﻿THE 
  CIRCUMSTANCES 
  OP 
  AN 
  ECLIPSE 
  OF 
  THE 
  SUN. 
  301 
  

  

  moon 
  from 
  the 
  sun 
  on 
  the 
  apparent 
  relative 
  orbit. 
  To 
  the 
  square 
  of 
  

   n 
  add 
  the 
  square 
  of 
  M, 
  and 
  from 
  the 
  sum 
  subtract 
  the 
  square 
  of 
  M', 
  

   denoting 
  the 
  remainder 
  by 
  N 
  2 
  . 
  Multiply 
  N 
  2 
  by 
  30, 
  and 
  divide 
  the 
  

   product 
  by 
  the 
  square 
  of 
  n, 
  extending 
  the 
  quotient 
  to 
  one 
  decimal 
  

   figure. 
  This 
  quotient 
  will 
  be 
  an 
  interval 
  in 
  minutes 
  of 
  time, 
  which, 
  

   being 
  added 
  to 
  the 
  time 
  T, 
  or 
  subtracted 
  from 
  it, 
  according 
  as 
  T' 
  is 
  

   later 
  or 
  earlier 
  than 
  T, 
  will 
  give 
  the 
  time 
  of 
  greatest 
  obscuration. 
  

  

  7. 
  Taking 
  now 
  T', 
  to 
  represent 
  the 
  time 
  of 
  greatest 
  obscuration, 
  

   find 
  for 
  this 
  time 
  the 
  values 
  of 
  p, 
  q, 
  H, 
  u, 
  C, 
  c, 
  and 
  v. 
  Also, 
  when 
  

   taking 
  c 
  from 
  the 
  traverse 
  table, 
  take 
  the 
  corresponding 
  difference 
  of 
  

   latitude, 
  and 
  marking 
  it 
  with 
  the 
  same 
  sign 
  as 
  C, 
  denote 
  it 
  by 
  g. 
  

   With 
  (/-J- 
  g) 
  as 
  the 
  argument, 
  take 
  from 
  Table 
  IX., 
  to 
  two 
  figures, 
  

   the 
  correction 
  of 
  r. 
  Subtract 
  this 
  correction 
  from 
  r, 
  and 
  denote 
  the 
  

   remainder 
  by 
  r'. 
  To 
  r 
  add 
  2.73, 
  the 
  moon's 
  reduced 
  semidiameter, 
  

   and 
  denote 
  the 
  sum 
  by 
  k. 
  Now 
  adding 
  together 
  the 
  squares 
  of 
  

   (p 
  — 
  u 
  ) 
  and 
  (q 
  — 
  v), 
  take 
  the 
  root 
  of 
  the 
  sum, 
  and 
  denote 
  it 
  by 
  m. 
  

   Then 
  will 
  m 
  express 
  the 
  least 
  distance 
  of 
  the 
  centres. 
  Multiply 
  

   (k 
  — 
  m) 
  by 
  6, 
  and 
  divide 
  the 
  product 
  by 
  r'. 
  The 
  quotient 
  will 
  ex- 
  

   press 
  the 
  digits 
  eclipsed 
  ; 
  on 
  the 
  northern 
  limb 
  if 
  (q 
  — 
  v) 
  is 
  -{-, 
  but 
  

   on 
  the 
  southern 
  if 
  it 
  is 
  — 
  . 
  If 
  m 
  is 
  equal 
  to, 
  or 
  greater 
  than 
  k, 
  the 
  

   eclipse 
  will 
  not 
  be 
  visible 
  at 
  the 
  given 
  place. 
  

  

  From 
  the 
  square 
  of 
  k 
  subtract 
  the 
  square 
  of 
  m, 
  and 
  taking 
  the 
  root 
  

   of 
  the 
  remainder, 
  denote 
  it 
  by 
  h. 
  Then, 
  as 
  n 
  : 
  h 
  : 
  : 
  60 
  minutes 
  : 
  an 
  

   interval 
  of 
  time, 
  which 
  being 
  subtracted 
  from 
  the 
  time 
  of 
  greatest 
  ob- 
  

   scuration, 
  and 
  added 
  to 
  it, 
  will 
  give 
  approximate 
  times 
  of 
  the 
  beginning 
  

   and 
  end 
  of 
  the 
  eclipse. 
  

  

  8. 
  Taking 
  T' 
  equal 
  the 
  approximate 
  time 
  of 
  beginning, 
  find 
  as 
  before, 
  

   for 
  this 
  time, 
  the 
  values 
  of 
  jj, 
  q, 
  H, 
  n, 
  C, 
  c, 
  g, 
  i>, 
  r\ 
  and 
  k. 
  Also 
  w 
  7 
  ith 
  

   the 
  sun's 
  declination 
  as 
  a 
  course, 
  and 
  u 
  as 
  a 
  distance, 
  find 
  the 
  corres- 
  

   ponding 
  departure, 
  marking 
  it 
  with 
  the 
  same 
  sign 
  as 
  u, 
  when 
  the 
  de- 
  

   clination 
  is 
  -f-, 
  but 
  with 
  a 
  contrary 
  sign 
  when 
  the 
  declination 
  is 
  — 
  , 
  

   and 
  denote 
  it 
  by 
  E. 
  Then 
  with 
  C 
  and 
  E, 
  respectively 
  as 
  arguments, 
  

   take 
  the 
  corresponding 
  quantities 
  from 
  Table 
  III., 
  marking 
  each 
  with 
  

   the 
  same 
  sign 
  as 
  its 
  argument, 
  and 
  denote 
  them 
  by 
  u' 
  and 
  v. 
  Then 
  

   will 
  u' 
  and 
  v' 
  express 
  the 
  horary 
  changes 
  of 
  the 
  parallaxes 
  in 
  right 
  as- 
  

   cension 
  and 
  declination, 
  at 
  the 
  time 
  T'. 
  

  

  VOL. 
  V. 
  4 
  A 
  

  

  