254 NOTE ON DELAUNAY'S EXPRESSION [30 



the radius vector, can only be indirectly inferred from observation through 

 the parallax, to the sine of which it is inversely proportional. 



Hence the accuracy of the theoretical values of the longitude and 

 latitude can be much more severely tested by observation than that of the 

 radius vector. 



Delaunay has, on account of this circumstance, found the analytical 

 expressions for the longitude and latitude with a much greater degree of 

 accuracy than that for the reciprocal of the radius vector. 



In the two former coordinates he has taken into account generally the 

 terms of the 7th order, and in cases where the convergence of the series 

 is found to be slow, he has included terms of the 8th and 9th orders. 

 In the reciprocal of the radius vector, however, he has confined his attention 

 to terms of the 5th order. Consequently, while the coefficients of the 

 inequalities in longitude and latitude as found by him are generally only 

 a small fraction of a second in error, the inequalities in the reciprocal of 

 the radius vector are not found with sufficient precision to give even the 

 parallax itself with all the accuracy which is desirable. 



The coefficients of the inequalities of the parallax given by me in 

 Vol. xiu. of the Monthly Notices, p. 263 (see p. 109 above), are considerably 

 more accurate than those of Delaunay. 



In the paper just referred to, I have given the coefficients to hun- 

 dredths of a second only, and, as I have there stated, terms with coefficients 

 less than Q"'05 have been omitted except when they can be included in 

 the same table with larger terms. 



It may be worth while to give here a more complete view of the 

 values of the coefficients of parallax which I obtained in 1853. These 

 results are exhibited to thousandths of a second, as the calculation gave 

 them, although the figures in the last place of decimals are not to be 

 depended upon. 



I add, for the sake of comparison, Delaunay's coefficients of the cor- 

 responding terms as given in the Connaissance des Temps for 1869, and 

 also the coefficients of Hansen's theory as transformed by Professor Newcomb. 

 The several arguments are expressed in Delaunay's notation*. 



* In the following table the arguments are also given in Damoiseau's notation, which 

 has been employed in paper 18 (see p. 109 above). 



