108 



March 29, 1832. 



GEORGE RENNIE, Esq. Vice President, in the Chair. 



The following Report, drawn up by the Rev. William Whewell, 

 M.A. F.R.S., and John William Lubbock, Esq. M.A. V.P. and 

 Treasurer R.S., on Professor Airy's Paper, read before the Royal 

 Society on November 24, 1831, and entitled, " On an Inequality of 

 Long Period in the Motions of the Earth and Venus," was read. 



Report. 



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 investi- 

 gation was an acceleration of the mean motion of Jupiter and a re- 

 tardation 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 re- 

 ferred 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 in- 

 vestigation 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 in- 

 equality 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 di- 

 visor ; and by the augmentation arising from this and other parts of 

 the process, the term receives a multiplier of about 2,200,000. Tn 

 the corresponding step of the investigation of the great inequality of 



