182 ON THE MOTION OF THE MOON'S NODE [24 



which enter into the value of the above determinant are of the fourth, 

 eighth, twelfth, &c. orders, considering, as usual, the ratio of the mean 

 motion of the Sun to that of the Moon as a small quantity of the first 

 order ; and the author has taken into account all the terms of lower orders 

 than the sixteenth. The ratio of the motion of the perigee to that of 

 the Moon thus obtained is true to twelve or thirteen significant figures. 

 The author compares his numerical result with that deduced from Delaunay's 

 analytical formula, which gives the ratio just mentioned developed in a 

 series of powers of m, the ratio of the mean motions of the Sun and 

 Moon. The numerical coefficients of the successive terms of this series 

 increase so rapidly that the convergence of the series is slow, so that the 

 terms calculated do not suffice to give the first four significant figures of 

 the result correctly, although by induction, a rough approximation may be 

 made to the sum of the remaining terms of the series. 



I have been led to dwell thus particularly on Mr Hill's investigation 

 because my own researches in the Lunar Theory have followed, in some 

 respects, a parallel course, sed longo intervallo. 



I have long been convinced that the most advantageous way of treating 

 the Lunar Theory is, first, to determine with all desirable accuracy the 

 inequalities which are independent of the eccentricities e and e', and the 

 inclination 2sin~'y, and then, in succession, to find the inequalities which 

 are of one dimension, two dimensions, and so on, with respect to those 

 quantities. 



Thus the coefficient of any inequality in the Moon's coordinates would 

 be represented by a series arranged in powers a.nd products of e, e', and y, 

 and each term in this series would involve a numerical coefficient which 

 is a function of m alone and which may be calculated for any given value 

 of m without the necessity of developing it in powers of m. The variations 

 of these coefficients which would result from a very small change in m 

 might be found either independently or by making the calculation for two 

 values of m differing by a small quantity. 



This method is particularly advantageous when we wish to compare 

 our results with those of an analytical theory such as Delaunay's, in which 

 the eccentricities and the inclination are left indeterminate, since each 

 numerical coefficient so obtained could be compared separately with its 

 analytical development in powers of m. 



