314 A Short History of Astronomy [CM. xi. 



communication, and scarcely any important advance was 

 made by the one which was not at once utilised and 

 developed by the other. 



The central problem was that of the secular alterations 

 in the elements of a planet's orbit regarded as a varying 

 ellipse. Three of these elements, the axis of the ellipse, 

 its eccentricity, and the inclination of its plane to a fixed 

 plane (usually the ecliptic), are of much greater importance 

 than the other three. The first two are the elements on 

 which the size and shape of the orbit depend, and the first 

 also determines (by Kepler's Third Law) the period of 

 revolution and average rate of motion of the planet ; * the 

 third has an important influence on the mutual relations of 

 the two planets. The other three elements are chiefly of 

 importance for periodical inequalities. 



It should be noted moreover that the eccentricities and 

 inclinations were in all cases (except those specially men- 

 tioned) considered as small quantities ; and thus all t e 

 investigations were approximate, these quantities and the 

 disturbing forces themselves being treated as small. 



245. The basis of the whole series of investigations was a 

 long paper published by Lagrange in 1766, in which he 

 explained the method of variation of elements, and gave 

 formulae connecting their rates of change with the disturbing 

 forces. 



In his paper of 1773 Laplace found that what was true of 

 Jupiter and Saturn had a more general application, and 

 proved that in the case of any planet, disturbed by any 

 other, the axis was not only undergoing no secular change 

 at the present time, but could not have altered appreciably 

 since " the time when astronomy began to be cultivated." 



In the next year Lagrange obtained an expression for the 

 secular change in the inclination, valid for all time. When 

 this was applied to the case of Jupiter and Saturn, which on 

 account of their superiority in size and great distance from 

 the other planets could be reasonably treated as forming 

 with the sun a separate system, it appeared that the changes 

 in the inclinations would always be of a periodic nature, so 



* This statement requires some qualification when perturbations 

 are taken into account. But the point is not very important, and 

 is too technical to be discussed. 



