TSA 
ON THEORETICAL DYNAMICS. 3 
‘8. The idea of a force function seems to have originated in the problems 
of physical astronomy. Lagrange, in a memoir ‘ On the Secular Equation 
of the Moon,’ crowned by the French Academy of Sciences in the year 
1774, expressed the attractive forces, decomposed in the directions of the 
axes of coordinates, by the partial differential coefficients of one and the same 
function with respect to these coordinates. And it was in these problems 
natural to distinguish the forces into principal and disturbing forces, and 
thence to separate the force function into two parts, a principal force function 
and a disturbing function. The problems of physical astronomy led also to. 
the idea of the variation of the arbitrary constants of a mechanical problem. 
For as a fact of observation the planets move in ellipses the elements o 
which are slowly varying; the motion in a fixed ellipse was accounted for 
by the principal force, the attraction of the sun; the effect of the disturbing 
force is to produce a continual variation of the elements of such elliptic 
orbit. Euler, in a memoir published in 1749 in the ‘ Memoirs of the Academy 
of Berlin’ for that year, obtained differential equations of the first order for 
two of the elements, viz. the inclination and the longitude of the node, by 
making the arbitrary constants which express these elements in the fixed 
orbit to vary: this seems to be the first attempt at the method of the varia- 
tion of the arbitrary constants. Euler afterwards treated the subject in a 
more complete manner, and the method is also made use of by Lagrange in 
his ‘ Memoir on the Perturbations of the Planets’ in the Berlin Memoirs for 
1781, 1782, 1783, and by Laplace in the ‘Mécanique Céleste,’ t. i. 1799. 
The method in its original form seeks for the expressions of the variations 
of the elements in terms of the differential coefficients of the disturbing 
function with respect to the coordinates. As regards one element, the longi- 
tude of the epoch, such expression (at least in a finite form) was first ob- 
tained by Poisson in his memoir of 1808, to be spoken of presently ; but I 
am not able to refer to any place where such expressions in their best form 
are even now to be found; the question seems to have been unduly passed 
over in consequence of the new form immediately afterwards assumed by the 
method. It was very early observed that the variation of one of the ele- 
ments, 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 co- 
ordinates 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 diffe- 
rential 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 @ 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 In- 
equalities 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 re- 
marked, 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 formule made use of are those involving the differential 
coefficients of the disturbing function with respect to the coordinates; and 
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