4 REPORT—1857. 
there is nothing which can be considered an anticipation of Lagrange’s 
idea of the investigation, a@ 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 17th 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 me- 
moir :—“ Ayant montré 4 M. Laplace mes formules et mon analyse, il me 
montra de son cété en méme temps des formules analogues qui donnent 
les variations des élémens elliptiques par les différences partielles d'une 
méme fonction, relatives 4 ces élemens. J'ignore comment il y est par- 
venu ; mais je présume qu'il les a trouvées par une combinaison adroite des 
formules qu’il avait donné dans la ‘ Mécanique Céleste.’” This is, in fact, 
the character of Laplace’s analysis for the demonstration of the formule. 
- 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 ap- 
pears from the title. The author obtains formule 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 of the integrals, and 
the memoir may be considered as the foundation of the general theory. 
The equations of motion are considered under the form,— 
dx _1l+m ra OO 
dr dx’ 
d’y_1+m,_ dQ 
dé? ae dy’ 
az _1+m 2 dO 
dt* 7 dz’ 
and it is assumed that the terms in Q being neglected, the problem is com- 
pletely solved, viz. uy that the three coordinates, x,y, 2, and their differential 
coefficients, 2’, y', z', are each of them given as functions of ¢, and of the 
constants of integration a, b,c, f, g,h; the disturbing function Q is conse- 
quently also given as a function of ¢, and of the arbitrary constants. The 
velocities are assumed to be the same as in the undisturbed orbit. This 
gives the conditions 
dz=0, dy=0, dz=0; 
and then the equations of motion give 
yd _ dO sdy_ dO, jz _ a0 
dt dx dt dy dt dz 
equations in which dx, &c. denote the variations x, Ho arising from the 
variations of the arbitrary constants, viz., tv Oa +% ab + &ce. The 
differential coefficients ad &c., can of course be expressed by means of 
