﻿Adams. 
  — 
  To 
  Calculate 
  Distances 
  by 
  Reciprocal 
  Vertical 
  Angles. 
  135 
  

  

  Feet. 
  

  

  Taking 
  Bessel's 
  value 
  of 
  the 
  equatorial 
  radius 
  (E) 
  = 
  20928597 
  

  

  And 
  Bessel's 
  value 
  of 
  the 
  polar 
  semi-axis 
  (P) 
  = 
  20853054 
  

  

  The 
  value 
  of 
  1" 
  on 
  the 
  meridian 
  at 
  lat. 
  39° 
  = 
  101-164 
  

  

  The 
  value 
  of 
  1" 
  on 
  the 
  prime 
  vertical 
  at 
  lat. 
  89° 
  = 
  101-575 
  

  

  .-. 
  The 
  mean 
  value 
  of 
  1" 
  at 
  all 
  azimuths 
  in 
  lat. 
  39° 
  = 
  101-370 
  

  

  A<j;ain, 
  the 
  value 
  of 
  1" 
  on 
  the 
  meridian 
  at 
  lat. 
  44° 
  = 
  101-252 
  

  

  And 
  the 
  value 
  of 
  1" 
  on 
  the 
  prime 
  vertical 
  at 
  lat. 
  41° 
  = 
  101-604 
  

  

  .-. 
  The 
  mean 
  value 
  of 
  1" 
  at 
  all 
  azimuths 
  at 
  lat. 
  44° 
  = 
  101-428 
  

  

  And 
  the 
  mean 
  value 
  of 
  1" 
  at 
  all 
  azimuths 
  at 
  lat. 
  39° 
  = 
  l()l-370 
  

  

  .*. 
  The 
  mean 
  value 
  of 
  1" 
  at 
  all 
  azimuths 
  for 
  both 
  Islands 
  of 
  N.Z. 
  = 
  101-899 
  

   Or 
  say, 
  101-4 
  feet. 
  

  

  It 
  will 
  thus 
  be 
  seen 
  that, 
  by 
  using 
  this 
  mean 
  value, 
  the 
  results 
  would 
  be 
  

   sometimes 
  slightly 
  in 
  excess 
  of 
  the 
  true 
  values, 
  and 
  sometimes 
  slightly 
  in 
  

   defect 
  ; 
  but 
  in 
  any 
  case 
  the 
  difference 
  would 
  only 
  amount 
  to 
  about 
  -g- 
  per 
  

   cent., 
  and 
  may 
  therefore 
  in 
  ordinary 
  practice 
  be 
  neglected. 
  

  

  With 
  regard 
  to 
  the 
  co-efficient 
  of 
  refraction 
  which 
  I 
  have 
  adopted, 
  it 
  

   may 
  be 
  thought 
  that 
  -j^^- 
  is 
  too 
  small, 
  as 
  in 
  most 
  works 
  on 
  surveying 
  it 
  is 
  

   stated 
  to 
  be 
  from 
  -^V 
  to 
  y^^. 
  

  

  The 
  reason 
  I 
  have 
  used 
  -^-^ 
  is 
  because 
  I 
  find 
  it 
  more 
  in 
  accordance 
  with 
  

   actual 
  observations 
  in 
  hilly 
  countr}'- 
  in 
  New 
  Zealand. 
  

  

  The 
  factor 
  177"3, 
  as 
  stated 
  above, 
  is 
  obtained 
  by 
  takiug 
  the 
  value 
  of 
  1" 
  

   on 
  the 
  earth's 
  surface 
  as 
  153-6 
  links, 
  and 
  the 
  refraction 
  as 
  -^-^ 
  of 
  the 
  con- 
  

   tained 
  arc; 
  but 
  if 
  it 
  is 
  required 
  to 
  obtain 
  the 
  distance 
  in 
  any 
  other 
  

   denomination, 
  such 
  as 
  feet, 
  metres, 
  miles, 
  etc., 
  for 
  any 
  other 
  values 
  of 
  ter- 
  

   restrial 
  curvature 
  and 
  refraction, 
  this 
  may 
  easily 
  be 
  done 
  by 
  means 
  of 
  the 
  

   following 
  formula 
  : 
  — 
  

  

  Let 
  V 
  = 
  value 
  of 
  1" 
  on 
  the 
  earth's 
  surface, 
  in 
  the 
  given 
  denomination 
  

   ,, 
  7?i 
  = 
  co-efficient 
  of 
  refraction 
  

   ,, 
  i^= 
  the 
  factor 
  required 
  ;- 
  then 
  

   J? 
  V 
  __ 
  

  

  ^ 
  — 
  1— 
  2ni 
  

  

  Example, 
  Suppose 
  i; 
  = 
  30-89 
  metres 
  and 
  m—-Qn\ 
  

  

  then 
  j:2|;^ 
  ="^8~ 
  ~ 
  ^^' 
  '^^ 
  factor 
  required. 
  

  

  It 
  must 
  be 
  borne 
  in 
  mind 
  that 
  this 
  method 
  is 
  only 
  approximate, 
  as 
  the 
  

   observed 
  vertical 
  angles 
  are 
  liable 
  to 
  an 
  error 
  of 
  2" 
  or 
  3" 
  even 
  when 
  an 
  

   8-inch 
  theodolite 
  is 
  used, 
  and 
  a 
  mean 
  of 
  several 
  observations 
  taken. 
  

  

  Supposing 
  the 
  average 
  error 
  of 
  each 
  double 
  observation 
  to 
  be 
  6" 
  or 
  6" 
  

   then 
  the 
  error 
  in 
  the 
  calculated 
  distance 
  would 
  be 
  5 
  or 
  6 
  times 
  177 
  links, 
  

   say 
  about 
  10 
  chains. 
  This 
  would 
  be 
  1 
  per 
  cent, 
  in 
  a 
  distance 
  of 
  1000 
  

   chains, 
  which 
  is 
  the 
  usual 
  distance 
  between 
  geodesical 
  stations 
  in 
  New 
  

   Zealand. 
  

  

  The 
  chief 
  advantage 
  of 
  this 
  method 
  is 
  that 
  the 
  observations 
  are 
  not 
  

   subject 
  to 
  a 
  ratio 
  of 
  error 
  in 
  proportion 
  to 
  the 
  distance. 
  Most 
  approximate 
  

   methods, 
  by 
  telemeters, 
  etc., 
  although 
  tolerably 
  correct 
  for 
  short 
  distances, 
  

   fail 
  altogether 
  when 
  applied 
  to 
  long 
  distances 
  ; 
  but 
  this 
  method 
  gives 
  pi-o= 
  

  

  