IN PHYSICAL ASTRONOMY. 
41 
fore they may be included in the same inequality, either in the expression for 
the parallax or in that for the mean longitude. 
In the elliptic theory 
e' 2 = e-{ l — sin 2 1 sin 2 (v — -nr) } See Phil. Trans. 1831, p. 56. 
These equations of condition are true, however far the approximation be 
carried; provided only, that the arbitrary quantities e and sin / be determined 
so as not to contain the mass of the sun implicitly. 
The determination of the coefficients of the arguments t + z, t — x + z, and 
2t — 2x 2z will require particular attention in the numerical calculation. 
According to the analysis of M. Poisson (Journal de l’Ecole Poly technique, 
vol. viii. and M6moires de l’Academie des Sciences, vol. i.), the coefficient of the 
argument t — x -f- z in the quantity J' d R equals zero. Conversely therefore 
this theorem may furnish an equation of condition between some of the coeffi- 
cients. According to M. Damoiseau, the coefficient of this argument in the 
expression for the longitude is only 2" , 05, and the argument 2t — 2x 2 z is 
insensible. The expressions which I gave, Phil. Trans. 1830, p. 334, are well 
adapted for finding in the theory of the moon, in which the square of the dis- 
turbing force is so sensible, by means of the variation of the elliptic constants, 
the coefficient of any inequality which arises from the introduction of a small 
divisor, these expressions being true, however far the approximation is carried. 
It may be seen in the authors themselves, or in the excellent history of phy- 
sical astronomy by M. Gautier *, that the methods of Clairaut, D’Alembert 
and Euler, do not resemble in any respect those which I have employed. Both 
Clairaut and D’Alembert, by means of the differential equation of the second 
order in which the true longitude is the independent variable, obtained the 
expression for the reciprocal of the radius vector in terms of cosines of the true 
longitude. They substituted this value in the differential equation which de- 
termines the time, and obtained by integration the value of the mean motion 
in terms of sines of the true longitude. By the reversion of series they then 
found the true longitude in terms of sines of the mean motion. The method 
* Essai Historique sur le Probleme des Trois Corps, p. 53. 
MDCCCXXXII. 
G 
