SECT. III. INEQUALITY OF JUPITER AND SATUEN. 25 



of gravitation are so perfectly confirmed by observation, that 

 analysis has become one of the most certain means of discovering 

 the planetary irregularities, either when they are too small, or 

 too long in their periods, to be detected by other methods. 

 Jupiter and Saturn, however, exhibit inequalities which for a 

 long time seemed discordant with that law. All observations, 

 from those of the Chinese and Arabs down to the present day, 

 prove that for ages the mean motions of Jupiter and Saturn have 

 been affected by a great inequality of a very long period, forming 

 an apparent anomaly in the theory of the planets. It was long 

 known by observation that five times the mean motion of Saturn 

 is nearly equal to twice that of Jupiter ; a relation which the 

 sagacity of La Place perceived to be the cause of a periodic 

 irregularity in the mean motion of each of these planets, which 

 completes its period in nearly 918 years, the one being retarded 

 while the other is accelerated ; but both the magnitude and 

 period of these quantities vary, in consequence of the secular 

 variations in the elements of the orbits. Suppose the two planets 

 to be on the same side of the sun, and all three in the same 

 straight line, they are then said to be in conjunction (N. 83). 

 Now, if they begin to move at the same time, one making exactly 

 five revolutions in its orbit while the other only accomplishes 

 two, it is clear that Saturn, the slow-moving body, will only 

 have got through a part of its orbit during the time that Jupiter 

 has made one whole revolution and part of another, before they 

 be again in conjunction. It is found that during this time their 

 mutual action is such as to produce a great many perturbations 

 which compensate each other, but that there still remains a portion 

 outstanding, owing to the length of time during which the forces 

 act in the same manner ; and, if the conjunction always happened 

 in the same point of the orbit, this uncompensated inequality in 

 the mean motion would go on increasing till the periodic times 

 and forms of the orbits were completely and permanently 

 changed : a case that would actually take place if Jupiter accom- 

 plished exactly five revolutions in the time Saturn performed 

 two. These revolutions are, however, not exactly commensur- 

 able ; the points in which the conjunctions take place are in 

 advance each time as much as 8'37 ; so that the conjunctions 

 do not happen exactly in the same points of the orbits till after 

 a period of 850 years ; and, in consequence of this small ad- 



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