470 Astronomical Societj/. 



while the action of the stars is at least of the order of their 

 squares or even cubes. 



The 3rd 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 fol- 

 lowed by M. Laplace, which he considers defective and in 

 some measure 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 litde from that given in p. 98 of the third volume 

 of the Mecanique Celeste, and in p. 32 of the tables of Mer- 

 cury published by M. Lindenau, yet this apparent accordance 

 is merely a consequence of the excessive smallness of the nu- 

 merical coefficient of the term in question, and that his object 

 has rather been to rectify the analytical fornnilse than the nu- 

 merical results, by taking into consideration all the terms of 

 the same order ; without which he considers it very possible to 

 commit material errors in the final results of such operations. 



The 4th chapter has for its object an examination of M.La- 

 place's method of taking account of the square of the disturbing 

 force in the theory of the great inequality' 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 consequence, it may be observed, one might naturally pre- 

 sume from the form of the multiplier itself, into which the 

 simple ratio of the masses only enters as a factor). 



[? = -i5v/^?.] 



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

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

 verify every part of the developments and calculations ; and on 

 reducing his formulae to numbers, obtains (not, as he says, 

 without surprise) a final result, of a contrary sign to that of 

 Laplace, and only one third of its amount, the coefficients of 

 the terms of the great inequality arising from the square of 

 the disturbing force being according to M. Plana 

 — l"-9200 and + 5"'5Ti5 for Jupiter 

 + 25"-1036 and — 12"'8932 for Saturn. 



The 



