3] NUMERICAL DEVELOPMENTS IN THE LUNAR THEOEY. 105 



"It is to be remarked that it is only the series proceeding by powers 

 of m in Delaunay's Theory which have a slow rate of convergence, so 

 that it is probable that all the sensible corrections required by Delaunay's 

 coefficients would be found among the terms of low order in e, e', and y. 



" The differential equations which would require solution in these 

 successive operations after the determination of the inequalities indepen- 

 dent of eccentricities and inclination would be all linear and of the same 

 form. 



"It is many years since I obtained the values of these last-named 

 inequalities to a great degree of approximation, the coefficients of the 

 longitude expressed in circular measure, and those of the reciprocal of the 

 radius vector, or of the logarithm of the radius vector, being found to 

 ten or eleven places of decimals. 



"In the next place I proceeded to consider the inequalities of 

 latitude 



" I have also succeeded in reducing the determination of the in- 

 equalities of longitude and radius vector which involve the first power of 

 the lunar eccentricity to the solution of a differential equation of the 

 second order, but my method is much less elegant than that of Mr Hill." 



This pronouncement explains the purpose of the following developments.] 

 Taking the equations of Lecture IV., 



#6 2drd0 _3 , 2 . ,._ , _ ,, 



Suppress the epochs e, e', and define a so that 



and take the value of m, 



n' 



m = - = 0'0748013, 

 n 



then the following quantities substituted in the equations satisfy them to 

 ten or eleven places of decimals : 



A. II. 14 



