44] ROYAL ASTRONOMICAL SOCIETY TO M. CHARLES DELAUNAY. 331 



has supplied more accurate values of the elements, their corrected values 

 can be at once substituted in the same formula?, without requiring any 

 additional work. 



On the other hand, if the numerical values of the elements be intro- 

 duced into the calculations from the first, then if it is desired to introduce 

 corrected values of the elements, much additional investigation will be 

 required for the purpose. 



No doubt the labour required in order to obtain a given amount of 

 numerical accuracy by this method is very much greater than is required 

 when each coefficient, instead of consisting of a series of terms, is reduced 

 to a simple numerical quantity, but the great theoretical advantage of 

 knowing the composition of every coefficient in terms of the elements well 

 repays the additional labour. 



The degree of convergence of the series obtained for the several co- 

 efficients is in general sufficiently rapid, but in some few of the coefficients, 

 on the contrary, the convergence is so slow, at least in the leading terms, 

 that it is necessary to take into account terms which are analytically of 

 a higher order than those to which the approximation is in general limited. 



Thus Plana, who proposed to himself to determine the lunar inequalities 

 completely to the fifth order, found it necessary in special cases to carry 

 the approximation to the seventh and even to the eighth order, and in 

 several cases he also added an estimated value of the remainder of the 

 series founded on the observed law of diminution of the calculated terms. 



Soon after the publication of Plana's great work, Sir John Lubbock 

 formed the plan, which he partly carried out in his various tracts on the 

 theory of the Moon, of verifying Plana's results by a totally different 

 method, starting from differential equations in which the time is taken as 

 the independent variable, and thus avoiding the necessity of reversion of 

 series. 



Later, M. de Pontecoulant undertook the same work on a similar plan, 

 and carried it out more completely in the fourth volume of his Theorie 

 Analytique de Systeme du Monde. 



These works, while they corrected some errors which had crept into 

 Plana's computations, confirmed their wonderful general accuracy, and with 

 some few exceptions they do not extend the approximation beyond the 

 order to which Plana restricts himself. 



422 



