SECOND REPORT— 1832. 
r 
170 
as nearly the state of physical astronomy. The method of in¬ 
vestigating the perturbations of the radius vector and longitude 
and latitude of a planet, and of expressing them by means of a 
single function, was well understood. The treatise in which this 
(and nearly everything relating to planetary perturbation,) is 
given with the greatest extension, is the Mecanique Celeste , a 
work which contains, without any acknowledgement, a vast quan¬ 
tity of the labours of preceding and contemporary writers. The 
method commonly referred to by the Germans is Klugel’s, given 
in vols. 10 and 12 of the Gottingen Transactions : it differs little 
from that of the Mecanique Celeste. The general conception of 
the variation of elements had long been formed, and expressions 
had been given for each variation; but as they depended on 
differential coefficients of the perturbing function with regard 
to the coordinates of the disturbed planet, and not with regard 
to the elements themselves, they could not easily be applied to 
the planets. Still it was possible to use them, and Laplace has 
used them in one instance. The theory of the secular varia¬ 
tions of the elements, the limits of variation of the eccentricity 
and inclination, the unlimited variation of the perihelion and 
node, and the permanency of the axis major, were (to a certain 
degree of approximation,) well understood. The perturbations 
depending on the second order of the disturbing force were 
well understood by Laplace. The long inequality of Jupiter 
and Saturn (a discovery which has been stated, though not 
quite correctly, to have “ banished empiricism from astronomy,”) 
had been calculated, and even the terms of the second order 
had been included (by proper application of the expressions 
for the variation of the elements): the acceleration of the moon’s 
mean motion had also been explained, and the inequality de¬ 
pending on the sun’s parallax had been pointed out, as well as 
that depending on the earth’s ellipticity. And (which appears 
to me the greatest step of all,) the remarkable relation between 
the motions of Jupiter’s three first satellites, which exists in 
consequence of their mutual perturbations, and depends on 
the second order of the disturbing force, had been explained. 
These theories had been numerically applied to all the planets, 
the terms depending on the second and third powers of the 
eccentricities being (unnecessarily) developed by a method dif¬ 
ferent from that used for the first powers. The lunar theory 
was almost perfect. The general methods of computing the 
perturbations of comets had been well explained by Lagrange. 
With regard to the figure of the planets, Laplace’s remarkable 
theory had appeared. The theorems for precession, change 
of obliquity of ecliptic, (depending on the change of both the 
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