OF URANUS AND NEPTUNE. lvii 



give rise to a portion of the long inequality. We shall now examine separately the effects of 

 these four causes of disturbance; for since the circular disturbing forces can only produce 

 rapidly periodical variations of the major axis, it is in those circumstances, in which elliptic 

 motion differs from circular, that the origin of the long inequality of the mean motions 

 must be sought. 



64. Let AA' (fig. 3) be the line of conjunctions, the planets coming to conjunction at 

 A once in every revolution of P'; ad the major axis of P's orbit, a the perihelion; and let 

 the planets start from conjunction at A. 



From A through B to A' the circular tangential force depending on 9.PSP" is retardative, 

 and at a maximum when P is at B. P moving from perihelion to aphelion is in advance of 

 its mean place ; therefore, when between A and B, it is nearer to that position, in which the 

 tangential force is at a maximum, than it would have been if its motion had been uniform ; and 

 it is therefore acted upon by a greater retarding force. This is equivalent to being acted upon 

 both by the circular tangential force, and an additional retarding force (which is indicated in 

 the figure by a negative sign). From B to d P is still in advance of its mean place ; but the 

 circular tangential force is now decreasing, and P is further from the position in which that 

 force is at a maximum ; therefore it is acted on by a smaller retarding force than it would 

 have been if its motion had been uniform : this is equivalent to being acted on by an 

 additional accelerating force (indicated by a + sign). Between d and A' the circular tangential 

 force is still retarding, but P is behind its mean place, and therefore the additional force is 

 now retarding. Of course there is no additional force arising from this cause at d, since 

 there the mean and true places of P coincide. 



From A' through Pi to A the circular tangential force is accelerating and at a maximum 

 at B ' . Therefore while P moves from A' to B', being behind its mean place, it is further 

 from the position in which the tangential force is at a maximum, and therefore it is acted 

 upon by an additional retarding force. From B' to a the true place of P is nearer to the 

 position of maximum force, and therefore the additional force is accelerating : while from a 

 to A the true place is before the mean, and the additional force is retarding, and vanishes 

 at a. The additional force also vanishes when B or B' is half way between the mean and 

 true places of P. 



From this cause, therefore, in this position of the line of conjunction we have a pre- 

 ponderance of retarding force; and the amount of this excess depends upon the angle 

 aSA, and is zero when AA' coincides with ad, and at a maximum when AA' coincides 

 with bb', in which position the whole of the additional force is negative. Now since the 

 periodic time of P is somewhat less than twice that of P, each conjunction takes place at 

 a point a little in advance of the preceding, so that the point A performs a complete revo- 

 lution in about 4047 years. While therefore A revolves from a through b to d the addi- 

 tional force is retarding; and it is accelerating throughout the second half of the revolution 

 of A. Consequently during the former half of a complete revolution of the point of 

 conjunction the major axis is continually decreasing and the mean angular velocity in- 

 creasing, and vice versa during the latter half. 



[H] 



