195] REPORT ON THE RECENT PROGRESS OF THEORETICAL DYNAMICS. 
159 
observed that the variation of one of the elements, viz. the mean distance, was 
expressible in a remarkable form by means of the differential coefficients of the 
disturbing function taken with respect to the time t, in so far as it entered into the 
function through the coordinates of the disturbed planet. I am not able to say at what 
time, or whether by Euler, Lagrange, or Laplace, it was observed that such differential 
coefficient with respect to the time was equivalent to the differential coefficient of the 
disturbing function with respect to one of the elements. But however this may be, 
the notion of the representation of the variations of the elements by means of the 
differential coefficients of the disturbing function with respect to the elements had 
presented itself à posteriori, and was made use of in an irregular manner prior to 
the year 1800, and therefore some eight years at any rate before the establishment 
by Lagrange of the general theory to which these forms belong. 
4. Poisson’s memoir of the 20th of June, 1808, “ On the Secular Inequalities of 
the Mean Motion of the Planets,” was presented by him to the Academy at the age 
of twenty-seven years. It contains, as already remarked, an expression in finite terms 
for the variation of the longitude of the epoch. But the memoir is to be considered 
rather as an application of known methods to an important problem of physical 
astronomy, than as a completion or extension of the theory of the variation of the 
planetary elements. The formulae made use of are those involving the differential 
coefficients of the disturbing function with respect to the coordinates ; and there is 
nothing which can be considered an anticipation of Lagrange’s idea of the investigation, 
à priori, of expressions involving the differential coefficients with respect to the 
elements. But, as well for its own sake as historically, the memoir is a very important 
one. Lagrange, in his memoir of the 17th of August, 1808, speaks of it as having 
recalled his attention to a subject with which he had previously occupied himself, but 
which he had quite lost sight of; and Arago records that, on the death of Lagrange, 
a copy in his own handwriting of Poisson’s memoir was found among his papers ; and 
the memoir is referred to in, and was probably the occasion of, Laplace’s memoir also 
of the 17 th of August, 1808. 
5. With respect to Laplace’s memoir of the 17th of August, 1808, it will be 
sufficient to quote a sentence from the introduction to Lagrange’s memoir :—“ Ayant 
montré à M. Laplace mes formules et mon analyse, il me montra de son côté en 
même temps des formules analogues qui donnent les variations des élémens elliptiques 
par les différences partielles d’une même fonction, relatives à ces élémens. J’ignore 
comment il y est parvenu ; mais je présume qu’il les a trouvées par une combinaison 
adroite des formules qu’il avait données dans la Mécanique Céleste.” This is, in fact, the 
character of Laplace’s analysis for the demonstration of the formulae. 
6. In Lagrange’s memoir of the 17th of August, 1808, “On the Theory of 
the Variations of the Elements of the Planets, and in particular on the Variations of 
the Major Axes of their Orbits,” the question treated of appears from the title. The 
author obtains formulae for the variations of the elements of the orbit of a planet in 
terms of the differential coefficients of the disturbing function with respect to the 
elements ; but the method is a general one, quite independent of the particular form