OP URANUS AND NEPTUNE. lxvii 



place, and therefore the force producing the long inequalities is less than it would have been ; 

 or there is an additional force tending to diminish the mean motion of P and increase that 

 of P'. This force vanishes at a since the mean longitudes, and therefore the point of 

 conjunction, are restored to their undisturbed state. Consequently, the new force goes 

 through all its changes while A performs half a revolution, and therefore in half the periodic 

 time of the principal disturbing force. 



82. It is clear that the long inequality in the longitude of the perihelion will produce 

 a similar force (of much greater importance, because the change of place of perihelion is much 

 greater than the change in the place of conjunction) ; for this perturbation of perihelion alters 

 the relative position of the perihelion and line of conjunction in the same manner as the 

 perturbation of mean longitudes does. But the resulting disturbance is nearly neutralized 

 by another arising from the variation of the eccentricity. For when A is at a the eccentricity 

 is least (Art. 69), and is less than its mean value while A revolves from a to b, at which 

 point it is restored to its mean value. On this account therefore the disturbing force is 

 diminished while it has been increased by the simultaneous recess of perihelion. 



83. Since the perturbations of e and • 5r depend upon the position of A with respect 

 to the perihelion, the shifting of A will produce inequalities in those elements depending on 

 the square of the disturbing force similar to those produced in the mean motions. 



A similar explanation may be given of those terms of the square of the disturbing force 

 which depend upon the ellipticity of the orbit of P ; and the explanation of the perturbations 

 of P by 2* may be applied to those of P 1 by P. 



