44] ROYAL ASTRONOMICAL SOCIETY TO M. CHARLES DELAUNAY. 339- 



converge slowly to take into account terms of the eighth, and even of the 

 ninth order. 



Those who have had any experience in calculations of this nature will 

 readily understand how enormously the labour required has been increased 

 by thus adding two orders more to those which Plana has considered. It 

 is not merely that the terms of higher orders are far more numerous than 

 those of the lower, but also that each of the terms of the former kind is 

 much more difficult to calculate, since it arises from a much greater number 

 of combinations of terms of the inferior orders. 



This enormous labour, which has occupied M. Delaunay for nearly twenty 

 years, has been performed by him without assistance from any one. Indeed, 

 from the nature of the calculations which are required, it would not have 

 been easy to obtain any effective assistance. In order to insure accuracy, 

 M. Delaunay has omitted no means of verification, and he has performed 

 all the calculations, without exception, at two separate times, with a suf- 

 ficient interval between them to prevent any special risk of committing the 

 same error twice in succession. 



The volumes before us are perfect models of orderly arrangement. Not- 

 withstanding the great length and complication of the calculations, the 

 whole work is so disposed that any part of it may be specially examined 

 with the utmost readiness by any one who may wish to test its accuracy. 



Finally, the analytical expressions which have been obtained for the 

 Moon's coordinates are converted into numbers, by substituting for the 

 elements the most accurate numerical values which the comparison of theory 

 with observation has made known. 



Such is an imperfect sketch of M. Delaunay's labours on the Theory 

 of the Moon contained in these two magnificent volumes, the former of 

 which appeared in 1860, and the latter in 1867. As I have already stated, 

 they do not include a complete theory of the Moon, but only that which 

 is by far the most difficult and complicated part of that theory, viz., the 

 investigation of the perturbations due to the direct action of the Sun 

 supposing its apparent motion about the Earth to be purely elliptic. Of 

 the investigations which are required to take into account the remaining 

 very small causes of disturbance, and which are intended by M. Delaunay 

 to be included in a supplementary volume, some of the most important 

 have been already completed by him, particularly the calculation of the 



432 



