14-4 LAPLACE. 



scribes arouud the suu ; that a diuiinntiou of the ecceutricity of the 

 ellipse inevitably induces au increase in the velocity of our satellite, and 

 vice versa; finally, that this cause suffices to explain the numerical value 

 of the acceleration Avhich the mean motion of the moon has experienced 

 from the earliest ages down to the present time.* 



The origin of the inequalities in the mean motions of Jupiter and 

 Saturn will be, I hope, as easy to conceive. 



Mathematical analysis has not served to represent in finite terms the 

 values of the derangements which each planet experiences in its move- 

 ment from the action of all the other planets. In the present state of 

 science, this value is exhibited in the form of an indefinite series of terms 

 diminishing rapidly in magnitude. In calculation it is usual to neglect 

 such of those terms as correspond in the order of magnitude to quantities 

 beneath the errors of observation. But there are cases in which the 

 order of the term in the series does not decide whether it be small or 

 great. Certain numerical relations between the i^rimitive elements of 

 the disturbing and disturbed planets may impart sensible values to terms 

 which usually admit of being neglected. This case occurs in the per- 

 turbations of Saturn produced by Jupiter, and in those of Jupiter pro- 

 duced by Saturn. There exists between the mean motions of these two 

 great planets a simple relation of commensurability, five times the mean 

 motion of Saturn being, in fact, very nearly- equal to twice the mean 

 motion of Jupiter. It happens in consequence that certain terms, which 

 •would otherwise be very small, acquire from this circumstance consider- 

 able values. Hence arise, in the movements of these two planets, ine- 

 qualities of long duration, which require more than 900 years for their 

 complete development, and which represent with marvelous accuracy all 

 the irregularities disclosed by observation. Is it not astonishing to find 

 in the commensurability of the mean motions of two planets a cause of 

 perturbation of so influential a nature ; to discover that the definitive 

 solution of an immense difficulty, which baffled the genius of Euler, and 

 ■which even led persons to doubt w hether the theory of gravitation was 

 capable of accounting for all the iihenomena of the heavens, should de- 

 pend upon the fortuitous circumstance of five times the mean motion of 

 Saturn being equal to twice the mean motion of Jupiter 1 The beauty 



* Mr. Adams has recently detected a remarkable oversif^ht committed by Laplace and bis 

 successors in the analytical investigation of the expression for this inequality. The cft'ect of 

 the rectification rendered necessary by the researches of Mr. Adams will be to diminish by 

 about one-sixth the co-efficient of the principal term of the secular inequality. This co-effi- 

 cient has for its multiplier the square of the number of centuries which have elapsed from a 

 given epoch; its value was found by Laplace to be 10". 18. Mr. Adams has ascertained 

 that it must be diminished by 1".G6. This result has recently been verified by the researches 

 of M. Plana. Its effect will be to alter in some degree the calculations of ancient eclipses. 

 The astronomer royal has stated, in his last annual report to the board of visitors of the 

 Royal Observatory, (June, 1856,) that steps have recently been taken at the observatory for 

 calculating the various circumstances of those phenomena, upon the basis of the more correct 

 data furnished by the researches of Mr. Adams. — Translator. 



