24] IN A PAETICULAR CASE. 183 



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 co- 

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



The differential equations which would require solution in these suc- 

 cessive operations after the determination of the inequalities independent 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, 

 or rather the disturbed value of the Moon's coordinate perpendicular to the 

 Ecliptic, omitting the eccentricities as before, and taking account only of 

 the first power of y. 



In this case the differential equation for finding z presents itself natur- 

 ally in the form to which Mr Hill reduces, with so much skill, the equations 

 depending on the first power of the eccentricity of the Moon's orbit. 



In solving this equation I fell upon the same infinite determinant as 

 that considered by Mr Hill, and I developed it in a similar manner in a 

 series of powers and products of small quantities, the coefficient of each 

 such term being given in a finite form. 



The terms of the fourth order in the determinant were thus obtained 

 by me on the 26th December 1868. I then laid aside the further in- 

 vestigation of this subject for a considerable time, but resumed it in 1874 

 and 1875, and on the 2nd of December in the latter year I carried the 

 approximation to the value of the determinant as far as terms of the 

 twelfth order, or to the same extent as that which has been attained by 

 Mr Hill. I have also succeeded in reducing the determination of the 

 inequalities 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. 



Immediately after Mr Hill's paper reached me, I wrote to him expressing 

 my opinion of its merits, and telling him what I had done in the same 

 direction, and I received from him a very cordial and friendly letter in reply. 



