66 Proceedings of Philosophical Societies, [Jan. 



elusion by another and direct method. He then applies his 

 reasoning to numerical examples, and computes the quantity by 

 which the moon's mean distance from the earth is permaueiitly 

 altered by the sun's action, which he finds to be about 1-lOOth 

 of the radius of the globe of the moon, in augmentation, the 

 corresponding increase of the periodic time being about one- 

 fourth of a day. The excentricity too undergoes an aUeration 

 in its mean quantity from the same cause, equal to about 0*0007 

 of its actual amount. 



A similar artifice in the use of an arbitrary constant added in 

 one of the necessary integrations for arriving at the first term of 

 the motion of the moon's perigee, M. Plana observes, has ena- 

 bled 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 

 superfluous constant, by the aid of an equation he deduces for 

 the variable portion of the moon's radius vector. 



The 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 eflfect of 

 perturbation. The other he states to have been first employed 

 by Lagrange in the volume of the Memoirs of the Academy of 

 Berlin for 1783. 



The author next proceeds to examine those parts of the theory 

 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 Mtcariique 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 perturbations of the 

 planets due to the ellipticity of the sun by the formulse 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 

 elliptic 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. 



Tiie second chapter of this paper is devoted to the consider- 

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

 variations of the planetary system. The expressions for the 

 secular 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 'J where Laplace has f-, 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 mapy hundreds of centuries ; so that these discre- 



