THE PLANETS URANUS AND NEPTUNE.

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is considerably more complicated) somewhat after the mannerindicated by the arrows in the diagram, and thus have alter-nately accelerated and retarded the velocity which Uranus would otherwise have possessed.*

The problem of investigating where that planet might bewas altogether an indeterminate one, and the number ofequations of condition and unknown quantities involved, verynumerous. It was evident that a larger planet further off, ora smaller one at a nearer distance, might equally well produce

Fig . XCII.—Illustrating the perturbation of motion of Uranus by Neptune .

the observed result. Nor was it possible to make a preliminary

* In Herschel’s “ Outlines of Astronomy,” sec. 760 et seq., a full ex-planation of the theory of the action of Neptune upon Uranus is given.It is there shown, by a beautiful and lucid geometrical process, that thedirect, or tangential, effect of Neptune upon the velocity of Uranus ,which is all that Fig . XCII. indicates, would be of comparatively littleimportance except for a moderate number of years on each side of thedate of its opposition in 1822, near to which time its effect wouldfirst experience a rapid increase, followed by an equally rapid decrease.It is, moreover, proved that Neptune would also exert another effectnormal to the path of Uranus , by which it would draw it from time totime nearer to, or farther from, the Sun, and thus indirectly alter itsvelocity, by changing the power of the Sun’s attraction upon it. Thislatter effect is, however, much slower in action, less important, and moredifficult of explanation than the other.