﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  315 
  

  

  two 
  rectangular 
  planes 
  of 
  symmetry 
  ; 
  if 
  these 
  planes 
  are 
  

   chosen 
  as 
  XZ- 
  and 
  YZ-plane, 
  then 
  1 
  = 
  0. 
  In 
  case 
  the 
  planes 
  

   halving 
  the 
  angles 
  between 
  the 
  planes 
  of 
  symmetry 
  just 
  

   mentioned 
  are 
  also 
  planes 
  of 
  symmetry, 
  p=—q. 
  The 
  relation 
  

   between 
  £ 
  and 
  <f> 
  takes 
  this 
  form 
  : 
  

  

  Kl-0«rff=sr{(l-0 
  + 
  r. 
  

  

  When 
  we 
  turn 
  the 
  axes 
  of 
  coordinates 
  through 
  an 
  angle 
  

   of 
  45°, 
  then 
  the 
  relation 
  between 
  f 
  and 
  </>' 
  must 
  have 
  the 
  

   same 
  form, 
  but 
  q' 
  and 
  r' 
  will 
  not 
  have 
  in 
  general 
  the 
  values 
  

   of 
  q 
  and 
  r. 
  

  

  The 
  relation 
  between 
  f 
  and 
  <p 
  may 
  be 
  written 
  as 
  follows 
  : 
  

  

  r(W) 
  S 
  in 
  2 
  </>=(l-?)f(l-0-r. 
  

  

  According 
  to 
  § 
  41, 
  ?(1 
  — 
  f) 
  sin 
  2 
  is 
  an 
  invariant. 
  There- 
  

   fore 
  according 
  to 
  § 
  46, 
  i/r 
  being 
  45° 
  now, 
  we 
  have 
  

  

  r(i-r)sm 
  2 
  f=(i- 
  ? 
  ){i-r(i-r)co^f}-n 
  

  

  After 
  reduction 
  : 
  

  

  r(i-r)cosv=^'(i-o+ 
  ?Z1 
  ^ 
  : 
  . 
  

  

  This 
  may 
  be 
  written 
  : 
  

  

  where 
  

  

  » 
  1 
  i 
  <7 
  — 
  l 
  + 
  4r 
  

   ?= 
  ? 
  ' 
  r= 
  ^4r- 
  (40) 
  

  

  § 
  50. 
  The 
  case 
  that 
  /(f) 
  is 
  a 
  linear 
  function 
  of 
  J 
  (§ 
  48) 
  

   occurs 
  for 
  Z 
  = 
  andp=— 
  <? 
  when 
  

  

  — 
  4r^ 
  = 
  ^ 
  2 
  . 
  

  

  There 
  are 
  three 
  possibilities, 
  namely, 
  q 
  = 
  0, 
  q=co 
  , 
  and 
  

   q=—4:r. 
  

  

  (1) 
  = 
  0. 
  The 
  sides 
  of 
  the 
  enveloping 
  rectangles 
  form, 
  

   according 
  to 
  § 
  48, 
  where 
  m 
  = 
  0, 
  angles 
  of 
  45° 
  with 
  the 
  axes 
  

   of 
  coordinates. 
  For 
  r=\ 
  g=l 
  sin 
  = 
  continually 
  ; 
  the 
  

   rectangles 
  have 
  contracted 
  into 
  straight 
  lines 
  (periodic 
  form 
  

   of 
  motion). 
  For 
  r=0 
  cos 
  = 
  continually; 
  the 
  two 
  

   enveloping 
  rectangles 
  have 
  coincided 
  to 
  a 
  square. 
  

  

  (2) 
  q 
  = 
  co. 
  This 
  case 
  may 
  be 
  deduced 
  from 
  the 
  foregoing 
  

   case 
  by 
  turning 
  the 
  axes 
  through 
  an 
  angle 
  of 
  45° 
  (q't=0). 
  

  

  (3) 
  q= 
  — 
  4r. 
  We 
  turn 
  the 
  axes 
  of 
  coordinates 
  through 
  such 
  

   an 
  angle 
  that 
  the 
  sides 
  of 
  an 
  enveloping 
  rectangle 
  are 
  parallel 
  

  

  