REPORT ON ASTRONOMY. 173 
sions for the variation of the elements are of course assumed 
as a foundation for the first investigations. 
In the Berlin Memoirs , 1824, Bessel has given a method of 
investigating separately the effects of perturbation produced 
by a planet’s action on the sun and its action on another planet. 
Thi s was done in consequence of the agitation of the question, 
to which I have before alluded, whether the absolute force of 
the planet on these bodies was the same ; a question first 
started (I believe,) by John Tobias Mayer, Gott. Trcms. 
1804-1808. The physical investigation consists merely in taking 
the two terms of the perturbing function separately: this 
paper however is remarkable for the mathematical part of the 
process, which by a mixture of general integration and definite 
integration, assisted by special Tables, seems well adapted to 
the accurate calculation of planetary inequalities. The sub¬ 
jects of investigation are the perturbations of radius vector, lon¬ 
gitude, and latitude (that of the longitude being expressed in¬ 
dependently of the radius vector,) to the first order of the dis¬ 
turbing force. 
In the Phil. Trans. 1830, 1831, and 1832, Mr. Lubbock 
has given four papers on the general problem of perturbations. 
The object of the first of these is to give expressions for the 
variation of the elements which shall be true to all orders of 
the disturbing force, (which however holds with regard to La¬ 
place’s and Lagrange’s expressions,) togetherwith equations in 
which the eccentric anomaly is the independent variable. In 
the second it was shown that the perturbations of the recipro¬ 
cal of the radius vector might be found more readily than those 
of the radius vector itself. The rest of these papers (relating to 
perturbations in general,) is occupied with expansions, and with 
theorems equivalent to those of Laplace, but in a different 
form. In the Phil. Trans. 1832, Mr. Ivory has also given an 
investigation of the perturbation of elements, and Mr. Lubbock 
has shown the identity of the results obtained by perturbation 
of the elements and by perturbation of the co-ordinates : it is 
not the object of these papers to extend the theory of pertur¬ 
bations. In the Munchen Denkschriften and the Turin 
Memoirs , Pfaff and Cisa de Gresy have given various expres¬ 
sions, which however are only equivalent to those of preceding 
writers. 
In the Milan Ephemeris for 1818, and the Memoirs for 1823, 
Carlini and Laplace have shown that some of the series by 
which a planet’s place is expressed in terms of the mean longi¬ 
tude, cease to be convergent when the eccentricity exceeds 
0*62. In the Berlin Memoirs 1816-1817, Bessel has expressed 
