243,244] Long Inequalities 313 



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 con- 

 siderable effect.* 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 



* If n, n' are the mean motions of the two planets, the expression 

 for the disturbing force contains terms of the type = CQS (n pn' p') t, 



where/*, />' are integers, and the coefficient is of the order p t^ />' 

 in the eccentricities and inclinations. If now p and />' are such 

 that np r^f n' p' is small, the corresponding inequality has a perio 1 

 2 IT I (n p <^> n' p' '), and though its coefficient is of order/) f^/*', it 

 has the small factor np r*~s n p' (or its square) in the denominator and 

 may therefore be considerable. In the case of Jupiter and Saturn, 

 for example, n = 109257 in seconds of arc per annum, n' = 43,996; 

 5 n' 2n = 1,466; there is therefore an inequality of the third order, 



with a period (in years) = = 900. 

 1,400 



