17] ON NEW TABLES OF THE MOON'S PARALLAX. 91 



A strange fatality seems to have attended all Burckhardt's calculations 

 respecting the Moon's Parallax. In the Connaissance des Temps for the 

 year xv of the Republic, he gives a comparison between the values furnished 

 by Mayer's and Laplace's theories, and he concludes that the error of the 

 former may sometimes amount to 7". 



But this difference is caused almost wholly by an error in his own 

 transformation of Laplace's expression. In the formation of Mayer's Argu- 

 ments of Evection and Variation, the true longitude of the Sun is employed, 

 but Burckhardt appears to have inadvertently used the mean longitude 

 instead of it, an error which is the exact converse of the one above noticed 

 with respect to his own Tables. 



After examining Burckhardt's Table of Parallax, I was naturally led to 

 scrutinize more closely the results of the theories of Damoiseau, Plana, and 

 Pontecoulant, with respect to the same subject. Although the differences 

 between these were very trifling when compared with the errors of Burck- 

 hardt, still they were greater than we had a right to expect, considering 

 the close agreement which existed with respect to the equations of longitude. 

 In the theories of Damoiseau and Plana, the expression for the projection 

 of the Moon's radius vector on the Ecliptic in terms of her true longitude 

 is required in order to find the relation between that longitude and the 

 time, and therefore no pains have been spared to obtain it with accuracy ; 

 but in the subsequent operations and transformations necessary in order to 

 deduce the expression for the Parallax in terms of the time, the same care 

 has not been employed. In Ponte'coulant's theory the time is taken as 

 the independent variable, and consequently the analytical expression for the 

 Parallax in the form required is obtained immediately, and is developed 

 to as great an extent as the corresponding expression for the longitude, 

 yet in the conversion of his formula into numbers he neglects all the 

 terms beyond the fifth order, so that several of the resulting coefficients 

 are sensibly in error. 



I have endeavoured to supply these defects and omissions. 



In the seventeenth volume of the Astronomische Nachrichten, M. Hansen 

 gives the expression which he has obtained for the logarithm of the sine 

 of the horizontal Parallax, by means of his new method of treating the 

 Lunar Theory. I have transformed this expression with the care which its 

 great value deserves, so as to compare it with the results of the former 



theories. 



122 



