﻿the 
  Viscosity 
  of 
  Glacier 
  Tee. 
  109 
  

  

  p 
  

  

  which 
  satisfies 
  V"w~ 
  « 
  

  

  V 
  

   p 
  

   K 
  = 
  — 
  — 
  , 
  K 
  being 
  the 
  central 
  surface 
  velocity, 
  

  

  1 
  1 
  

   and 
  the 
  tunnel 
  is 
  the 
  ellipse 
  with 
  semi-axes 
  — 
  j= 
  9 
  *tj. 
  

  

  To 
  this 
  may 
  be 
  added 
  any 
  function 
  of 
  x 
  and 
  y 
  which 
  

   satisfies 
  \?' 
  2 
  f(d', 
  y) 
  = 
  Q. 
  

  

  In 
  order 
  to 
  find 
  the 
  best 
  functions 
  to 
  use, 
  we 
  notice 
  that 
  

   since 
  the 
  tunnel 
  must 
  be 
  symmetrical 
  in 
  order 
  to 
  admit 
  of 
  

   the 
  half 
  of 
  it 
  being 
  considered 
  a 
  channel, 
  even 
  powers 
  only 
  

   may 
  be 
  used 
  for 
  y. 
  If 
  the 
  powers 
  of 
  y 
  are 
  the 
  square 
  and 
  

   fourth 
  only, 
  the 
  whole 
  function 
  will 
  be 
  a 
  quadric 
  in 
  if, 
  and 
  

   so 
  will 
  offer 
  no 
  analytical 
  difficulties. 
  

  

  Limiting 
  ourselves 
  to 
  even 
  powers 
  of 
  y 
  not 
  exceeding 
  the 
  

   fourth, 
  we 
  find 
  the 
  most 
  general 
  expression 
  for 
  the 
  curves 
  of 
  

   equal 
  velocity 
  (to 
  = 
  constant) 
  to 
  be 
  

  

  = 
  C-«or 
  ! 
  -/fy 
  2 
  + 
  7.i' 
  + 
  S0c 
  3 
  -3V) 
  + 
  €(a? 
  4 
  -6.i-y+y 
  4 
  ) 
  

  

  +K« 
  B 
  -ifay+& 
  | 
  0% 
  . 
  . 
  (29) 
  

  

  or 
  = 
  (e 
  + 
  5$% 
  4 
  - 
  (£ 
  + 
  3Sa 
  + 
  Geo; 
  2 
  + 
  10£>: 
  V 
  

  

  + 
  (C 
  + 
  yx-aj? 
  + 
  8a? 
  + 
  €J*+& 
  i 
  ) 
  . 
  . 
  (30) 
  

  

  to 
  

  

  where 
  C 
  = 
  l—^ 
  and 
  the 
  tunnel, 
  for 
  which 
  w 
  = 
  0, 
  is 
  given 
  

  

  byC 
  = 
  l. 
  K 
  

  

  As 
  this 
  equation 
  has 
  not 
  previously 
  been 
  named 
  it 
  is 
  

   proposed 
  to 
  call 
  it 
  Parr's 
  equation, 
  and 
  the 
  curves 
  represented 
  

   by 
  it 
  Parr's 
  curves. 
  

  

  The 
  equation 
  represents 
  a 
  great 
  variety 
  of 
  curves, 
  and 
  the 
  

   complete 
  analysis 
  has 
  not 
  been 
  made 
  as 
  yet. 
  For 
  present 
  

   purposes, 
  in 
  order 
  to 
  obtain 
  a 
  channel 
  approximating 
  closely 
  

   to 
  a 
  natural 
  valley, 
  the 
  equation 
  must 
  be 
  of 
  the 
  form 
  

  

  ( 
  t 
  r 
  4- 
  a) 
  2 
  (a? 
  — 
  a) 
  2 
  (x 
  + 
  c) 
  when 
  y 
  = 
  0, 
  

  

  or 
  = 
  & 
  + 
  ex* 
  - 
  2a*a* 
  - 
  2a' 
  'ex 
  2 
  + 
  a\c 
  + 
  a% 
  

  

  and 
  a. 
  7, 
  8, 
  e, 
  and 
  J 
  may 
  be 
  determined 
  by 
  equating 
  co- 
  

   efficients, 
  /3 
  and 
  c 
  being 
  arbitrary. 
  

  

  As 
  we 
  are 
  only 
  concerned 
  with 
  the 
  shape 
  and 
  not 
  the 
  

   actual 
  size 
  of 
  the 
  curve, 
  we 
  may 
  put 
  the 
  semi-width 
  of 
  the 
  

   channel 
  equal 
  to 
  unity, 
  or 
  a 
  = 
  l, 
  and 
  combining 
  this 
  with 
  

  

  