﻿a 
  New 
  Type 
  of 
  Rough. 
  Surface. 
  499 
  

  

  where 
  

  

  Assuming 
  the 
  equations 
  of 
  the 
  path 
  to 
  be 
  given 
  by 
  

  

  # 
  = 
  cos 
  (scot 
  a), 
  ?/=sin 
  (s 
  cot 
  a), 
  . 
  . 
  . 
  (5) 
  

   we 
  have, 
  for 
  points 
  on 
  the 
  curve, 
  

  

  a 
  2 
  = 
  4P 
  2 
  + 
  

  

  sec 
  2 
  a 
  

  

  XI. 
  Z 
  = 
  ytana-2#P, 
  

   fl 
  . 
  m=— 
  (a?tana 
  + 
  2yP), 
  

   and 
  O 
  . 
  n 
  = 
  1. 
  

  

  Also 
  we 
  find, 
  for 
  downward 
  motion, 
  

  

  cZs 
  

  

  -y- 
  -— 
  — 
  sin 
  a, 
  

  

  d# 
  . 
  , 
  , 
  v 
  

  

  — 
  - 
  = 
  cos 
  asm 
  (scot 
  a), 
  

  

  c/s 
  v 
  / 
  ? 
  

  

  -— 
  = 
  —COS 
  a 
  COS 
  (s 
  cot 
  a), 
  

   Tj 
  = 
  ~ 
  COS" 
  a 
  COS 
  (s 
  Cot 
  a), 
  

  

  -y^- 
  = 
  — 
  cos 
  2 
  a 
  sin 
  (2 
  cot 
  a). 
  

  

  Substituting 
  these 
  in 
  the 
  equation 
  (4) 
  and 
  putting 
  for 
  a 
  

   and 
  y 
  their 
  values 
  given 
  by 
  (5), 
  we 
  see 
  that 
  the 
  equation 
  

  

  -7- 
  = 
  — 
  -j 
  1 
  — 
  a 
  cot 
  a 
  . 
  cos 
  a 
  \f 
  4P 
  2 
  + 
  sec 
  2 
  a 
  }■ 
  

   dz 
  l 
  r 
  } 
  

  

  must 
  be 
  true 
  in 
  order 
  that 
  the 
  curve 
  described 
  by 
  the 
  

   particle 
  may 
  be 
  represented 
  by 
  the 
  equations 
  (5). 
  But 
  

   differentiating 
  (2) 
  we 
  find 
  that 
  the 
  above 
  equation 
  is 
  true. 
  

  

  The 
  equations 
  (5) 
  also 
  satisfy 
  the 
  equation 
  (1). 
  

  

  Therefore 
  it 
  is 
  proved 
  that, 
  if 
  a 
  particle 
  be 
  placed 
  on 
  the 
  

   surface 
  (1) 
  and 
  projected 
  with 
  n 
  suitable 
  velocity 
  along 
  the 
  

   helix 
  (5), 
  it 
  will 
  continue 
  to 
  describe 
  that 
  curve. 
  

  

  