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APPENDIX.

into conjunction with that planet. It may be easily shown that Uranus will require to advance through an arc equal to 15° 15', before it againcomes up with Neptune . The same is true with respect to the thirdconjunction of the planets, and thus it is evident that the line of conjunc-tion on each of such occasions will advance through an arc equal to 15° 15'in the direction of the motions of the two planets.

(44.) It has been remarked (41), that the alteration induced in theelements of either planet on the occasion of its conjunction with thedisturbing planet, will depend upon the position of the line of apsidesof the orbit of each planet with respect to the line of conjunction. Letus consider the effect which Neptune tends to produce on the eccen-tricity and the longitude of the perihelion of Uranus . If the line ofapsides of the disturbed planet be oblique with respect to the line ofconjunction, it is manifest that the variation of the eccentricity cannoteffect a complete compensation, and therefore a change will ultimately beproduced in the value of that element, which will be maintained untilthe planet again arrives within the sphere of the disturbing force onthe occasion of approaching the next conjunction. It is easy to see,therefore, that in such a case the eccentricity of the orbit would undergoan alteration at each successive conjunction. Let us suppose, however,that the eccentricity constantly retains the same value (which is actuallytrue in so far as regards that part of the eccentricity which is essentiallyco-existent with the disturbing force, and which is the only part we arehere considering), and let us enquire into the circumstances under whichthis condition of permanence can be maintained. Now it appears from (24),(30), and (32), that if the line of apsides coincide with the line of conjunction,the eccentricity will not undergo any change from the action either of theradial or the tangential force. It is manifest, however, that this coin-cidence can only be maintained by a progressive motion of the line ofapsides, equal to that of the line of conjunction. Now if the perihelionof the planet be turned towards the disturbing body, it follows, from (24),that the line of apsides will regress, and therefore, instead of constantlycoinciding with the line of conjunction, it will immediately commence todeviate from that line by receding in the opposite direction. But if theaphelion of the planet be turned towards the disturbing body, the line ofapsides will then progress (24), from the action of the disturbing forces;and, by duly adjusting the eccentricity, the progression may be madeexactly equal to the progression of the line of conjunction. In this case,then, the variation of the eccentricity occasioned by the disturbing forcewill effect a complete compensation at each conjunction, leaving theeccentricity of the undisturbed portion of the orbit unaltered, while at thesame time the line of apsides will advance regularly in the direction ofthe planet’s motion with an angular velocity equal to that of the line ofconjunction, or at the rate of 15° 15' in each synodic revolution of the twoplanets.

(45.) The eccentricity above referred to, may be considered as aninequality of a perturbative character, inasmuch as the necessity for itsexistence depends wholly on the action of the disturbing force. It isright to bear in mind, however, that its maintenance is mainly due to thecentral force. The direct effect of the disturbing force is, merely to pro-duce a slight variation of the eccentricity (which, however, effects a com-plete compensation at each conjunction), and to cause a constant progres-sion of the line of apsides.