﻿Ada^.is. 
  — 
  To 
  Calculate 
  Distances 
  h)j 
  Beciprncal 
  Yeriical 
  A^ijlex. 
  183 
  

  

  as 
  it 
  is 
  tlien 
  only 
  necessary 
  to 
  multiply 
  the 
  number 
  of 
  units 
  in 
  the 
  given 
  

   angle 
  hj 
  the 
  value 
  of 
  one 
  unit 
  at 
  the 
  earth's 
  surface, 
  in 
  order 
  to 
  find 
  the 
  

   distance, 
  or 
  " 
  Contained 
  Arc," 
  as 
  it 
  is 
  generally 
  called. 
  

  

  The 
  method 
  of 
  deducing 
  the 
  subtended 
  angle, 
  or 
  rather 
  the 
  length 
  of 
  the 
  

  

  " 
  Contahied 
  Arc," 
  Avhen 
  the 
  vertical 
  angle 
  from 
  each 
  station 
  to 
  the 
  otiier 
  is 
  

  

  given, 
  may 
  be 
  shown 
  as 
  follows, 
  dividing 
  the 
  problem 
  into 
  two 
  cases 
  — 
  first, 
  

  

  when 
  one 
  angle 
  is 
  an 
  elevation 
  ; 
  and, 
  secondly, 
  when 
  both 
  are 
  depressions. 
  

  

  Case 
  1. 
  — 
  WJieii 
  one 
  anfjle 
  is 
  an 
  elevation, 
  and 
  tlie 
  other 
  a 
  ('cprcssion. 
  

  

  To 
  investigate 
  a 
  method 
  of 
  ascertaining 
  distances 
  by 
  

   means 
  of 
  reciprocal 
  vertical 
  angles 
  : 
  — 
  Let 
  A 
  and 
  B 
  in 
  

   fig. 
  1 
  represent 
  the 
  two 
  stations, 
  and 
  C 
  the 
  centre 
  of 
  

   the 
  earth. 
  Draw 
  A 
  F 
  a 
  horizontal 
  line 
  at 
  A, 
  and 
  B 
  G 
  

   a 
  horizontal 
  line 
  at 
  B, 
  and 
  B 
  H 
  parallel 
  to 
  A 
  F. 
  

   Then 
  A 
  B 
  G 
  is 
  the 
  true 
  angle 
  of 
  depression 
  at 
  B, 
  and 
  

   B 
  A 
  F' 
  is 
  the 
  true 
  angle 
  of 
  elevation 
  at 
  A, 
  and 
  G 
  B 
  A 
  

   ~ 
  Fl 
  B 
  A 
  (or 
  B 
  A 
  F) 
  = 
  C 
  (the 
  contained 
  arc). 
  Thus 
  

   we 
  see 
  that 
  the 
  difference 
  between 
  the 
  true 
  angles 
  of 
  

   elevation 
  and 
  depression 
  is 
  equal 
  to 
  the 
  "contained 
  

   arc," 
  and 
  taking 
  the 
  mean 
  value 
  of 
  1" 
  on 
  the 
  earth's 
  surface 
  = 
  101-4 
  feet 
  

   or 
  153-6 
  links, 
  we 
  could 
  thus 
  obtain 
  the 
  distance 
  between 
  the 
  two 
  stations 
  

   in 
  feet 
  or 
  links. 
  

  

  But 
  the 
  observed 
  angles 
  are 
  not 
  the 
  true 
  angles, 
  as 
  they 
  are 
  both 
  affected 
  

   with 
  refraction. 
  

  

  Let 
  A 
  K 
  and 
  B 
  K 
  represent 
  the 
  apparent 
  direction 
  of 
  each 
  station 
  from 
  

   the 
  other. 
  

  

  Let 
  D 
  represent 
  the 
  true 
  angle 
  of 
  dejiression, 
  G 
  B 
  A. 
  

   Let 
  i? 
  represent 
  the 
  true 
  angle 
  of 
  elevation 
  B 
  A 
  F. 
  

   Let 
  C 
  represent 
  the 
  angle 
  A 
  C 
  B 
  or 
  contained 
  arc 
  ; 
  and 
  

   Let 
  R 
  represent 
  the 
  angle 
  of 
  refraction 
  =^ 
  K 
  A 
  B 
  or 
  KB 
  A. 
  

   G 
  B 
  K 
  will 
  be 
  the 
  apparent 
  angle 
  of 
  depression 
  = 
  D 
  — 
  11. 
  

   And 
  K 
  A 
  F 
  will 
  be 
  the 
  apparent 
  angle 
  of 
  elevation 
  = 
  E 
  -\- 
  Ph 
  

   and 
  the 
  difference 
  between 
  the 
  observed 
  angles 
  of 
  depression 
  and 
  elevation 
  

   will 
  be 
  (Z) 
  - 
  i?) 
  - 
  (i? 
  + 
  i7) 
  = 
  D 
  - 
  E 
  - 
  2R 
  = 
  C 
  - 
  2R. 
  

  

  Now, 
  assuming 
  R 
  to 
  be 
  ^^ 
  of 
  the 
  contained 
  arc, 
  — 
  2R 
  will 
  be 
  C 
  — 
  -^-^ 
  C 
  

  

  Therefore 
  the 
  difference 
  between 
  the 
  observed 
  angles 
  of 
  elevation 
  and 
  

   depression 
  will 
  be 
  If 
  C 
  ; 
  so, 
  by 
  multiplying 
  the 
  number 
  of 
  seconds 
  in 
  -}-| 
  C 
  

   by 
  \^ 
  X 
  153-6, 
  we 
  shall 
  get 
  the 
  number 
  of 
  links 
  in 
  the 
  contained 
  arc, 
  or 
  

   the 
  distance 
  between 
  A 
  and 
  B. 
  (Note 
  jf 
  x 
  153-6 
  = 
  177-3). 
  If 
  the 
  dis- 
  

   tance 
  between 
  A 
  and 
  B 
  is 
  required 
  in 
  feet 
  instead 
  of 
  links, 
  then 
  multiply 
  

   by 
  If 
  X 
  101-4 
  = 
  117. 
  

  

  