Royal Astronomical Society, 523 



be referred to Newton himself, the accelerating forces which act on 

 a celestial body are conceived to be divided into two parts, one of 

 which renders integrable the differential equations between the co- 

 ordinates and the time, and gives the elliptic orbit which the body 

 would describe about a centre of force if there was no disturbance ; 

 while the arbitrary quantities introduced by this first integration are 

 supposed to be rendered variable by the other part, and their varia- 

 tions determined by means of differential equations of the first order, 

 whose integrals (usually obtained by successive approximation) give 

 the elements of the true perturbed orbit, from which the radius 

 vector, longitude, and latitude of the body at any given time are com- 

 puted. 



The first example of this method of computing the planetary per- 

 turbations was given by Euler in the Berlin Memoirs for 1749, 

 where he obtains the differential equations of the first order of the 

 inclination and longitude of the node by varying the arbitrary con- 

 stants which express these two elements in the elliptic orbit. But 

 though Euler afterwards succeeded in finding expressions for the 

 variations of some of the other elements, the complete development 

 of the method, and its application not only to physical astronomy, 

 but to the general theory of mechanics, is due to Lagrange ; and it 

 forms the distinguishing feature, so far as dynamics are concerned, 

 of the beautiful system of mathematical analysis which that illus- 

 trious geometer has bequeathed to science in the Me'canique Analy- 

 tique. 



The method of analysis which we are now considering, is attended 

 with peculiar advantages when applied to the determination of the 

 secular inequalities of the orbits, in the development of which the 

 greatest triumphs have been achieved of which physical astronomy 

 can boast since the discoveries of Newton. It was by this means 

 that Lagrange demonstrated that the greater axes of the planetary 

 orbits are affected by no inequalities independent of the configuration 

 of the bodies, and consequently that amidst all the fluctuations of the 

 system, the mean distances of the planets from the sun, and there- 

 fore also their mean motions, remain for ever and unchangeably the 

 same. It was by the same means Laplace formed exact expressions 

 for the secular variations of the eccentricities and inclinations* and 

 thence proved that the changes of those elements must always be 

 inconsiderable ; that they do not increase indefinitely with the time, 

 but after a longer or shorter period again resume their former values. 

 These conclusions, which were confirmed by the subsequent and 

 more complete analysis of Poisson, lead immediately to what may 

 be regarded as the most remarkable triumph of modern science, 

 namely, the stability of the solar system ; for they show that, how- 

 ever the motions and positions of the several planets and satellites 

 may be deranged and disturbed by their mutual perturbations, the 

 variations which take place in the magnitudes and forms and posi- 

 tions in space of the different orbits are not only periodic, but con- 

 fined within narrow limits. 



But, although in the hands of these great masters of analysis the 



