work to consider the perturbations produced by Jupiter
and Saturn, to see if they had been with perfect accuracy allowed for,
or whether some minute improvements could be made sufficient to destroy
the irregularities. He introduced several fresh terms into these
perturbations, but none of them of sufficient magnitude to do more than
slightly lessen the unexplained perturbations.

He next examined the various hypotheses that had been suggested to
account for them:--Was it a failure in the law of gravitation? Was it
due to the presence of a resisting medium? Was it due to some unseen but
large satellite? Or was it due to a collision with some comet?

All these he examined and dismissed for various reasons one after the
other. It was due to some steady continuous cause--for instance, some
unknown planet. Could this planet be inside the orbit of Uranus? No, for
then it would perturb Saturn and Jupiter also, and they were not
perturbed by it. It must, therefore, be some planet outside the orbit of
Uranus, and in all probability, according to Bode's empirical law, at
nearly double the distance from the sun that Uranus is. Lastly he
proceeded to examine where this planet was, and what its orbit must be
to produce the observed disturbances.

[Illustration: FIG. 94.--Uranus's and Neptune's relative positions.

The above diagram, drawn to scale by Dr. Haughton, shows the paths of
Uranus and Neptune, and their positions from 1781 to 1840, and
illustrates the _direction_ of their mutual perturbing force. In 1822
the planets were in conjunction, and the force would then perturb the
radius vector (or distance from the sun), but not the longitude (or
place in orbit). Before that date Uranus had been hurried along, and
after that date it had been retarded, by the pull of Neptune, and thus
the observed discrepancies from its computed place were produced. The
problem was first to disentangle the outstanding perturbations from
those which would be caused by Jupiter and Saturn and all other known
causes, and then to assign the place of an outer planet able to produce
precisely those perturbations in Uranus.]

Not without failures and disheartening complications was this part of
the process completed. This was, after all, the real tug of war. So many
unknown quantities: its mass, its distance, its excentricity, the
obliquity of its orbit, its position at any time--nothing known, in
fact, about the planet except the microscopic disturban