104 SOME METHODS OF COMPUTING THE RATIO O^ 



outset by correcting the tabular places of the sun for parallax 



(47) and by applying to R, R, and R' a small correction = — earth's 

 radius X cosine of the sun's zenith distance at the time of each 



(48) observation. By neglecting terms of the second order in either 

 of the above equations, approximate values are obtained of the 

 ratios y or -j, which will be more or less accurate according as 

 the assumed values of ^ and ^^ vary from the true. 



II. For the correction of these latter quantities different meth- 

 ods may be employed, in the several cases which may occur ; 



(49) in all of which the object in view is to obtain from the use of er- 

 roneous values new values, which shall be nearer the truth than 

 those from which they have been derived ; the convergence of 

 the successive approximations depending on the amount of helio- 

 centric motion in the intervals between the observations. 



Using in (4) 0' = 0" = O, 



(50) = ^% sin. ^ — f^] E sin. + ^ sin. ©' — R' sin. 0". 



(51 ) = t^ p sin. 6 — jR sin. + z/' R' sin. 0'. 



And in a similar manner, 



(.52) = — ^ p' sin. e'—j7T^ R sin. e + ^-^R' sin. e' ~ R" sin. 0". 



(53) = — j^ g' sin. 6'— J R sin. O -\- j' R' sin. 0'. 



(54) = p" sin. 5" _ ^ B sin. + ^^ R' sin. 0'— R" sin. 0". 



(55) = q" sin. 6" — jR sin. O -\- J R' sin. 0'. 



These may be expressed in terms of right ascension and dec- 



(56) lination. As, for instance, putting in (13), A^a, D = d, A = a", 

 and D' = 6" ; or denoting by C the right ascension of the point 

 in the equator, where it is intersected by the great circle passing 

 through the first and last places of the comet, and putting in 

 (13), A = a, D = d, A'=C', D' = 0: 



