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numbers, then part of the periodic disturbing force produces a secular change in their motions, acting continually in the same direction; though he pointed out that such a case did not occur in the solar system. If moreover the times of revolution are nearly proportional to two whole numbers (neither of which is very large), then part of the periodic disturbing force produces an irregularity that is not strictly secular, but has a very long period; and a disturbing force so small as to be capable of being ordinarily overlooked may, if it is of this kind, be capable of producing a considerable effect.^[1] Now Jupiter and Saturn revolve round the sun in about 4,333 days and 10,759 days respectively; five times the former number is 21,665, and twice the latter is 21,518, which is very little less. Consequently the exceptional case occurs; and on working it out Laplace found an appreciable inequality with a period of about 900 years, which explained the observations satisfactorily.

The inequalities of this class, of which several others have been discovered, are known as long inequalities, and may be regarded as connecting links between secular inequalities and periodical inequalities of the usual kind.

244. The discovery that the observed inequality of Jupiter and Saturn was not secular may be regarded as the first step in a remarkable series of investigations on secular inequalities carried out by Lagrange and Laplace, for the most part between 1773 and 1784, leading to some of the most interesting and general results in the whole of gravitational astronomy. The two astronomers, though living respectively in Berlin and Paris, were in constant

1. If $n$, $n'$ are the mean motions of the two planets, the expression for the disturbing force contains terms of the type $={\begin{matrix}sin\\cos\end{matrix}}(np\pm n'p')t$, where $p$, $p'$ are integers, and the coefficient is of the order $p\sim p'$ in the eccentricities and inclinations. If now $p$ and $p'$ are such that $n\,p\sim n'\,p'$ is small, the corresponding inequality has a period $2\,\pi /(n\,p\sim n'\,p)$, and though its coefficient is of order $p\sim p'$, it has the small factor $n\,p\sim n'p'$ (or its square) in the denominator and may therefore be considerable. In the case of Jupiter and Saturn, for example, $n=109,257$ in seconds of arc per annum, $n'=43,996$; $5n'-2n=1,466$; there is therefore an inequality of the third order, with a period (in years) $={\frac {360^{\circ }}{1,466''}}=900$.