DISCREPANCIES EXPLAINED. 



those of Neptune by Q, those of M. Le Terrier's planet by 0, 

 and those of Mr. Adams' planet by . 



It will be observed that the distances of the two planets assigned 

 by MM. Le Verrier and Adams, as laid down in the diagram, 

 differ less from the distance of the planet Neptune than the 

 mean distances given in their elements differ from the mean dis- 

 tance of Neptune. This is explained by the eccentricities of the 

 orbit, which, in the elements of both astronomers, are consider- 

 able, being nearly an eighth in one and a ninth in the other, 

 and by the positions of the supposed planets in their respective 

 orbits. 



If the masses of the three planets were equal, it is clear that 

 the attraction with which Le Terrier's planet would act upon 

 Uranus, would be less than that of the true planet, and that of 

 Adams' planet still more so, each being less in the same ratio as 

 the square of its distance from Uranus is greater than that of 

 Neptune. But if the planets are so adjusted that what is lost by 

 distance is gained by the greater masses, this will be equalised, 

 and the supposed planet will exert the same disturbing force as the 

 actual planet, so far as relates to the effects of variation of dis- 

 tance. It is true that, throughout the arcs of the orbits over which 

 the observations extend, the distances of the three planets in simul- 

 taneous positions are not everywhere in exactly the same ratio, while 

 their masses must necessarily be so ; and, therefore, the relative 

 masses, which would produce perfect compensation in one position, 

 would not do so in others. This cause of discrepancy would ope- 

 rate, however, under the actual conditions of the problem, in a 

 degree altogether inconsiderable, if not insensible. 



But another cause of difference in the disturbing action of the 

 real and supposed planets would arise from the fact, that the 

 directions of the disturbing forces of all the three planets are 

 different, as will be apparent on inspecting the figure in which 

 the degree of divergence of these forces at each position of the 

 planets is indicated ; but it will be also apparent, that this diver- 

 gence is so very inconsiderable that its effect must be quite insen- 

 sible in all positions in which Uranus can be seriously affected. 

 Thus, from 1800 to 1815, the divergence is very small. It 

 increases from 1815 to 1835; but it is precisely here, near the 

 epoch of heliocentric conjunction, which took place in 1822, 

 that all the three planets cease to have any direct effect in 

 accelerating the motion of Uranus. When the latter planet 

 passes this point sufficiently to be sensibly retarded by the dis- 

 turbing action, as is the case after 1835, the divergence again 

 becomes inconsiderable. 



From these considerations it will therefore be understood, that 



183 



