1826.] Astronomical Society. 67 



nancies are practically of no importance. He remarks too that 

 this cause of perturbation prevents the equations between the 

 squares of theexcentricities, the masses, and square roots of the 

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

 tary 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 the 

 action of the stars is at least of the order of their squares or 

 even cubes. 



The third chapter is devoted to the evaluation of those terms 

 in the theory of the perturbations of Mercury by the Earth 

 whose coefficient, being divided by the square of the difference 

 between the mean motion of Mercury and four times that of the 

 Earth, may acquire a notable value by the smallness of its 

 divisor, the author first examines the indirect method followed 

 by M. Laplace, which he considers defective and in some mea- 

 sure illusory, and then substitutes a method of his own. After 

 going through all the very laborious calculations of the analytical 

 and 'numerical values of the coefficients he arrives at a final 

 result, of which he remarks that although it differs very little 

 from that given in p. 98 of the third volume of the M^canicjue 

 Celeste, and in p. 32 of the tables of Mercury published by 

 M. Lindenau, yet this apparent accordance is merely a conse- 

 quence of the excessive smallness of the numerical coefficient 

 of the term in question, and that his object has rather been to 

 rectify the analytical formulee than the numerical results, by 

 taking into consideration all the terms of the same order ; with- 

 out which he considers it very possible to commit material 

 errors in the final results of such operations. 



The fourth chapter has for its object an examination of M. 

 Laplace's method of taking account of the squareof the disturb- 

 ing force in the theory of the great inequahty of Jupiter and 

 Saturn. 



In this investigation the author is led to conclude, that the 

 equation connecting the reciprocal perturbations of the mean 

 motions of two planets, and by which the one may be derived 

 from the other by a simple multiplication, holds good only when 

 the first powers of the disturbing forces are considered (a con- 

 sequence, it may be observed, one might naturally presume 

 from the form of the multiplier itself, mto which the simple 

 ratio of the masses only enters as a factor). 



M. Plana gives this part of his paper with the fullest possible 

 detail, in order, he observes, to enable astronomers to verify 



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