432 RISE AND PROGRESS OF 



engaged the attention of mathematicians at an early period. 

 Astronomers had long observed that Saturn's mean motion was 

 subject to a slow retardation, and it was discovered by Halley 

 that the motion of Jupiter was in like manner accelerated. It 

 was natural, then, to look for the cause of these phenomena in the 

 mutual action of the two planets; and accordingly the French 

 Academy soon proposed the theory of Jupiter and Saturn, as the 

 subject of their prize-essay. Its difficulties, however, were such 

 as to baffle the genius of Euler, who engaged in the research. 

 Laplace and Lagrange afterwards entered on the investigation, 

 and advanced, almost side by side, to its complete solution. In 

 the course of these researches it was proved that the mean motions 

 of the planets were subject to no secular inequalities; and accord- 

 ingly the retardation of Saturn's motion, which was supposed to 

 be of that nature, appeared to be at variance with the law of 

 universal gravitation. It was finally shown, however, by Laplace, 

 that the mean motions of Jupiter and Saturn were subject to an 

 inequality of a very long period ; and that the retardation of the 

 one planet, and the acceleration of the other, would in time cease, 

 and the effects be reversed. This remarkable irregularity in the 

 motion of these two planets was found to depend upon the near 

 commensurability of their mean motions. A phenomenon of the 

 same kind, depending upon a similar cause, is found to take place 

 in the motion of the earth and of Yenus ; and these two bodies 

 also are subject to an inequality of a long period, lately brought 

 to light by Professor Airy. 



The principle above referred to namely, that the mean 

 motions of the planets are subject to no secular inequalities was 

 discovered by Lagrange, who was the first writer to examine with 

 attention the variations of the elements of the planetary orbits. 

 The importance of the discovery is obvious : if the mean dis- 

 tances and mean motions were under the influence of secular 

 inequalities if Saturn should continue to be retarded, and Jupiter 

 to be accelerated the one would recede from the sun, until it 

 finally fell under the influence of some distant body of the starry 

 universe ; while the other would approach indefinitely, and at last 

 reach the central mass. This however is not the case : the inequa- 

 lities which exist are all periodical, and the changes which occur 

 are continually repaired. It is in this sense that the planetary 

 system is said to be stable. 



