lxvi THE THEORY OF THE LONG INEQUALITY 



b'aJ, ba'A', and negative while it describes the arcs A b, J'b'. The resulting inequalities 

 of P are therefore of opposite sign to the corresponding inequalities of P. 



Other terms are produced by the alteration of the tangential force depending on PSP 

 due to the deviation of P' from circular motion. Now this force is of the same sign as the 

 corresponding force acting on P, and therefore the additional force arising from the variation 

 in the longitude of P' will be of the same sign as the corresponding force acting on P, and 

 therefore the disturbance will be of the same sign. But it is otherwise with the alteration 

 of this force due to the fluctuation in the radius vector of P'. The tangential force depending 

 on PSP consists of two parts, one part arising from the attraction of P on S, and the other 

 from the attraction of P on P. Now it is this latter part only which is altered by the 

 fluctuation of the radius vector of P' ; and this part is accelerating while P moves through 

 ABA', and retarding while it moves through A'B'A, and is increased by a diminution of 

 SP. The resulting inequality is therefore of opposite sign to the corresponding term for P 

 investigated in Art. 76. 



Again the radial force acts inwards while P describes B AB, and outwards in the other 

 half of the orbit. Therefore resolving this along the tangent the additional force is retarding 

 through the arcs aAB, a'A'B', and accelerating through B'a, Ba. Therefore there is a 

 preponderance of negative force diminishing the axis and increasing the mean motion. 



Lastly, the velocity is above the mean in the arc b' B'A where the tangential force is 

 accelerating, and below the mean in the arc b B A' where it is retarding, therefore the effect 

 of accelerating force preponderates increasing the major axis and diminishing the mean motion. 

 These are all the perturbations of the mean motion of P' of the first order. 



The effect of the normal force upon the epoch may be investigated as in Art. 72. 



80. By the same reasoning as that in Art. 69 it may be shewn that the tangential force 

 depending on PSP increases the eccentricity (in the position of A assumed in the figures) 

 and causes the perihelion to recede. And the normal force depending on PSP produces a 

 perturbation of these elements of an opposite character. 



81. Since the disturbing forces alter the mean longitudes of the planets, the position of 

 the line of conjunction will be different from what it would have been if there had been no 

 disturbing force. But the disturbing forces themselves have been shewn to depend upon the 

 position of the line of conjunction with respect to the apses. These forces therefore will be 

 altered by their own action ; and this alteration gives rise to another long inequality which 

 we shall now examine. 



Let the planets start from conjunction at the perihelion of P's orbit. Then (taking 

 account only of the disturbance arising from the ellipticity of the orbit of P) Ps angular 

 velocity is least and P's greatest ; P immediately begins to lose in longitude and P to gain ; 

 the progressive motion of the line of conjunctions therefore is accelerated, and the disturbing 

 force (which is greatest when A is at b) is increased ; i. e. there is an additional force tending 

 to increase the mean motion of P and diminish that of P. When A is near b this new force 

 vanishes. When it has passed b the place of conjunction is still in advance of its undisturbed 



