﻿Transactions, 
  — 
  Miscellanenm, 
  

  

  Case 
  2. 
  — 
  When 
  both 
  angles 
  are 
  depr6Ssions. 
  

  

  Using 
  the 
  same 
  notation 
  as 
  before, 
  except 
  tliat 
  D 
  

   and 
  d 
  represent 
  the 
  true 
  angles 
  of 
  depression, 
  and 
  

   D 
  — 
  R, 
  d 
  — 
  R 
  the 
  observed 
  angles 
  of 
  depression 
  ; 
  

   then 
  D 
  -\- 
  d 
  -{- 
  F 
  = 
  2 
  right 
  angles, 
  also 
  G 
  ~\- 
  F 
  = 
  2 
  

   right 
  angles 
  .*. 
  D 
  + 
  cZ 
  = 
  C 
  ; 
  and 
  D 
  — 
  R 
  -\- 
  d 
  — 
  R 
  

   = 
  C 
  — 
  2 
  Ji. 
  That 
  is, 
  the 
  sum 
  of 
  both 
  angles 
  of 
  de- 
  

   pression 
  = 
  C 
  - 
  2R 
  = 
  C 
  - 
  j% 
  C 
  = 
  If 
  C, 
  and 
  || 
  C 
  

  

  X 
  xf 
  X 
  158-6 
  (or 
  sum 
  of 
  observed 
  depressions 
  in 
  seconds 
  multiplied 
  by 
  

   177-8) 
  = 
  distance 
  in 
  links 
  between 
  A 
  and 
  B. 
  If 
  the 
  distance 
  between 
  A 
  

   and 
  B 
  is 
  required 
  in 
  feet, 
  then 
  multiply 
  by 
  117 
  instead 
  of 
  177'3. 
  

   The 
  above 
  results 
  expressed 
  in 
  words 
  give 
  the 
  following 
  

  

  Practical 
  Ride. 
  

   Take 
  the 
  sum 
  of 
  the 
  observed 
  vertical 
  angles 
  when 
  both 
  are 
  depressions 
  ; 
  

   or 
  their 
  difference 
  Avlien 
  one 
  is 
  an 
  elevation, 
  and 
  reduce 
  this 
  sum 
  or 
  differ- 
  

   ence 
  to 
  seconds 
  ; 
  multiply 
  by 
  177'3, 
  and 
  the 
  result 
  will 
  be 
  the 
  approximate 
  

   distance 
  between 
  the 
  two 
  stations 
  in 
  links. 
  Note. 
  — 
  If 
  the 
  distance 
  be 
  

   required 
  in 
  feet, 
  then 
  multiply 
  by 
  117. 
  

  

  Or 
  the 
  following 
  general 
  rule 
  will 
  apply 
  to 
  all 
  cases 
  : 
  — 
  Subtract 
  180° 
  

   from 
  the 
  observed 
  zenith 
  distances, 
  reduce 
  the 
  remainder 
  to 
  seconds, 
  and 
  

   multiply 
  by 
  177-3, 
  the 
  result 
  will 
  be 
  the 
  approximate 
  distance 
  between 
  the 
  

   two 
  stations 
  in 
  links. 
  

  

  In 
  the 
  preceding 
  investigation, 
  I 
  have 
  assumed 
  the 
  mean 
  value 
  of 
  1" 
  on 
  

   the 
  earth's 
  surface 
  = 
  101-4 
  feet, 
  and 
  I 
  shall 
  now 
  show 
  what 
  is 
  the 
  greatest 
  

   error 
  that 
  can 
  be 
  introduced 
  in 
  any 
  case 
  by 
  this 
  assumption. 
  

  

  The 
  radius 
  of 
  curvature 
  on 
  the 
  meridian 
  varies 
  with 
  the 
  latitude 
  from 
  a 
  

   minimum 
  at 
  the 
  Equator 
  (=— 
  ) 
  to 
  a 
  maximum 
  at 
  the 
  Pole 
  /"e"V 
  

  

  And 
  the 
  radius 
  of 
  curvature 
  of 
  the 
  Prime 
  Vertical 
  also 
  varies 
  mth 
  the 
  

   latitude 
  from 
  a 
  minimum 
  at 
  the 
  Equator 
  (= 
  E.) 
  to 
  a 
  maximum 
  at 
  the 
  Pole 
  

  

  {-%-)■ 
  

  

  Also, 
  the 
  radius 
  of 
  curvature 
  in 
  any 
  latitude 
  varies 
  with 
  the 
  Azimuth 
  

   from 
  a 
  minimum 
  on 
  the 
  meridian 
  to 
  a 
  maximum 
  on 
  the 
  2)rim8 
  vertical. 
  

  

  Still 
  the 
  limits 
  of 
  variation 
  are 
  so 
  small, 
  compared 
  with 
  the 
  ordinary 
  

   errors 
  of 
  observation, 
  that 
  in 
  general 
  practice 
  it 
  is 
  sufficient 
  to 
  assume 
  

   101-4 
  feet 
  as 
  the 
  mean 
  value 
  of 
  1" 
  on 
  the 
  surface 
  of 
  the 
  earth 
  for 
  New 
  

   Zealand. 
  

  

  The 
  following 
  are 
  the 
  precise 
  values 
  for 
  latitudes 
  39° 
  and 
  44°, 
  taking 
  39° 
  

   as 
  the 
  mean 
  latitude 
  of 
  tlie 
  North 
  Island 
  of 
  New 
  Zealand, 
  and 
  44° 
  as 
  the 
  

   mean 
  latitude 
  of 
  the 
  South 
  Island, 
  

  

  