  1 
  + 
  p 
  > 
  =0 
  r 
  3 
  „ 
  

  

  -X 
  „ 
  . 
  A 
  

  

  V-=^ 
  + 
  ^ 
  x 
  V=o 
  

  

  X 
  

  

  We 
  must 
  notice 
  that 
  the 
  calculation 
  is 
  exact 
  only 
  in 
  case 
  

   <£> 
  = 
  for 
  £=?i 
  and 
  £=£2* 
  if 
  </>=7r 
  for 
  one 
  of 
  these 
  values 
  of 
  

   £ 
  or 
  for 
  both, 
  then 
  the 
  calculation 
  is 
  not 
  exact 
  ; 
  however, 
  it 
  

   is 
  not 
  difficult 
  to 
  verify 
  that 
  the 
  system 
  of 
  equations 
  we 
  

   arrive 
  at 
  is 
  always 
  the 
  same. 
  

  

  § 
  36. 
  Strict 
  relation. 
  — 
  In 
  case 
  the 
  strict 
  relation 
  v 
  — 
  2n 
  x 
  = 
  

   is 
  satisfied 
  we 
  have 
  p' 
  = 
  ; 
  the 
  relation 
  between 
  £ 
  and 
  cf> 
  

   runs 
  : 
  

  

  £ 
  yl-fcos 
  (£> 
  = 
  k. 
  

  

  The 
  enveloping 
  parabolas 
  have 
  their 
  focus 
  at 
  0. 
  

  

  Fig. 
  15 
  gives 
  a 
  representation 
  ; 
  besides 
  the 
  enveloping 
  

   parabolas 
  some 
  osculating 
  curves, 
  among 
  which 
  the 
  double 
  

   curves, 
  have 
  been 
  drawn 
  ; 
  the 
  figure 
  corresponds 
  to 
  a 
  posi- 
  

   tive 
  value 
  of 
  k. 
  The 
  enveloping 
  parabola, 
  which 
  has 
  its 
  

   opening 
  turned 
  downwards, 
  is 
  the 
  \ 
  parabola 
  ; 
  of 
  the 
  two 
  

   remaining 
  parabolas, 
  the 
  f 
  2 
  parabola 
  has 
  its 
  top 
  nearest 
  to 
  

   the 
  X-axis 
  (? 
  2 
  being 
  >?,). 
  

  

  The 
  periodic 
  form 
  of 
  motion 
  we 
  have 
  for 
  the 
  maximal 
  

   valne 
  of 
  **(^). 
  Then 
  £=&=§, 
  sin 
  <f> 
  = 
  () 
  invariably. 
  The 
  

   motion 
  takes 
  place 
  in 
  a 
  parabola, 
  in 
  which 
  the 
  fi 
  and 
  f 
  8 
  

  

  