100 SOME METHODS OF COMPUTING THE RATIO OF 



(13) = [sin. {A' — a) tan. D — sin. (A' —A) tan. 5 + sin. (a— A) tan. D'] ^-—j q cos. 6 



— [sin. (^'— a')tan.J9 — sin.(^' — ^)tan.e' + sin.(a'— ^)tan.2)'] t^p'cos.fl' 

 + [sin. (.4' — a")tan. D— sin. {A'— A) tan. 6" -\- sin. («"— .4)tan.I>'] 9"cos.fl" 



— [sin.(^'— ©)tan.D— sin.(^'— ^)tan.0+sin.(© — ^)tan. iy]E^^Bcos.0 

 + [sin.(^'— ©')tan.I>— sin.(^'— J)tan.0'+sin.(©'— ^)tan.2)']|^B'cos.0' 



— [sin. (^'—©") tan. U— sin. {A'— A) tan.0"+sin. (©"—.4) tan. D'] R"cos. 6". 

 (i3o) The sum of the last three terms being by (5) and (8) of the 



second order in r. 



(14) By introducing in (13) the proper values of the arbitrary quan- 

 tities A, A,' D, and D', we can obtain all the equations wliich can 

 be derived from (4), by (10), in a form in which right ascen- 

 sions and declinations may be employed directly in the computa- 

 tions, but not always with advantage over the simpler expressions 

 of (10), except in those approximations in which the effect of 

 parallax, and the quantities J and J' are neglected ; because the 



(15) angles 6, 6', &c., in the equations derived directly by (10) from 

 (4) have a definite geometrical meaning, which is an advantage 

 in computation. 



(16) Putting 0' and in (4) and (5) = 0, they become 



(17) = i;^' o sin. e+ q" sin. e" — ^-^ R sin. — R" sin. ©". 



(18) = ^^^ 9 sin. d-j-e" sin. 6" — J R sin. 0. 



(19) 6, 0, &c., being here perpendiculars upon the great circle joining 

 the middle places of the sun and comet. 



(17) and (18) may be expressed in terms of the original values 

 of 6 and a, &c., by taking in (13) for the points B and B' (11) 

 the places of the sun and comet at the second observation, that 

 \s,A = a\ D=d', A' = e', D' = 0', which give 



(20) 0=[sin. (O' — «) tan. 5'— sin. (Q'— «') ta"- ^ + sin. {« — «') tan. 0'] ^^ p cos. (9 



4- [sin.(0'— «")tan. (9'— sin. (©'—«') 'an. 5"+sin («"— a')tan. 0'] p"cos.e" 



— [sin.(0'— 0)tan.e'— sin.(0'— a')tan.0-f-sin.(O— a')tan.0']^'Bcos. 



— [sin.(0'— 0")tan.fl'— sin.(©'— a')tan.0"+sin.(O"— o')tan.0']E"cos.0". 



