﻿Xakuu.—To 
  calculati 
  Distances 
  by 
  Ueciprocai 
  Vertical 
  Angles, 
  189 
  

  

  If 
  no 
  logaritlimic 
  or 
  trigonometrical 
  tables 
  are 
  at 
  hand, 
  the 
  difference 
  of 
  

   altitude 
  may 
  be 
  found 
  as 
  follows 
  : 
  — 
  

  

  As 
  -00000485 
  represents 
  the 
  value 
  of 
  sin 
  1" 
  arc 
  1" 
  or 
  tang 
  1" 
  (true 
  to 
  

   the 
  last 
  figure), 
  and 
  as 
  the 
  tangents 
  of 
  small 
  angles 
  vary 
  very 
  nearly 
  as 
  the 
  

   number 
  of 
  seconds 
  contained 
  in 
  the 
  angle, 
  we 
  may 
  suhstitu.te 
  for 
  the 
  tangent 
  

   of 
  the 
  angle 
  the 
  number 
  of 
  seconds 
  multipled 
  by 
  '00000485. 
  

  

  In 
  practice, 
  the 
  operation 
  may 
  be 
  shortened 
  by 
  combining 
  the 
  two 
  

   multipKers 
  together 
  ; 
  thus, 
  -00000485 
  X 
  117 
  = 
  -0005675. 
  

  

  (Note. 
  — 
  In 
  order 
  to 
  show 
  how 
  very 
  nearly 
  the 
  sines, 
  arcs, 
  and 
  tangents 
  

   agree 
  for 
  the 
  first 
  two 
  degrees, 
  their 
  values 
  at 
  two 
  degrees 
  are 
  given, 
  for 
  the 
  

   sake 
  of 
  comparison. 
  

  

  Diff. 
  

   Thus 
  sin 
  2° 
  = 
  -0348995] 
  „. 
  _ 
  ■,.„ 
  

   arc 
  2° 
  = 
  -03490661 
  '^ 
  - 
  -^4 
  

  

  tang 
  2° 
  = 
  -0349208 
  1 
  142 
  = 
  3" 
  

  

  Therefore, 
  the 
  arc 
  of 
  2° 
  ^ 
  sin 
  2° 
  00' 
  01^", 
  and 
  the 
  tangent 
  of 
  

   2° 
  = 
  arc 
  of 
  2° 
  00' 
  03". 
  

  

  Also, 
  in 
  obtaining 
  the 
  tangent 
  of 
  2° 
  by 
  multiplying 
  -00000485 
  X 
  60 
  

   X 
  60 
  X 
  2, 
  the 
  result 
  is 
  -0349200, 
  or 
  just 
  -^ 
  of 
  a 
  second 
  below 
  the 
  true 
  value. 
  

  

  Similarly 
  the 
  tangent 
  of 
  1°, 
  found 
  in 
  the 
  same 
  manner, 
  is 
  -0174600, 
  or 
  

   just 
  1" 
  above 
  its 
  true 
  value 
  ; 
  but 
  the 
  value 
  used 
  for 
  tang 
  1", 
  viz., 
  -00000485, 
  

   is 
  shghtly 
  in 
  excess 
  of 
  its 
  true 
  value, 
  which 
  is 
  -0000048481368, 
  etc.) 
  

   Then 
  the 
  difference 
  of 
  altitude 
  may 
  be 
  found 
  by 
  the 
  following 
  rules 
  :— 
  

  

  Case 
  1. 
  — 
  When 
  one 
  angle 
  is 
  in 
  an 
  elevation. 
  

  

  EuLE. 
  — 
  Take 
  the 
  difference 
  of 
  the 
  observed 
  vertical 
  angles, 
  and 
  also 
  half 
  

   the 
  sum, 
  both 
  reduced 
  to 
  seconds 
  ; 
  multiply 
  them 
  together, 
  and 
  their 
  product 
  

   by 
  -0005675 
  ; 
  the 
  result 
  will 
  be 
  the 
  difference 
  of 
  altitude 
  between 
  the 
  two 
  

   stations 
  in 
  feet. 
  

  

  Case 
  2. 
  — 
  When 
  both 
  angles 
  are 
  dejn'essions. 
  

  

  EuLE. 
  — 
  Take 
  the 
  sum 
  of 
  the 
  observed 
  verticle 
  angles, 
  and 
  also 
  half 
  the 
  

   difference, 
  both 
  reduced 
  to 
  seconds, 
  multiply 
  them 
  together, 
  and 
  their 
  pro- 
  

   duct 
  by 
  -0005675 
  ; 
  the 
  result 
  will 
  be 
  the 
  difference 
  of 
  altitude 
  between 
  the 
  

   two 
  stations 
  in 
  feet. 
  

  

  Or, 
  if 
  zenith 
  distances 
  are 
  used, 
  the 
  following 
  general 
  rule 
  will 
  apply 
  in 
  

   all 
  cases 
  : 
  — 
  

  

  EuLE. 
  — 
  Subtract 
  180° 
  from 
  the 
  sum 
  of 
  the 
  observed 
  zenith 
  distances 
  and 
  

   reduce 
  the 
  remainder 
  to 
  seconds 
  ; 
  then 
  take 
  half 
  the 
  difference 
  of 
  the 
  

   observed 
  zenith 
  distances 
  and 
  reduce 
  it 
  to 
  seconds 
  ; 
  multiply 
  the 
  two 
  

   quantities 
  together, 
  and 
  the 
  product 
  by 
  -0005875, 
  and 
  the 
  result 
  will 
  be 
  the 
  

   difference 
  of 
  altitude 
  between 
  the 
  two 
  stations 
  in 
  feet. 
  

  

  