318 A Short History of Astronomy [CH. xi. 



246. A few of Laplace's numerical results as to the secular 

 variations of the elements- may serve to give an idea of 

 the magnitudes dealt with. 



The line of apses of each planet moves in the same 

 direction ; the most rapid motion, occurring in the case of 

 Saturn, amounted to about 15" per annum, or rather less 

 than half a degree in a century. If this motion were to 

 continue uniformly, the line of apses would require no less 

 than 80,000 years to perform a complete circuit and return 

 to its original position. The motion of the line of nodes 

 (or line in which the plane of the planet's orbit meets that 

 of the ecliptic) was in general found to be rather more 

 rapid. The annual alteration in the inclination of any orbit 

 to the ecliptic in no case exceeded a fraction of a second ; 

 while the change of eccentricity of Saturn's orbit, which 

 was considerably the largest, would, if continued for four 

 centuries, have only amounted to TTy V^- 



247. The theory of the secular inequalities has been 

 treated at some length on account of the general nature of 

 the results obtained. For the purpose of predicting the 

 places of the planets at moderate distances of time the 

 periodical inequalities are, however, of greater importance. 

 These were also discussed very fully both by Lagrange and 

 Laplace, the detailed working out in a form suitable for 

 numerical calculation being largely due to the latter. From 

 the formulae given by Laplace and collected in the Mecanique 

 Celeste several sets of solar and planetary tables were 

 calculated, which were in general found to represent closely 

 the observed motions, and which superseded the earlier 

 tables based on less developed theories.* 



248. In addition to the lunar and planetary theories 

 nearly all the minor problems of gravitational astronomy 

 were rediscussed by Laplace, in many cases with the aid 

 of methods due to Lagrange, and their solution was in all 

 cases advanced. 



The theory of Jupiter's satellites, which with Jupiter form 



* Of tables based on Laplace's work and published up to the time 

 of his death, the chief solar ones were those of von Zach (1804) and 

 Delambre (1806) ; and the chief planetary ones were those of 

 Lalande (i770 f Lindenau for Venus, Mars, and Mercury (181013), 

 and of BoHvard for Jupiter, Saturn, and Uranus (1808 and 1821). 



