﻿110 
  On 
  the 
  Viscosity 
  of 
  Glacier 
  Ice, 
  

  

  = 
  1 
  and 
  solving 
  the 
  equation 
  for 
  y, 
  we 
  obtain 
  as 
  the 
  section 
  

   of 
  the 
  channel 
  

  

  yio 
  , 
  , 
  a 
  2 
  6 
  

  

  , 
  a 
  /80 
  , 
  96 
  7 
  / 
  00 
  80\ 
  '~ 
  

  

  ^~™ 
  . 
  10 
  

  

  2+ 
  — 
  a? 
  

  

  c 
  

  

  H 
  !sg 
  F 
  H^(i; 
  +ltf 
  „y-(Bg±«). 
  +t 
  ,- 
  t 
  , 
  +(y 
  _ 
  < 
  , 
  

  

  — 
  — 
  — 
  ) 
  (.31) 
  

  

  and 
  the 
  surface 
  velocity, 
  when 
  y 
  = 
  0, 
  is 
  given 
  by 
  

  

  i 
  y 
  =^^ 
  + 
  ,i' 
  4 
  -^' 
  3 
  -2^ 
  2 
  + 
  1 
  ^ 
  + 
  l, 
  . 
  . 
  (32) 
  

  

  C 
  C 
  G 
  K 
  ' 
  

  

  which 
  depends 
  on 
  c 
  only. 
  

   The 
  depth 
  of 
  the 
  centre 
  is 
  

  

  V< 
  

  

  If 
  a 
  is 
  the 
  semi-width 
  of 
  the 
  channel, 
  and 
  b 
  its 
  depth 
  at 
  

   the 
  centre, 
  we 
  have 
  

  

  a 
  2 
  ' 
  ' 
  a 
  4 
  6 
  

  

  «=-- 
  /3=-^' 
  (33) 
  

  

  and 
  T7- 
  P 
  / 
  a 
  £ 
  S 
  

  

  -s(SpJ- 
  • 
  • 
  • 
  ■ 
  • 
  <M 
  

  

  The 
  curves 
  given 
  by 
  the 
  above 
  equation 
  will 
  be 
  found 
  to 
  

   approximate 
  very 
  closely 
  indeed 
  to 
  natural 
  glaciated 
  valleys, 
  

   most 
  of 
  which 
  can 
  be 
  " 
  fitted 
  " 
  very 
  accurately 
  by 
  a 
  proper 
  

   choice 
  of 
  the 
  constants 
  fi 
  and 
  c. 
  The 
  Hintereis 
  valley, 
  as 
  

   given 
  by 
  Bliimcke 
  and 
  Hess, 
  agrees 
  very 
  well 
  with 
  /3 
  = 
  3'5, 
  

   c 
  = 
  2, 
  as 
  shown 
  in 
  our 
  preceding 
  paper. 
  

  

  If 
  <•= 
  qo 
  , 
  the 
  channel 
  becomes 
  symmetrical, 
  and 
  the 
  curves 
  

   are 
  special 
  cases 
  of 
  an 
  equation 
  considered 
  by 
  Grratz 
  *, 
  but 
  

   he 
  unfortunately 
  did 
  not 
  discover 
  this, 
  the 
  most 
  important 
  

   special 
  case, 
  but 
  only 
  considered 
  the 
  curvilinear- 
  square 
  and 
  

   cruciform 
  tunnels 
  included 
  in 
  the 
  equation. 
  

  

  In 
  order 
  to 
  obtain 
  a 
  useful 
  curve 
  c 
  must 
  be 
  greater 
  than 
  1, 
  

   and 
  yS 
  greater 
  than 
  a 
  certain 
  limiting 
  value 
  which 
  depends 
  

   upon 
  the 
  value 
  of 
  c, 
  but 
  which 
  is 
  greater 
  than 
  2 
  unless 
  c 
  is 
  

   infinite. 
  

  

  * 
  " 
  Ueber 
  die 
  Bewegung 
  von 
  Flussigkeiten 
  in 
  Rohren," 
  Zeitschrift 
  

   fur 
  Mathematik 
  unci 
  Physik, 
  xxv. 
  p. 
  5. 
  

  

  