REPORT ON ASTRONOMY. 
171 
equator and the ecliptic), &c., were almost complete. It will 
be seen that comparatively little has been added since the be¬ 
ginning of the century. 
In 1808, Laplace presented to the French Bureau des Lon¬ 
gitudes a Supplement to the third volume of the Mecanique 
Celeste. Lagrange immediately after produced equivalent re¬ 
sults obtained in a different way ( Memoires de VInstitut 1808). 
These essays may be considered as completing the theory of 
planetary perturbations. Their object was to express the va¬ 
riation of all the elements by differential coefficients of the per¬ 
turbing function (supposed to be expanded in terms of the ele¬ 
ments and the time), taken with respect to the elements only, 
and multiplied only by functions of the elements. The pertur¬ 
bations of the elements can therefore be found from the usual 
expansion of the perturbing function: and then the true posi¬ 
tion in longitude and latitude can be found by using the ele¬ 
ments corresponding to that time as if they were invariable. It 
has been objected, that whereas we want only three quantities 
(perturbations of radius vector, longitude, and latitude,) we in 
fact investigate six (those of the six elements). I believe, how¬ 
ever, that in any case the investigation is not more difficult, 
and in many cases the saving of time is very great. For instance, 
in an inequality of long period, which is always accompanied 
by other terms ; if the method of variation of elements is used, 
the development of one term only of the perturbing function is 
sufficient: if the original methods were used, the development 
of several terms would be necessary, and the treatment of each 
of these would be more troublesome. But it is principally 
with regard to terms of the second and higher orders of the 
disturbing force that its advantage is felt: it is necessary to 
substitute in the expressions values of the elements as near as 
possible to the true ones, and the method therefore becomes a 
very simple successive approximation, no reference to the lon¬ 
gitude &c. being necessary till the whole is completed. 
To reduce the calculation of perturbations to a mere me¬ 
chanical operation, nothing was wanting but the expansion of 
the perturbing function. This was given in part by Burck- 
hardt in the Memoir es, 1808 : the terms depending on the in¬ 
clination were not included ; but those depending on the ec¬ 
centricities and their combinations were given to the sixth 
order. 
On the variation of constants generally (in mechanics as well 
as astronomy), and the secular variation of the elements, the 
most able papers are by Lagrange, Memoir es, 1808 and 1809 ; 
Poisson, Journal de VEcole Polytechnique, vol. 8, and Me- 
