lxiv 



THE THEORY OF THE LONG INEQUALITY 



sign to that in the next. The force depending on 3PSP goes through its changes three 

 times in two revolutions of P, or once and a half in one revolution ; therefore in two 

 successive revolutions, the planet is, at every point of its orbit, presented to equal and 

 opposite values of this force ; so that the additional forces are equal and opposite. Similar 

 reasoning will apply to the forces depending on all higher multiples of PSP, and will shew 

 that all disturbances must be compensated in one synodical revolution of P and P at the most; 

 though it is not to be inferred that they require so much as one synodical revolution. 



75. Next, let us suppose the orbit of P to be elliptical, while that of P remains circular ; 

 and consider the alteration which this will make in those circular forces the period of which 

 is nearly equal to the period of P; that is, the forces depending on the argument PSP'. 

 This part of the circular tangential force is retarding during one revolution of P, and 

 accelerating in the next. 



Let ad (fig. 5) be the line of apses of the orbit of P ; a the perihelion ; and let the 

 planets start from conjunction at A. 



Then the circular tangential force is retarding on P, while P moves through ABA', and 

 accelerating while P moves through A'B'A, and arrives at a maximum when the mean place 

 of P is at B and B'. While P 1 moves from A to B it is before its mean place ; there- 

 fore PSP is less than it would otherwise have been, and therefore the retarding force is 

 less, or, in other words, there is an additional accelerating force on P. While P' describes 

 Bd it is still before its mean place, and therefore PSP' is less than it would have been ; but 

 the retarding force is past its maximum, so that a diminution of PSP" increases the force: 

 therefore there is an additional retarding force. Through the arc a A' the angle PSP is 

 increased, and therefore the retarding force diminished ; or the additional force is now posi- 

 tive. Similarly, it is positive in the arc A'B', negative in B>a, and positive in aA. Hence 

 there is a preponderance of force (which is positive in the assumed position of A) depending on 

 the position of the line of conjunction with respect to the apses of the orbit of P, which 

 produces a long inequality in the mean motion and epoch of P. 



76. Another inequality of the same sign arises from the fluctuation in the distance of P 

 from S. While P describes the arc b'B" A (fig. 6), the tangential accelerating force depending 

 on PSP 1 is increased by the diminution of SP'; and in the arc bBA' the force, which is 

 then retarding, is diminished by the increase of SP". Therefore in each case there is an 

 additional accelerating force. Similarly, in the arcs Ab, A'b', there is an additional retarding 

 force; so that there is a preponderance of force of the same sign as that investigated in the last 

 Article*. 



* When the orbit of P' is alone supposed to be elliptical 



d(R) dR „ d*R , d /dR\ _ , 



■V = —7- V+ —r—rr, Vid,'+ 3-; I -r- I Via', 

 at ad<p acUj>d<p r da \ad<pj 



R = -m 



(PR 



ad<pd<t>' 



Vi<t>' = -m' 





(*-#'); 



cos($-<£').2e'sin/8', 



= + m'n [A l -—J e' sin (X + ro'), &c. 

 d /dR\ _ , , /dA, 2a\ . ,, „ . . 



JS\fik%) Via =~ mn (^■ + ; ? i)« u >(*-*M« cos 0' 



m'n I . dA, 2a\ , . 

 = + -2-(^. + ,«^- -^)«sin(X + OT ') ( &c. 



For the position of the line of conjunction assumed in the 

 figures \+w' is negative ; therefore both the above terms indi. 

 cate a preponderance of accelerating force. 



