44] ROYAL ASTRONOMICAL SOCIETY TO M. CHARLES DELAUNAY. 333 



Academy of Sciences have done both M. Delaunay and themselves the 

 honour of publishing among the volumes of their Memoirs. It is for this 

 great work that your Council have awarded to M. Delaunay the Society's 

 medal. 



Strongly impressed with the advantages of determining the coefficients 

 of the lunar inequalities in the analytical form, both as affording a solution 

 more complete in itself and more satisfactory to the mind, as well as one 

 offering facilities for the comparison of the results of different investigations, 

 M. Delaunay did not hesitate to follow the example set in this respect 

 by M. Plana, notwithstanding the immense length of the necessary calcu- 

 lations. M. Delaunay's results are thus obtained in a form which makes 

 them directly comparable with those of M. Plana, while the methods employed 

 in obtaining them are wholly different. 



M. Delaunay chooses the time as the independent variable, and takes 

 as his starting-point the differential equations furnished by the theory of 

 the variation of the arbitrary constants. In an able Memoir which appeared 

 in 1833, Poisson had advocated the employment of these equations in 

 the theory of the Moon's motion, and he applied them to the discussion 

 of some special points of that theory. These equations had been long used, 

 almost exclusively, for the determination of the perturbations of the planets, 

 and they offer peculiar advantages in the treatment of the secular in- 

 equalities and those of long period. In the case of the Moon, however, 

 in consequence of the large perturbations caused by the disturbing force of 

 the Sun, the ordinary mode of integrating these equations by successive 

 approximations soon leads to calculations of inextricable complexity. In 

 fact, these equations give the differential coefficients of the several elliptic 

 elements taken with respect to the time, in terms of the elements them- 

 selves. In the case of the planets, where the disturbing forces are so small 

 compared with the predominant central force of the Sun, very approximate 

 values of the disturbed elements may be found by substituting in the 

 values of the differential coefficients, the undisturbed instead of the disturbed 

 values of the elements, and then integrating. 



The perturbations of the elements thus found are said to be due to 

 the first power of the disturbing force. If now the approximate values of 

 the disturbed elements be substituted in the differential equations, and these 

 be again integrated, we shall obtain a second approximation to the values 

 of the disturbed elements, and the additional terms thus found are said 

 to depend on the square of the disturbing force. In the theories of the 



