44] ROYAL ASTRONOMICAL SOCIETY TO M. CHARLES DELAUNAY. 335 



the elements of each pair being conjugate to each other. Then the differ- 

 ential coefficient of any element with respect to the time is simply equal 

 to the partial differential coefficient of the disturbing function taken with 

 respect to the element which is conjugate to the former, the partial 

 differential coefficients which occur in the two equations corresponding to a 

 pair of conjugate elements being affected with opposite signs. 



The disturbing function may be readily developed in a series of periodic 

 terms involving cosines of angles, each of which is formed by the combination 

 of multiples of the Moon's mean longitude, the distance of the Moon's 

 perigee from its node, and the longitude of the node, together with angles 

 which depend on the position of the disturbing bodies. The disturbing 

 function likewise contains a non-periodic term, which, as well as the co- 

 efficients of the periodic terms, are all functions of the major semi-axis, 

 the eccentricity and the inclination of the Moon's orbit. 



Since the mean longitude of the Moon involves the time multiplied 

 by the mean motion which is a function of one of the elements, it is 

 obvious that the differentiation with respect to this element will give rise 

 to terms in which the time occurs without its being included under a sine 

 or a cosine. Such terms would render the equations veiy inconvenient for 

 the determination of the lunar inequalities ; and M. Delaunay accordingly 

 avoids the introduction of them by taking the mean longitude itself instead 

 of the epoch of mean longitude, as one of his elements, while by the simple 

 yet novel expedient of adding to the disturbing function a non-periodic 

 term which is a function of the major semi-axis alone and is independent 

 of the disturbing forces, he preserves to the differential equations the same 

 very simple form which they had at first. After this modification of the 

 disturbing function, the time no longer enters into it explicitly except in 

 so far as it is introduced by the values of the coordinates of the disturbing 

 bodies, and consequently the difficulty which was before met with completely 

 disappears. 



The six elements employed by M. Delaunay are thus, the Moon's mean 

 longitude, the distance of the perigee of its orbit from the node, and the 

 longitude of the node, which for distinction may be called the three angular 

 elements, and three other elements which are respectively conjugate to the 

 former, and which are determinate functions of the major semi-axis, the 

 eccentricity and the inclination of the orbit. 



The three coordinates of the Moon at any time are given in terms of 

 the three angular elements and of the quantities last mentioned. 



