﻿1875.] 
  

  

  On 
  Multiple 
  Contact 
  of 
  Surfaces. 
  

  

  509 
  

  

  mediately 
  expanding 
  to 
  the 
  same 
  volume 
  as 
  it 
  previously 
  occupied, 
  and 
  

   the 
  viscosity 
  of 
  the 
  material, 
  which 
  also 
  renders 
  it 
  slow 
  to 
  expand. 
  Both 
  

   these 
  causes 
  are, 
  however, 
  rather 
  connected 
  with 
  the 
  effect 
  of 
  the 
  speed 
  

   of 
  the 
  roller 
  on 
  the 
  resistance 
  than 
  with 
  the 
  residual 
  resistance, 
  which, 
  so 
  

   far 
  as 
  the 
  surfaces 
  are 
  perfectly 
  true 
  and 
  perfectly 
  hard, 
  appears 
  to 
  be 
  

   due 
  to 
  the 
  friction 
  which 
  accompanies 
  the 
  deformation, 
  and 
  is 
  hence 
  

   called 
  rolling-friction. 
  

  

  No 
  attempt 
  has 
  yet 
  been 
  made 
  to 
  "investigate 
  the 
  laws 
  of 
  rolling-fric- 
  

   tion, 
  although 
  the 
  author 
  hopes 
  to 
  continue 
  the 
  investigation 
  in 
  this 
  direc- 
  

   tion 
  as 
  soon 
  as 
  he 
  has 
  obtained 
  the 
  necessary 
  apparatus. 
  

  

  At 
  the 
  end 
  of 
  the 
  paper 
  attention 
  is 
  called 
  to 
  certain 
  phenomena 
  con- 
  

   nected 
  with 
  railway-wheels, 
  which 
  it 
  is 
  thought 
  now, 
  for 
  the 
  first 
  time, 
  

   receive 
  an 
  explanation. 
  Thus 
  the 
  surprising 
  superiority 
  of 
  steel 
  rails 
  

   over 
  iron 
  in 
  point 
  of 
  durability 
  is 
  explained 
  as 
  being 
  due 
  as 
  much 
  to 
  the 
  

   fact 
  that 
  their 
  hardness 
  prevents 
  the 
  wearing-action, 
  i. 
  e. 
  the 
  slipping, 
  

   as 
  that 
  it 
  enables 
  them 
  better 
  to 
  withstand 
  the 
  wear. 
  Also 
  the 
  slipping 
  

   beneath 
  the 
  wheel 
  explains 
  the 
  wear 
  of 
  the 
  rails 
  in 
  places 
  where 
  brakes 
  

   are 
  not 
  applied 
  ; 
  and 
  the 
  severe 
  lateral 
  extension 
  beneath 
  the 
  wheel 
  is 
  

   thought 
  to 
  explain 
  the 
  scaling 
  of 
  wrought-iron 
  rails. 
  

  

  VI. 
  " 
  On 
  Multiple 
  Contact 
  of 
  Surfaces/-' 
  By 
  "William 
  Spottis- 
  

   woode, 
  M.A., 
  Treas. 
  U.S. 
  Received 
  May 
  24, 
  1875. 
  

  

  (Abstract.) 
  

  

  In 
  a 
  paper 
  " 
  On 
  the 
  Contact 
  of 
  Quadrics 
  with 
  other 
  Surfaces," 
  pub- 
  

   lished 
  in 
  the 
  Proceedings 
  of 
  the 
  London 
  Mathematical 
  Society 
  (May 
  14, 
  

   1874, 
  p. 
  70), 
  I 
  have 
  shown 
  that 
  it 
  is 
  not 
  in 
  general 
  possible 
  to 
  draw 
  a 
  

   quadric 
  surface 
  V 
  so 
  as 
  to 
  touch 
  a 
  given 
  surface 
  IT 
  in 
  more 
  than 
  two 
  

   points, 
  but 
  that 
  a 
  condition 
  must 
  be 
  fulfilled 
  for 
  every 
  additional 
  

   point. 
  The 
  equations 
  expressing 
  these 
  conditions, 
  being 
  interpreted 
  in 
  

   one 
  way, 
  show 
  that 
  two 
  points 
  being 
  taken 
  arbitrarily, 
  the 
  third 
  point 
  

   of 
  contact, 
  if 
  such 
  there 
  be, 
  must 
  lie 
  on 
  a 
  curve, 
  the 
  equation 
  whereof 
  is 
  

   there 
  given. 
  The 
  same 
  formulae, 
  interpreted 
  in 
  another 
  way, 
  serve 
  to 
  

   determine 
  the 
  conditions 
  which 
  the 
  coefficients 
  of 
  the 
  surface 
  V 
  must 
  

   fulfil 
  in 
  order 
  that 
  the 
  contact 
  may 
  be 
  possible 
  for 
  three 
  or 
  more 
  points 
  

   taken 
  arbitrarily 
  upon 
  it 
  ; 
  and, 
  in 
  particular, 
  the 
  degrees 
  of 
  these 
  con- 
  

   ditions 
  give 
  the 
  number 
  of 
  surfaces 
  of 
  different 
  kinds 
  which 
  satisfy 
  the 
  

   problem. 
  

  

  In 
  another 
  paper, 
  " 
  Sur 
  les 
  Surfaces 
  Osculatrices 
  " 
  (Comptes 
  Eendus, 
  

   6 
  Juillet, 
  1874, 
  p. 
  24), 
  the 
  corresponding 
  conditions 
  for 
  the 
  osculation 
  of 
  

   a 
  quadric 
  with 
  a 
  given 
  surface 
  are 
  discussed. 
  

  

  In 
  the 
  present 
  paper 
  I 
  have 
  regarded 
  the 
  question 
  in 
  a 
  more 
  general 
  

   way 
  ; 
  and 
  having 
  shown 
  how 
  the 
  formulae 
  for 
  higher 
  degrees 
  of 
  contact 
  

  

  