Ivi THE THEORY OF THE LONG INEQUALITY 



through all their changes in one synodic revolution, and the character and magnitude of which 

 depend upon the position of the line of conjunctions relatively to the major axis. The 

 tangential part of these additional forces produces the long inequality in the mean motions ; 

 but the eccentricities and perihelia are affected in a different manner; their long inequalities 

 will be seen to depend upon the circular part of the disturbing forces (i. e. that part which 

 is the same as if the orbits were circular). 



Let then P and P' be the planets supposed to revolve in circular orbits about S (fig. 2) ; 

 and let the periodic time of P' be twice that of P ; A the point of conjunction, and BB' 

 perpendicular to AA'. The disturbing forces acting on P are the attraction of P' on P in 

 the direction PP ', and the attraction of P' on S acting in a direction parallel to P'S. When 

 both these forces are resolved along the tangent and normal, the tangential force vanishes 

 when the angular separation of the planets PSP' — O or 180°, and also when S and P are 

 equally distant from P, i.e. when PSP' = cos~'^ct = 71°<| about; and is alternately retar- 

 dative and accelerative. Now this force may be conceived to be made up of several forces, 

 one depending upon the sin PSP, another upon the sin 2PSP, &c. A force depending on 

 sin 2PSP vanishes when PSF = 0°, 90°, 180°, 270°, 360°, and is at a maximum at points 

 half way between these, and is alternately retardative and accelerative. While therefore P 

 moves from 4 through B to A', it is retarded by this part of the tangential force (for the 

 angle PSP' is always half of the angle described by P from conjunction) ; and while it 

 moves through A'B'A, it is accelerated to the same amount. The tangential force which 

 depends upon sin PSP' is retardative during one revolution of P, and accelerative in the 

 next. The force depending on 9.PSP' is the more important of the two, since its law of 

 variation more nearly resembles that of the whole tangential force. In the same way the 

 normal force may be considered to be made up of a constant part, and of a part depending 

 upon cos PSP, another depending upon cos 2PSP', &c. The part depending upon cos 2PSP 

 acts outwards, while P moves from B' through A to B, and inwards during the other half of 

 a revolution, and is at a maximum at A and A'. The force depending upon cos PSP' acts 

 outwards during half a revolution of P before conjunction, and half a revolution after, 

 arriving at a maximum at conjunction ; and its direction is reversed in the next revolution 

 of P. 



63. Confining our attention at first to the forces depending upon the argument 2PSP 

 let us examine what alteration will be made in them when we suppose the orbit of P to 

 become elliptical, while that of P remains circular. They will be altered, 



1°. By reason of P being sometimes before, sometimes behind its mean place. 



2°. By the fluctuation of distance from S. 



3". The tangent to the elliptic orbit is not (generally) in the same direction as the tangent 

 to the circular orbit ; and, although this does not alter the value of the whole disturbing 

 force, it alters the values of the tangential and normal components. 



Since the momentary change of major axis is proportional to the tangential force multi- 

 plied by the linear velocity of the disturbed planet, a fluctuation in that velocity will produce 

 changes in the major axis other than those which exist in a circular orbit, and may therefore 



