590

APPENDIX.

]ion, the line of apsides will regress if the disturbing force should actafter the passage of the aphelion, but will progress if it should act at themean distance or before the passage pf the perihelion; while again, theeccentricity in each of these three cases will be increased by the disturbingforce.

(22.) We have seen that if the disturbing force should act when theplanet is at the perihelion, the effect is then thrown wholly upon theapsides, which rapidly progress ; that at the mean distance the apsidesalso progress, (though with less rapidity, the effect now being thrownpartly upon the apsides and partly on the eccentricity); but thatif the disturbing force acts when the planet arrives at the aphelion,the effect is again thrown wholly upon the apsides, which, however, in thiscase regress. Hence it is obvious, that there must be some intermediatepoint of the orbit between the mean distance and the aphelion at whichthe disturbing force produces no effect on the position of the line ofapsides. Similarly, it is manifest that there must exist some point betweenthe aphelion and the subsequent point of mean distance, at which theline of apsides does not undergo any change of position from the action ofthe disturbing force. It may be found by a simple investigation, to whichwe shall presently allude more particularly, that the two points in ques-tion are the extremities f g of the ordinate passing through h the upperfocus of the ellipse. In fact, as the planet revolvesfrom f to g through a, the line of apsides every-where progresses from the action of the disturbingforce, the amount of progression increasing fromnothing at f until it attains its maximum at a, andsubsequently diminishing until it vanishes again atg. Similarly, from g to f through b, the line ofapsides everywhere regresses from the same cause,the amount of regression being greatest at A, anddiminishing in either direction towards f and g.

(23.) We have seen that if, when the planetis revolving from the lower to the upper apse, thedisturbing force act a little after the passage of theperihelion, at the mean distance, or a little before the passage of theaphelion, the eccentricity is in each case diminished; but that, on theother hand, when the planet is revolving from the upper to the lowerapse, the eccentricity in each of the corresponding cases is increased bythe action of the disturbing force. Generally it may be shewn, that froma to b through g the eccentricity is everywhere diminished by the actionof the disturbing force, the amount of diminution increasing from nothingat a until it attains its maximum at g, and subsequently diminishing untilit vanishes at b ; and on the other hand, that from b to a through f theeccentricity is everywhere increased by the action of the disturbing force,the amount of increase being greatest at f, and diminishing from thatpoint towards a and b. Thus it appears, that when the variation in theposition of the apsides is greatest, the variation of the eccentricity is least,and vice versa.

(24.) If we suppose the disturbing force to act in the direction of theradius vector so as to diminish the central force at s, it will be found, in asimilar manner, that the effect both upon the eccentricity and the apsideswill now be precisely the reverse of that produced when the disturbingforce acts inwards. In this case the eccentricity will increase from a to

B