iv PREFACE. 



that the mode of considering the subject is well known, being that employed by 

 La Place, Herschel, De Pontecoulant, Encke, and perhaps others. The method 

 of forming the required derivations of the perturbative function from the analytical 

 development of that quantity, he has not seen elsewhere. 



With these improved elements and methods the work was recommenced in 1868. 

 The earlier investigations being merely provisional, it has not been deemed neces- 

 sary to present them in the present work. Some of the results, corrected for 

 errors of the older elements, are, however, given for the purpose of comparison. 



Although this investigation has absorbed the greater part of the author's leisure 

 for more than five years, it is only through the aid of the Smithsonian Institution 

 and Nautical Almanac that he has been enabled to bring it to a conclusion witliin 

 that time. At an early stage of the work Professor Henry responded favorably to 

 a request for aid by the employment of computers; it was, however, not found 

 practicable to use such aid until tlie perturbations had been completed, and the 

 provisional theory concluded. Then, the comparison of theory and observation, 

 and the construction of the tables, involved a large amount of mechanical compu- 

 tation, and on this part of the work a number of persons have been employed by 

 the Institution at various times, among whom may be mentioned Professor F. W. 

 Bardwell, of the University of Kansas, and Dr. C. L. F. Kampf, late of the Ob- 

 servatory of Leiden. Every part of the work has, however, been done under the 

 author's immediate direction, and, as nearly as possible, in the same way as if he 

 had done it himself, a result which, in one or two cases, has been attained only 

 by the expenditure of an amount of labor approximating that saved by the employ- 

 ment of the computer. 



In presenting the steps of the investigation, the end has been kept constantly in 

 view to render as easy as possible the detection and correction of any error, or the 

 introduction of any alteration in the elements or other data. It is, of course, 

 impossible to present the steps of the computation with any approach to fulness 

 without far transcending the limits of the printed work : The results given are, 

 therefore, those which it was supposed would be most useful to the future investi- 

 gator of the same subject. There is reason to believe that the original computa- 

 tions will ultimately become the property of the National Academy of Sciences, so 

 that they may always be referred to for the clearing up of any difficulty in tlie 

 printed text. 



The author's acknowledgments are due to Professor J. H. C. Coffin, Superin- 

 tendent of the Nautical Almanac, and Mr. E. J. Loomis, of the Nautical Almanac 

 Office, for reading the proof sheets of the last twelve tables during the absence of 

 the former abroad. 



WAsmNGTON, July 31, 1873. 



