Astronomical Societtf. 469 



has enabled M. Laplace to avoid an error in that research to 

 which his method seemed to expose him, and to obtain the 

 true result. But he proceeds to show that this artifice is not 

 necessary, and that the same result may be obtained without 

 the use of the supei-fluous constant, by the aid of an equa- 

 tion he deduces for the variable portion of the moon's radius 

 vector. 



Tlie method employed by M. Plana has the advantage, he 

 observes, of keeping distinctly in view throughout the whole 

 analysis the primitive elements, uninfluenced by the effect of 

 perturbation. The other he states to have been first em- 

 ployed by Lagrange in the volume of the Memoirs of the Aca- 

 demy of Berlin for 1783. 



The author next proceeds to examine those parts of the 

 theoiy of perturbations, which depend on the non-sphericity of 

 the central body, and in which he remarks that the use of a 

 similar artifice in the Mecanique Celeste is accompanied with 

 greater obscurity, as a portion only of the arbitrary constant 

 is retained. He therefore enters on the investigation without 

 the use of this artifice, and deduces the results for the pertur- 

 bations of the planets due to the ellipticity of the sun by the 

 formulae for the variation of the arbitrary constants. 



The author next applies the same method to the theory of 

 the perturbations of the seventh satellite of Saturn by the ellip- 

 tic figure of the planet ; and as he here arrives at final equa- 

 tions somewhat differing from those of M. Laplace, the whole 

 process is given in copious detail. 



The 2d chapter of this paper is devoted to the consideration 

 of the effect of the actions of the fixed stars on the secular 

 variations of the planetai'y system. The expressions for the se- 

 cular variations of the excentricity and aphelion which the 

 author brings out, agree perfectly with Laplace's in form, but 

 differ in the numerical coefficients, one of the terms having the 

 coefficient -^ where Laplace has |, and another — | where 

 Laplace makes it — 1. As he subsequently observes however, 

 the action of the stars cannot possibly become sensible till 

 after the lapse of many hundreds of centui'ies ; so that these 

 discrepancies are practically of no importance. He remarks too 

 that this cause of perturbation prevents the equations between 

 the squares of the excentricities, the masses, and square I'oots 

 of the axes, so often referred to as insuring the stability of the 

 planetary system, — as well as the similar one between the 

 squares of the tangents of the inclinations, the masses, and 

 square roots of the axes, — from being mathematically exact. 

 It will be noted, however, that these equations can only be 

 regarded as proved for the first powers of the disturbing forces, 



while 



