﻿about 
  a 
  Position 
  of 
  Equilibrium, 
  293 
  

  

  Geometrical 
  representation 
  of 
  the 
  motion 
  of 
  the 
  mechanism. 
  

  

  We 
  may 
  obtain 
  a 
  geometrical 
  representation 
  of 
  the 
  motion 
  

   by 
  making 
  use 
  of 
  a 
  representative 
  point, 
  which 
  moves 
  in 
  a 
  

   space 
  of 
  2, 
  .*>. 
  or 
  4 
  dimensions, 
  if 
  the 
  mechanism 
  we 
  have 
  in 
  

   view 
  has 
  resp. 
  2. 
  3, 
  or 
  4 
  degrees 
  of: 
  freedom. 
  From 
  what 
  is 
  

   said. 
  § 
  5 
  and 
  § 
  10, 
  it 
  follows 
  that 
  it 
  is 
  not 
  necessary 
  to 
  con- 
  

   sider 
  mechanisms 
  with 
  more 
  than 
  4 
  degrees 
  of 
  freedom. 
  

  

  Mechanism 
  with 
  two 
  degrees 
  of 
  freedom. 
  

  

  § 
  26, 
  For 
  an 
  arbitrary 
  mechanism 
  with 
  two 
  degrees 
  of 
  

   freedom 
  we 
  may 
  regard 
  as 
  a 
  representative 
  point 
  the 
  hori- 
  

   zontal 
  projection 
  of 
  a 
  material 
  point 
  which 
  moves 
  without 
  

   friction 
  yet 
  under 
  the 
  influence 
  of 
  gravitation 
  on 
  a 
  given 
  

   surface 
  in 
  the 
  vicinity 
  of 
  its 
  lowest 
  point. 
  The 
  surface 
  has 
  

   positive 
  curvature 
  in 
  the 
  vicinity 
  of 
  this 
  point 
  0, 
  plane 
  XY 
  

   i- 
  the 
  tangential 
  plane 
  in 
  0, 
  and 
  the 
  XZ- 
  and 
  YZ-planes 
  are 
  

   the 
  principal 
  sections 
  of 
  the 
  surface 
  in 
  that 
  point, 
  whilst 
  the 
  

   Z-axis 
  is 
  supposed 
  positive 
  upwards. 
  

  

  Then 
  we 
  have 
  : 
  

  

  V 
  = 
  </z 
  = 
  c 
  x 
  a? 
  + 
  c 
  B 
  f 
  + 
  d 
  x 
  x 
  z 
  + 
  d 
  2 
  x 
  2 
  n 
  + 
  d 
  Z 
  3cy 
  2 
  + 
  dtf 
  

  

  -f 
  e 
  x 
  x^ 
  + 
  e 
  2 
  x*y 
  + 
  ew 
  2 
  y 
  2 
  + 
  e±xy} 
  + 
  <? 
  5 
  2/ 
  4 
  -f 
  , 
  

  

  where 
  

  

  1 
  2 
  9 
  

  

  *rV 
  L 
  *-2"*- 
  2 
  R 
  

  

  v>27. 
  The 
  motion 
  of 
  the 
  representative 
  point 
  is 
  given 
  by 
  

  

  Rj 
  and 
  R 
  y 
  being 
  the 
  principal 
  radii 
  of 
  curvature 
  of 
  the 
  sur- 
  

   face 
  in 
  0. 
  

  

  The 
  expression 
  for 
  the 
  kinetic 
  energy 
  takes 
  the 
  form 
  : 
  

  

  (8, 
  p. 
  2S1; 
  13, 
  p. 
  285): 
  

  

  x=B 
  f 
  WScos(n 
  x 
  t+2nJ3 
  x 
  ), 
  

  

  ;>/=- 
  IVv 
  1 
  - 
  K 
  COsCtV 
  + 
  2yn 
  x 
  /3 
  !/ 
  ) 
  

  

  where, 
  according 
  to 
  § 
  17, 
  a 
  relation 
  exists 
  between 
  f 
  and 
  cf> 
  : 
  

   /(?,*)=0. 
  

  

  (f> 
  is 
  written 
  instead 
  of 
  2yn 
  x 
  ((3 
  x 
  — 
  l3y). 
  

  

  f, 
  fii, 
  and 
  y&, 
  have 
  for 
  every 
  moment 
  a 
  definite 
  value 
  ; 
  these 
  

   values 
  determine 
  a 
  certain 
  Lissajons 
  curve. 
  This 
  curve 
  we 
  

  

  