BIOGRAPHICAL NOTICE. xli 



gratulating Mr Hill upon his investigation, he mentions that his own researches had 

 followed in some respects a parallel course. In particular he remarks that the differential 

 equation for z, the Moon's coordinate perpendicular to the ecliptic, presents itself naturally 

 in the same form as that to which Mr Hill had so skilfully reduced his differential 

 equations. In solving this equation, which was therefore of Mr Hill's standard form, he 

 fell upon the same infinite determinaut as that considered by Mr Hill, and 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. This development was continued as far 

 as the terms of the fourth order in 1SG8; and in 1875, when he resumed the subject, 

 the approximation was extended to terms of the twelfth order, which is the same degree 

 of accuracy as that to which Mr Hill had carried his researches. On making the 

 reductions requisite in order to render the two results comparable, he found that they 

 were in agreement with the exception of one of the terms of the twelfth order, and that 

 this discrepancy was due to a simple error of transcription. He states that the calculations 

 by which he had found the value of the determinant were very different in detail from 

 those required by Mr Hill's method, but that he had not had time to copy them out 

 from his old papers and put them in order. In this communication, therefore, he confined 

 himself to making known the result which he had obtained for the motion of the Moon's 

 node. After giving an outline of the method pursued, including the equation derived 

 from the infinite determinant, he arrives at the formulae by means of which the value 

 of g, as dependent only upon in, was obtained to fifteen places of decimals. 



It is difficult to appreciate too highly the mathematical ability shown by Adams 

 and Hill in devising methods which did not require expansion in powers of m, and 

 which yielded with such wonderful accuracy these values of g and c. Apart, however, 

 from the mathematical and astronomical interest of the researches themselves, the co- 

 incidence of methods and ideas is very striking. But for the publication of Hill's memoir 

 it is probable that no account of these results of Adams's would have been published 

 in his lifetime, and it is not unlikely that he would never have put into writing his 

 views on the mathematical treatment of the lunar problem which give additional interest 

 to this short paper. As far back as 1853, in his memoir upon the secular acceleration, 

 he mentioned that the new terms in the expression of the Moon's coordinates occurred 

 to him some time before, when he was engaged in thinking over a new method of 

 treating the lunar theory, and it is well known that the theory itself, or problems 

 connected with it, constantly occupied his attention. In this paper of 1877 he states 

 that he had long been convinced that the most advantageous mode of treatment is by 

 first determining with all possible accuracy the inequalities which are independent of 

 e, e', and 7, and then in succession finding the inequalities which are of one dimension, 

 two dimensions, and so on with respect to these quantities. Thus, the coefficient of 

 any inequality in the Moon's coordinates would be represented by a series arranged in 

 powers and products of e, e, and 7 ; and each term in this series would involve a numerical 

 coefficient which is a function of m alone, and which admits of calculation for any given 

 value of m without the necessity of developing it in powers of m. This method is 

 particularly advantageous when the results are to be compared with those of an analytical 

 lunar theory such as Delaunay's, in which the eccentricities and the inclination are left 

 indeterminate, since each numerical coefficient admits of a separate comparison with its 



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