lxii THE THEORY OP THE LONG INEQUALITY 



and the mean longitude at epoch must be again adjusted, so that with the new eccentricity 

 and perihelion the planet may still have its true longitude at the same instant of time t. But 

 the mean motion calculated for that instant is the same as it was for the same instant in the 

 previous ellipse. Therefore, the variation of the epoch depends upon the alteration of the 

 eccentricity and perihelion, but not upon that of the mean angular velocity. If, however, the 

 mean motion were estimated in the former of the two ways described above, the mean 

 longitude at epoch would be different, and its variation would depend upon that of the mean 

 angular velocity, as well as upon that of the eccentricity and perihelion. 



71. Suppose a normal force to act outwards on P during a whole revolution. While 

 P moves from perihelion to aphelion the eccentricity is increased, and is diminished in the 

 other half of the orbit. While P is about perihelion the apsides recede, and while about 

 aphelion they advance. Therefore while P is moving from perihelion to mean distance 

 the equation of the centre corresponding to a given mean longitude is increased, both by 

 the increase of the eccentricity and because the point at which the equation is at a maximum 

 is brought nearer to the planet by the recess of perihelion. If therefore the mean longi- 

 tude were unaltered the planet would be in advance of its true place. Therefore the mean 

 longitude at epoch must be diminished in order that the planet may come to its true place 

 at the right time. And similarly the equation of the centre is increased, and therefore the 

 epoch must be diminished, while P moves from the extremity / of the latus rectum through 

 the upper focus H (fig. 7) to aphelion. While P moves from aphelion to the other extremity 

 t of the latus rectum, the equation of the centre is diminished by the diminution of the 

 eccentricity, and because the point at which it is at a maximum is kept further from the 

 planet by the advance of perihelion : and since the equation is now negative the planet would 

 be in advance of its true place if the epoch were not diminished. Similarly, the epoch 

 must be diminished while P moves from mean distance to perihelion. Between points on 

 either side at which the equation of the centre is at a maximum, and I or t the diminution 

 of epoch due to the alteration of eccentricity is partially neutralized by the effect due to 

 alteration of perihelion. A normal force acting outwards therefore in any part of the orbit 

 requires the epoch to be diminished on account of its action on the eccentricity and peri- 

 helion ; and similarly a normal force acting inwards requires it to be increased. 



72. Now by the same reasoning as in (Art. 64), it may be shewn that there is a small 

 additional normal force acting inwards in the arcs ABa, A'B'a, and outwards in the arcs 

 aA, a A'. And by the same reasoning as in (Art. 65), it follows that there is a normal force 

 acting inwards in the arcs B'Ab, BA'b', and outwards in the arcs bB, b'B'. In both cases 

 therefore the inward force exceeds the outward force by a quantity depending on the angle 

 aSA. In consequence of the action of this force the epoch will be at a maximum when 

 A is at b, and at a minimum at b', and is restored to its mean value, when A is at 

 a or a. 



Again, the circular tangential force must be resolved along the normal to the ellipse, 

 and will contribute a small normal force, which does not exist in the circular orbit. This 



