﻿[ 
  '34 
  ] 
  

  

  LXIII. 
  On 
  the 
  Resistance 
  of 
  a 
  Conductor 
  of 
  Uniform 
  Thickness 
  

   ivhose 
  Breadth 
  suddenly 
  changes, 
  and 
  on 
  the 
  Shapes 
  of 
  the 
  

   Stream-Lines 
  in 
  the 
  immediate 
  neighbourhood. 
  By 
  Charles 
  

   H. 
  Lees, 
  D.Sc, 
  F.R.S., 
  Professor 
  of 
  Physics 
  in 
  the 
  East 
  

   London 
  College, 
  University 
  of 
  London*. 
  

  

  A 
  KNOWLEDGE 
  of 
  the 
  resistance 
  of 
  a 
  conductor 
  whose 
  

   section 
  suddenly 
  changes 
  is 
  of 
  considerable 
  practical 
  

   importance, 
  but 
  mathematical 
  difficulties 
  have 
  prevented 
  an 
  

   exact 
  solution 
  of 
  the 
  problem. 
  Lord 
  Rayleigh 
  f 
  has 
  given 
  

   an 
  approximate 
  solution 
  of 
  the 
  case 
  in 
  which 
  a 
  cylinder 
  of 
  

   circular 
  section 
  is 
  joined 
  at 
  one 
  end 
  to 
  the 
  plane 
  surface 
  of 
  a 
  

   large 
  conducting 
  solid, 
  and 
  Professor 
  Hicks 
  % 
  has 
  solved 
  the 
  

   case 
  of 
  a 
  wire 
  of 
  small 
  diameter 
  ending 
  in 
  the 
  surface 
  of 
  a 
  

   conducting 
  sphere 
  § 
  . 
  

  

  The 
  mathematical 
  difficulties 
  of 
  the 
  problem 
  disappear 
  if 
  

   the 
  conductors 
  are 
  of 
  rectangular 
  section 
  and 
  one 
  dimension, 
  

   e.g. 
  the 
  thickness, 
  remains 
  constant, 
  while 
  the 
  other, 
  the 
  

   breadth, 
  suddenly 
  changes, 
  and 
  the 
  two 
  are 
  joined 
  together 
  

   either 
  with 
  their 
  axes 
  or 
  with 
  two 
  sides 
  colinear. 
  For 
  a 
  

   longitudinal 
  flow 
  through 
  the 
  conductors 
  the 
  former 
  case 
  

   becomes 
  the 
  latter 
  by 
  section 
  through 
  the 
  common 
  axis, 
  

   which 
  by 
  symmetry 
  is 
  a 
  line 
  of 
  flow. 
  

  

  Let 
  the 
  thickness 
  of 
  the 
  conductor 
  be 
  t, 
  the 
  breadth 
  of 
  the 
  

   wider 
  part 
  c, 
  that 
  of 
  the 
  narrower 
  b, 
  and 
  let 
  k 
  be 
  the 
  con- 
  

   ductivity 
  of 
  the 
  material. 
  Take 
  the 
  plane 
  of 
  the 
  conductor 
  

   as 
  the 
  z 
  plane, 
  fig. 
  1, 
  and 
  convert 
  the 
  outline 
  of 
  the 
  conductor 
  

  

  Fig. 
  1. 
  — 
  z 
  plane. 
  

  

  £ 
  = 
  + 
  oo 
  LP 
  

  

  -a. 
  

  

  by 
  means 
  of 
  the 
  Schwarzian 
  transformation 
  

  

  !=Ar'(r+iHr+«)-», 
  . 
  . 
  . 
  . 
  (i) 
  

  

  * 
  Communicated 
  bv 
  the 
  Physical 
  Society: 
  read 
  June 
  12, 
  1908. 
  

  

  t 
  Lord 
  Rayleigh, 
  Trans. 
  Roy. 
  Soc. 
  London, 
  lxi. 
  p. 
  77, 
  Nov. 
  1870 
  ; 
  

   and 
  ' 
  Scientific 
  Papers,' 
  vol. 
  i. 
  p. 
  5t5. 
  See 
  also 
  Phil. 
  Mag. 
  viii. 
  p. 
  481 
  

   (1904). 
  

  

  X 
  Professor 
  Hicks, 
  Messenger 
  of 
  Mathematics, 
  xii. 
  p. 
  183 
  (1883). 
  

  

  § 
  A 
  list 
  of 
  cases 
  which 
  have 
  been 
  worked 
  out 
  will 
  be 
  found 
  in 
  Professor 
  

   Auerbach's 
  articles 
  " 
  Flachenstrome 
  " 
  and 
  "Korperliche 
  Strome 
  " 
  in 
  

   Winkelmann's 
  Handbuch 
  der 
  Physik, 
  2nd 
  edit, 
  iv.pp. 
  238, 
  246. 
  

  

  