338 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [44 



coordinates of the Moon of an order inferior to the fourth. For this pur- 

 pose fifty-seven such operations are required to be performed. When these 

 have been effected, the periodic terms which remain in the disturbing 

 function are so small that their powers and products may be neglected, 

 and consequently the differential equations which determine the six elements 

 last introduced in terms of the time, may be integrated at once. Since 

 the values of the Moon's coordinates are known in terms of the elements 

 just mentioned and the time, we have only to substitute the values of the 

 elements that have been found, in order to determine the Moon's coordinates 

 in terms of the time. 



The values of the elements, however, that would be found in this way 

 are very complicated, and therefore the substitutions which would be required 

 in order to find the Moon's coordinates would be excessively long. M. De- 

 launay, accordingly, prefers to get rid of the remaining periodic terms 

 in the disturbing function, one by one, by means of transformations exactly 

 similar to those which have been already effected. In order to carry on 

 the approximation to the extent which he desires, M. Delaunay finds it 

 necessary to perform no less than 448 of these secondary operations, but 

 each such operation becomes very simple, since the squares of the coefficients 

 of the periodic terms under consideration may be neglected. 



Thus, at length, by means of 505 transformations, all the periodic terms 

 of the disturbing function are removed, and the problem is reduced to the 

 ideal case which was considered at the outset of our account of M. Delaunay's 

 method. 



After each transformation, by making the proper substitutions in the 

 expressions for the Moon's coordinates, those coordinates are obtained in 

 terms of the system of elements last introduced, so that finally the three 

 coordinates are known in terms of the three final constants and angles 

 which vary uniformly with the time. 



It has been already mentioned that Plana, in his great work on the 

 Lunar Theory, determined the analytical values of the coefficients of the 

 lunar inequalities as far as terms of the fifth order inclusive, and that he 

 only carried on the development to a greater extent hi cases where the 

 slowness of the convergence of the series appeared to him to render it 

 necessary to take into account terms of higher orders than the fifth. 



M. Delaunay has proposed to himself to carry on the approximation so 

 as to include all terms of the seventh order, and in cases where the series 



