332 LAPLACE. 



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 p, p/, the points whence they originally set 

 out; and the two succeeding conjunctions will also manifestly occur at 

 Q, Q! and K, a'. 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 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 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, q>, r, rl 

 respectively 8 in advance of Q, Q', E, 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 



