38 Analysis of the Mtcanique Celeste qf M. La Place, 



simple method, by which we immediately obtain the difFeren* 

 tial equations which determine the perturbations ordered ac- 

 cording to the powers and to the products of the excentricities 

 of the inclinations of the orbits. This method consists in sup- 

 posing the radius vector of the disturbed orbit expressed by 

 a function of the same form as that of the elliptic orbit ; the 

 quantity which enters into this function ia then found, given 

 as in the elliptic motion by a linear differetitial equation of 

 the second order with constant coefficients, .increased by a 

 last term depending on the action of the disturbing forces; 

 circumstances which permit us to apply to this equation the 

 methods of integration previously detailed : in the latter, re- 

 gard is had only to the first power of the disturbing force. 

 It is necessary, for what precedes, to develop a certain func- 

 tion of the masses and of the mutual distances of the bodies 

 of the system in a converging series of the sines and cosines 

 of angles increasing proportionally to the time. The au- 

 thor gives the method of attaining this, and employs, in 

 this calculation, in a very elegant manner, the characteristic 

 .of the integrals of finite differences : this permits him to ex- 

 press with much simplicity the development sought, and 

 the product of this development by the sine or cosine of any 

 angle. According to what precedes, he deterniines the 

 perturbations of motion in longitude, latitude, and those of 

 the radius vector of the orbit, carrying his precision as far 

 as quantities of the order of the eccentricities and of the 

 inclinations of the orbits, and demonstrates the convergency 

 of these results, whatever be the ratio of the distances of 

 the planets which we consider to the sun ; a circumstance 

 the more important to observe, since otherwise it would 

 have been impossible to express analytically the reciprocal 

 perturbations of the planets, with respect to which the rela- 

 tions of these distances approach to unity : he afterwards 

 collects these results which contain the whole theory of the 

 planets, when we neglect the squares and products of the 

 eccentricities, and of the inclinations of the orbits, which is 

 most frequently permitted. After having shown how we 

 could, if it were necessary, obtain a greater approximation, 

 he gives the means of estimating the degree of precision of 



the 



