174 
SECOND REPORT— 1832 . 
the co-efficients of these series by means of definite integrals: 
and in the Conn. des Temps 1825, Poisson has done nearly the 
same thing. This principle, I believe, has lately been extended 
by Cauchy. In the Conn, des Temps 1828, Laplace has given 
means of estimating the value of distant terms in the expansion 
of the perturbing functions. 
I am not acquainted with any other additions to the theories 
of elliptic motion or perturbation in general. 
With regard to the solar theory, Nicolai, in the Berliner Jakr- 
buch 1820, investigated the secular variations of the Earth’s or¬ 
bit, as a verification of those given by Lagrange and Laplace. 
In the Phil. Trans. 1828, the author of this Report announced 
the discovery of a small inequality of long period in the Earth’s 
motion produced by the action of Venus, and a correspond¬ 
ing inequality in the motion of Venus produced by the 
Earth : the details of the calculation are given in the Phil. 
Trans. 1832. And in the Milan Epliemeris , 1830 and 1831, 
(the latest volumes that I have been able to procure,) Carlini 
has given an investigation, not yet completed, of an inequality 
in the Earth’s motion, depending on the Sun’s distance from 
the Moon’s perigee. It has commonly been thought sufficient 
to consider the motion of the centre of gravity of the Earth and 
Moon the same as if their masses were united there : but it is 
quite conceivable that a small error in this may grow up into a 
sensible inequality; and this, I believe, is the subject of the 
investigation. 
The lunar theory has been much discussed. In the Conn, 
des Temps 1813, is a paper by Laplace on the inequality of 
long period, which, from observation, seems to exist in the 
Moon’s motion. In the Mecanique Celeste he had been dis¬ 
posed to attribute it to a term independent of the Earth’s form, 
which had been pointed out by Dalembert; in this paper he 
inclines to that depending on the difference of the northern and 
southern hemispheres. This inequality, with an empirical co¬ 
efficient, was adopted in Burekhardt’s Tables. In the Conn, des 
Temps 1823, Laplace re-investigated the equations depending 
on the Earth’s ellipticity, and on comparing their values with 
those found by Burg and Burckhardt from observations, fixed 
on as its value. The Institut having offered a prize for a 
complete lunar theory, in which the values of the co-efficients 
should be calculated from theory only, two that were sent 
in were deemed worthy of the prize, one by Damoiseau, the 
other by Carlini and Plana. The latter, I believe, is not 
printed ; the former is in the Savans Etrangers. The general 
method pursued by Damoiseau is the same as Laplace’s in the 
