On an inequality of long period in the motions of the Earth and Venus

Abstract

The object of this memoir is similar to that of Laplace’s celebrated investigation of the great inequality of Jupiter and Saturn, announced in the Memoirs of the Academy of Sciences for 1784, and given in the volume for the succeeding year. The occasion of that investigation was an acceleration of the mean motion of Jupiter and a retardation of that of Saturn,—which inequalities in the motions of the two planets Halley had discovered by a comparison of ancient and modern observations: and Laplace showed, in the Memoirs just referred to, that inequalities like those thus noticed would arise from the action of gravitation; that they would reach a considerable amount in consequence of twice the mean motion of Jupiter being very nearly equal to five times the mean motion of Saturn; and that their period would be nearly 900 years. The occasion of the investigation of Professor Airy was an inequality in the sun’s actual motion, as compared with Delambre’s Solar Tables, which appeared to result from a comparison of late observations with those of the last century,—as Professor Airy has explained in a memoir published in the Philosophical Transactions for 1828. This comparison having convinced him of the necessity of seeking for some inequality of long period in the earth’s motion, it was soon perceived that such an inequality would arise from the circumstance that 8 times the mean motion of Venus is very nearly equal to 13 times the mean motion of the earth. The difference is 1,675 centesimal degrees in a year,—from which it follows, that if any such inequality exist, its period will be about 240 years. To determine whether such an inequality arising from the action of gravitation, amounts to an appreciable magnitude, is a problem of great complexity and great labour. The coefficient of the term will be of the order 13 minus 8, or 5, when expressed in terms of the excentricities of the orbits of the Earth and Venus, and their mutual inclination; all which quantities are small; and the result would therefore, on this account, be very minute. But in the integrations by which the inequality is found, the small fraction expressing the difference of the mean motions of the planets enters twice as a divisor; and by the augmentation arising from this and other parts of the process, the term receives a multiplier of about 2,200,000. In the corresponding step of the investigation of the great inequality of Jupiter and Saturn, it was only necessary to take terms of the 3rd order of smallness, and the multiplier by which the terms are augmented has 30 2 instead of 240 2 for its factor.