ey
originally set out; and the two succeeding conjunctions will also
manifestly occur at Q, Q' and R, R'. Thus we see, that the conjunctions
will always occur in three given points of the orbit of each planet
situate at angular distances of 120 deg. from each other. It is also
obvious, that during the interval which elapses between the occurrence
of two conjunctions in the same points of the orbits, and which includes
three synodic revolutions of the planets, Jupiter will have accomplished
five revolutions around the sun, and Saturn will have accomplished two
revolutions. Now if the orbits of both planets were perfectly circular,
the retarding and accelerating effects of the disturbing force of either
planet would neutralize each other in the course of a synodic
revolution, and therefore both planets would return to the same
condition at each successive conjunction. But in consequence of the
ellipticity of the orbits, the retarding effect of the disturbing force
is manifestly no longer exactly compensated by the accelerative effect,
and hence at the close of each synodic revolution, there remains a
minute outstanding alteration in the movement of each planet. A similar
effect will he produced at each of the three points of conjunction; and
as the perturbations which thus ensue do not generally compensate each
other, there will remain a minute outstanding perturbation as the result
of every three conjunctions. The effect produced being of the same kind
(whether tending to accelerate or retard the movement of the planet) for
every such triple conjunction, it is plain that the action of the
disturbing forces would ultimately lead to a serious derangement of the
movements of both planets. All this is founded on the supposition that
the mean motions of the two planets are to each other as two to five;
but in reality, this relation does not exactly hold. In fact while
Jupiter requires 21,663 days to accomplish five revolutions, Saturn
effects two revolutions in 21,518 days. Hence when Jupiter, after
completing his fifth revolution, arrives at P, Saturn will have advanced
a little beyond P', and the conjunction of the two planets will occur at
P, P' when they have both described around S an additional arc of about
8 deg.. In the same way it may be shown that the two succeeding conjunctions
will take place at the points _q, q', r, r'_ respectively 8 deg. in advance
of Q, Q', R, R'. Thus we see that the points of conjunction will