* 245] Stability of the- Solar System 3 1 5 



that they could never pass beyond certain fixed limits, not 

 differing much from the existing values. The like result 

 held for the system formed by the sun, Venus, the earth, 

 and Mars. Lagrange noticed moreover that there were 

 cases, which, as he said, fortunately did not appear to exist 

 in the system of the world, in which, on the contrary, the 

 inclinations might increase indefinitely. The distinction 

 depended on the masses of the bodies in question ; and 

 although all the planetary masses were somewhat uncertain, 

 and those assumed by Lagrange for Venus and Mars almost 

 wholly conjectural, it did not appear that any reasonable 

 alteration in the estimated masses would affect the general 

 conclusion arrived at. 



Two years later (1775) Laplace, much struck by the 

 method which Lagrange had used, applied it to the dis- 

 cussion of the secular variations of the eccentricity, and 

 found that these were also of a periodic nature, so that the 

 eccentricity also could not increase or decrease indefinitely. 



In the next year Lagrange, in a remarkable paper of 

 only 14 pages, proved that whether the eccentricities and 

 inclinations were treated as small or not, and whatever the 

 masses of the planets might be, the changes in the length of 

 the axis of any planetary orbit were necessarily all periodic, 

 so that for all time the length of the axis could only fluctu- 

 ate between certain definite limits.- This result was, however, 

 still based on the assumption that the disturbing forces 

 could be treated as small. 



Next came a series of five papers published between 1781 

 and 1784 in which Lagrange summed up his earlier work, 

 revised and improved his methods, and applied them to 

 periodical inequalities and to various other problems. 



Lastly in 1784 Laplace, in the same paper in which he 

 explained the long inequality of Jupiter and Saturn, es- 

 tablished by an extremely simple method two remarkable 

 relations between the eccentricities and inclinations of the 

 planets, or any similar set of bodies. 



The first relation is : 



If the mass of each planet be multiplied by the square root 

 of the axis of its orbit and by the square of the eccentricity, 

 then the sum of these products for all the planets is invariable 

 save for periodical inequalities. 



