5a2 LAPLACE. 
4 
oscillations around a certain mean state. Let us now see 
in what way he succeeded in determining the absolute 
dimensions of the orbits. 
come into conjunction at Pp, pP/, the points whence they 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 dis- 
tances of 120° from each other. It is also obvious, that during the in- 
terval 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 out- 
standing alteration in the movement of each planet. A similar effect 
will be 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 derange- 
ment 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 Pp, Saturn will have 
advanced a little beyond P/, and the conjunction of the two planets 
will occur at Pp, P’! when they have both described around s an addi- 
tional arc of about 8°. In the same way it may be shown that the 
two succeeding conjunctions will take place at the points q, g/, r, r? 
respectively 8° in advance of Q, Q/, R, R/. Thus we see that the 
points of conjunction will travel with extreme slowness in the same 
direction as that in which the planets revolve. Now since the angular 
