REPORT ON ASTRONOMY. 
177 
Of the methods used by the German astronomers for the 
calculation of the perturbations of the small planets, I can give 
no complete account. I regret this the more, because the mag¬ 
nitude of their perturbations is far greater than those of any 
other planets. For though it may not appear, as far as their 
general theory has yet been carried, that they have equations 
as large as the great inequality of Saturn, which however is 
450 years in passing from one extreme value to the opposite, 
yet the magnitude of their perturbations in a given time, one 
year for instance, and the consequent irregularity of their mo¬ 
tion, is very much greater than that of Saturn. This only I 
can state, that the Germans do not generally compute the per¬ 
turbations of longitude, latitude, and radius vector, but the 
perturbations of the elements of the orbit; and these, I believe, 
entirely by mechanical quadratures ; in other words, by summa¬ 
tion instead of integration, in a method analogous to that which 
they use for comets. Perhaps in some calculations for Vesta, 
as in part of those by Encke, Berlin Memoirs 1826, they may 
use Tables and apply the perturbations directly to the radius 
vector, &c. : but even in this instance, the most important part 
of the perturbations, namely, those produced by Jupiter, are 
computed by quadrature, the elements being corrected for per¬ 
turbation : and Encke conceives this to be more accurate than 
the use of Tables. The intervals used here are of forty-two 
days each, and the fresh corrected elements are used after every 
sixth or seventh interval. 
The groundwork of Lagrange’s method tor the perturbation 
of comets ( Mec . Cel . tom. 4. liv. 9.) consists in estimating the 
disturbing forces resolved in the direction of three rectangular 
co-ordinates, finding the effect of these on the elements of the 
comet’s orbit, and performing the integration by quadratures. 
The method given by Bessel in the Untersuchungen uber die 
scheinbare und walire Balm des im Jahre 1807 erscliienenen 
grossen Kometen , referred to by the Germans as the standard 
work, consists in resolving the disturbing forces in the direc¬ 
tion of the comet’s radius vector, a perpendicular to the 
radius vector in the plane of the orbit, and a perpendicular 
to the orbit. The inclination and node are referred to the 
ecliptic. From these quantities, expressions are found for the 
variations of the elements, which are integrated by quadra¬ 
tures. This is the method used by Bessel in the calculations 
relative to Olbers’s comet, Berlin Mem. 1812-1818. For the 
comet of 1807, Bessel has calculated the quantities for every 
thirty days; and for Olbers’s comet he has taken intervals of 
twenty-five days during its visibility, of one year from 1815 to 
M 
