256 NOTE ON DELAUNAY'S EXPRESSION [30 



Hansen 



Argument. Delaunay. Adams. transformed by 



Delaunay. Damoiseau. Newcomb. 



D+i (t+x) -6-0971 -d'loe -o'-ioe 



3D 3t 0-0158 O'OOS 0'003 



3D-1 (3t-x) -0-0199 -0-036 -0'037 



3D + 1 (3t + x) 0-0025 0'002 



D-l (t-x) 0-0076 0-014 +0-011 



2D-21-1' (2t-2x-z) -0-0127 -O'OIS -0'019 



W + l (4t + x) 0-0185 0-032 0-043 



W-21-1' (4t-2x-z) 0-0159 0-030 0'032 



4D-1' (U-z) O'OllO 0-034 0'035 



In the above many very small coefficients have been omitted. 



As stated in my paper in the appendix to the Nautical Almanac for 

 1856, or in the Monthly Notices, Vol. xm. p. 177, my coefficients of parallax 

 were obtained by comparing the results of the theories of Damoiseau, Plana, 

 and Pontecoulant, and tracing out the origin of the discordances in the 

 cases where those results did not agree with each other. These coefficients 

 were also compared with those which I obtained by a transformation of 

 Hansen's preliminary results as given in a paper in Vol. xvn. of the Astro- 

 nomischc Naclinchten. 



In Pontecoulant's method the expression for the reciprocal of the radius 

 vector is first found, and then the expression for the longitude is derived 

 from it. Hence the analytical values of the coefficients of parallax, given 

 by Pontecoulant, Vol. iv. pp. 149 152, 281, 282, 336, 337, are at least as 

 accurate as the values of his coefficients of longitude. 



In his final expression, however, in pp. 568 572, in which the several 

 terms of the reciprocal of the radius vector are collected together, he neglects 

 all terms of orders higher than the 5th, and the same omission takes place 

 in the conversion of his coefficients of parallax into numbers. 



Accordingly these numerical values, which are calculated in pp. 599 601, 

 and collected together in p. 635, nearly coincide with the values of Delaunay, 

 but are on the whole still less accurate. 



It is greatly to be desired that some intrepid and competent calculator 

 would undertake to make the numerous substitutions which would be required 

 in order to find, by Delaunay's method, the expression for the reciprocal 



