﻿

  a 
  New 
  Type 
  of 
  Rough 
  Surface. 
  497 
  

  

  Then 
  we 
  proceed 
  to 
  investigate 
  the 
  path 
  of 
  a 
  heavy 
  particle 
  

   moving 
  on 
  this 
  surface, 
  the 
  coefficient 
  or* 
  friction 
  being 
  fi. 
  

   3. 
  The 
  equations 
  of 
  motion 
  are 
  

  

  y\ 
  — 
  inR 
  — 
  fiR 
  ~r 
  , 
  

  

  z 
  = 
  nR 
  — 
  fiR 
  -j 
  g, 
  

  

  where 
  R 
  is 
  the 
  normal 
  reaction 
  per 
  unit 
  mass 
  and 
  Z, 
  m, 
  n 
  are 
  

   the 
  direction-cosines 
  of 
  the 
  normal 
  to 
  that 
  side 
  of 
  the 
  surface 
  

   on 
  which 
  the 
  particle 
  lies. 
  

   Eliminating 
  R, 
  we 
  have 
  

  

  i+g 
  _&&+**) 
  

  

  , 
  dx 
  dy 
  dz 
  — 
  av 
  

  

  l-p- 
  m-p~ 
  n 
  -p 
  £ 
  

  

  Thus 
  we 
  have 
  

  

  >lv:6= 
  ( 
  i 
  "Ts- 
  l 
  )i(r 
  2+ 
  ^ 
  

  

  Writing 
  

  

  . 
  _ 
  d 
  I 
  dx\ 
  _ 
  d 
  / 
  dy\ 
  d 
  d 
  

  

  iC 
  ~dt\ 
  v 
  dsr 
  y 
  ~dt\ 
  v 
  tsr 
  dt- 
  v 
  ds> 
  

  

  and 
  simplifying, 
  we 
  get 
  

  

  „d 
  2 
  x 
  dv 
  ( 
  . 
  dx\ 
  dz 
  

  

  d?y 
  dv 
  ( 
  dt/\ 
  dz 
  

  

  ^d? 
  +mV 
  dS 
  + 
  { 
  m 
  -' 
  i 
  ds) 
  9 
  d 
  S 
  =0 
  - 
  

  

  Hence 
  we 
  obtain 
  

  

  9 
  dv 
  dz 
  

  

  i" 
  "T, 
  >T, 
  

  

  <-(-2-«2) 
  ('-'S9-M& 
  

  

  V 
  2 
  = 
  (1 
  

  

  dx 
  j 
  dy 
  

   dz 
  ds 
  ds 
  

   ds 
  ' 
  d 
  2 
  x 
  7 
  d 
  2 
  y 
  ' 
  

  

  m 
  d?~ 
  l 
  d? 
  

  

  \ 
  d 
  2 
  x 
  d 
  2 
  x 
  , 
  d 
  2 
  y 
  

  

  Jd? 
  m 
  d? 
  J* 
  

  

  (3) 
  

  

  