xv i INTRODUCTION. 



The mean angular distance between the perihelia of Jupiter and Uranus is 

 exactly 180° ; but the conditions of the variations of these elements are sufficiently 

 elastic to allow of a considerable deviation on each side of their mean positions. 

 The perihelion of Jupiter may differ from its mean place to the extent of 24° 10', 

 and that of Uranus to the extent of 47° 33'; and therefore the longitudes of the 

 perihelia of these two planets can differ from 180° 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 invari- 

 able 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° 38'; while that of Saturn may deviate from its mean place to the extent of 

 7° 7'. It therefore follows that their longitudes on the invariable plane can differ 

 from 180° by only 26° 45'. Their nearest possible approach is 153° 15', while their 

 present distance apart is 166° 27'. 



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

 two principal ones have periods of 613,900 years, and 418,060 years respectively. 

 Strictly speaking, the periods of the secular inequalities 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 principal inequalities of the several planetary 

 orbits are different, unless they are connected by some permanent relation, similar 

 to that Avhich 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 planet, and these 

 periods are, for the two principal inequalities, 51,280 years, and 56,303 years. In 

 like manner the principal inequalities in the eccentricities of Jupiter and Saturn 

 depend on their mutual attraction, and have a period of 69,141 years. The secular 

 inequalities of those orbits which have no vanishing elements are composed 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 composed of twenty-eight periodic terms, having coefficients of considerable 

 magnitude ; and this circumstance renders the law of their variations extremely 

 intricate. 



The method we have adopted for finding the coefficients of the corrections of 

 the constants, depending on finite variations of the different planetary masses, 

 consists in supposing that each planetary mass receives in succession a finite incre- 

 ment, and then finding the values of all the constants corresponding to this 

 increased mass in the same manner as for the assumed masses. By this means we 

 have a set of values corresponding to the assumed masses, and another set corre- 

 sponding to a finite increment to each of the planetary masses. Then, taking the 



