Ill 



order to which it is carried. In Professor Airy's investigation there 

 are, to each angular factor, twelve terms arising from the combina- 

 tions of e, e\f\ in that of the inequality of Jupiter to the 3rd order, 

 there are four such terms. 



The number of the coefficients C, of the series for R, which it is 

 requisite to calculate, depends both upon the number of effective 

 combinations of arcs and the number of effective combinations 

 of e, e' f f. Hence the number of such coefficients, and their 

 differentials, which Professor Airy's calculation demands, is very 

 considerable. In the calculation for Jupiter, Laplace uses 6 co- 

 efficients b, and 14 of their differentials, in which however 28 dif- 

 ferentials are virtually obtained. Professor Airy has occasion to use 

 70 values of C, the corresponding quantity in his process, and 98 of 

 its differential coefficients ; these quantities being calculated to a 

 number of places from 7 downwards. 



The calculation of the inequalities of the motions of Venus and 

 the Earth, from the numbers thus obtained, requires the combination 

 of these numbers with others depending on the excentricities, incli- 

 nation, perihelia, and nodes of the orbits ; and contains, as has al- 

 ready been said, 12 compound terms in the present investigation, 

 and 4 in that of Laplace. 



The greatest amount of the inequality thus explained by Laplace, 

 was 20' for Jupiter and 47' for Saturn. The effect of the inequality 

 examined in the memoir before us, would give an error in the geo- 

 centric longitude of Venus of between 20" and 30", if the mean mo- 

 tions of the Earth and Venus were determined, by comparing the 

 observations about Bradley's time with the observations of a few 

 years ago ; and if the result were applied to calculate the next 

 transit of Venus (in 1874). 



The method adopted by Professor Airy in this investigation offers 

 some peculiarities. There are two principal methods which may be 

 employed in such problems : one is the method of direct solution, 

 according to which the equations on which the inequalities depend 

 are solved directly, and the values obtained by the first approximate 

 solutions are substituted in the terms before neglected, in order 

 to obtain a new solution. The other method is that of the variation 

 of parameters (developed by Lagrange), according to which the 

 planet, at any moment of time, is conceived to be moving in an el- 

 lipse, and the alterations are investigated which the elements of this 

 ellipse must continually undergo, in order that the real motion may 

 result. The former of these methods is the one which has generally 

 been employed in calculating all inequalities of the planets except 

 secular ones, and is used by Laplace in the theory of Jupiter and 

 Saturn. In the present memoir the author has adopted the method 

 of the variation of parameters, and he states his opinion that this, or 

 some similar method, will ere long be adopted in the planetary thec~ 

 ries, to the exclusion of other methods. 



In one instance the author has introduced an alteration into the 

 formulae given for the variation of the elements by those who have 



