t although Neptune was found so near the place
where it was predicted, its orbit, after discovery, proved to be very
different from that which Adams and Le Verrier had supposed. You will
remember that both calculators assumed the distance from the sun, in
accordance with Bode's Law, to be nearly twice that of Uranus. The actual
planet was found to have a mean distance less than this by 25 per cent.,
an enormous quantity in such a case. For instance, if the supposed planet
and the real were started round the sun together, the real planet would
soon be a long way ahead of the other, and the ultimate disturbing effect
of the two on Uranus would be very different. To explain the difference,
we must first recall a curious property of such disturbances. When two
planets are revolving, so that one takes just twice or three times, or any
exact number of times, as long to revolve round the sun as the other, the
usual mathematical expressions for the disturbing action of one planet on
the other would assign an _infinite_ disturbance, which, translated into
ordinary language, means that we must start with a fresh assumption, for
this state of things cannot persist. If the period of one were a little
_longer_ than this critical value, some of the mathematical expressions
would be of contrary sign from those corresponding to a period a little
_shorter_. Now it is curious that the supposed planet and the real had
orbits on opposite sides of a critical value of this kind, namely, that
which would assign a period of revolution for Neptune exactly half that of
Uranus; and it was pointed out in America by Professor Peirce that the
effect of the planet imagined by Adams and Le Verrier was thus totally
different from that of Neptune. He therefore declared that the
mathematical work had not really led to the discovery at all; but that it
had resulted from mere coincidence, and this opinion--somewhat paradoxical
though it was--found considerable support. It was not replied to by Adams
until some thirty years later, when a short reply was printed in
_Liouville's Journal_. The explanation is this: the expressions considered
by Professor Peirce are those representing the action of the planet
throughout an indefinite past, and did not enter into the problem, which
would have been precisely the same if Neptune had been suddenly created in
1690; while, on the other hand, if Neptune had existed up till 1690 (the
time when Uranus was first observed, al