SECULAR VARIATIONS OF THE PLANETARY ORBITS. 271 



from its mean place to the extent of 24^ 10', and tliat of Uranus to the 

 extent of -H^ o'o' ; and therefore the longitudes of the perihelia of these 

 two planets can difter from ISO'^ to the extent of 71° 43'. The nearest 

 approach of the perihelia of these two planets, is, therefore, 108° 17'. 



In like manner the longitudes of the nodes of Jupiter and Saturn, on 

 the invariable plane, can suffer considerable deviations from their mean 

 positions. The actual position of Jupiter's node may differ from its 

 mean place to the extent of 19^ 3S' ; while that of S;iturn may deviate 

 from its mean place to the extent of 7° 7'. It therefore follows that 

 their longitudes on the invariable plane can ditter from 180^ by only 

 2G^ 45'. Their nearest possible approach is 153^ 15', Avhile their present 

 distance apart is IGG^ 27'. 



The inequalities in the eccentricity of Neptune's orbit are very small 

 and the two ])rincipal ones have periods of 013,900 years, and 418,Oi;o 

 years, respectively. Strictly spealcing, the periods of the secular inequali- 

 ties of the eccentricities and perihelia are the same for all the planets; 

 and the same remark is equally applicable to the nodes and inclinations. 

 But the princii)al inequalities of the several planetary orbits are different, 

 unless they are connected by «ome permanent relation, similar to that 

 which exists between the perihelia of Jupiter and Uranus, or the nodes 

 of Jupiter and Saturn. Thus the principal inequalities affecting the 

 inclination of the orbits of Jupiter and Saturn have the same periods for 

 each i)Ianet, and these periods are, for the two principal inequalities, 

 51,280 years, and 5G,303 years. In like manner the principal inequali- 

 ties in the eccentricities of Jupiter and Saturn depend on their mutual 

 attraction, and have a period of G9,141 years. The secular inequalities 

 of those orbits which have no vanishing elements are conq)osed of terms, 

 of very different orders of magnitude; and it frequently happens that 

 two or three of these terms are greater than the sum of all the remaining 

 ones. In such cases the variation of the corresponding element very 

 approximately conforms to a much simpler law, and the maxima and 

 minima repeat themselves according to definite and well-defined 

 cycles. But with regard to the orbits of Venus, the Earth, and Mars, 

 the rigorous expressions of the eccentricities and inclinations are com- 

 posed of twenty-eight periodic terms, having coeflicientsof consi<lerable 

 magnitude; and this circumstance renders the law of their variations 

 extremely intricate. 



The method we have adopted for finding the coefficients of the cor. 

 rections of the constants, depending on finite variations of the different 

 ])lanetary masses, consists in sui)posing that each planetary mass re- 

 ceives in succession a finite increment, and then finding the values of 

 all the coustauts corresponding to this increased mass in the same man- 

 ner as for the assumed masses. By this means we have a set of values 

 corresponding to the assumed masses, and another set corresponding to 



