PRErACE. 



The present Avork was undertaken as far back as the year 1859. But the labor 

 devoted to it at first amounted to little more than tentative efforts to obtain 

 numerical data of sufficient accuracy, and to decide upon a satisfactory method of 

 computing the general perturbations of the planet. The elements of Neptune 

 employed in the earlier computations were found to deviate too widely from tlie 

 truth to be used in computing the perturbations of Uranus with the first order of 

 accuracy, and it became necessary to correct them. This was done during the years 

 1864 and 1865, and the investigation was printed by the Smithsonian Institution 

 in the latter year. It was then found that the adopted elements of Uranus also 

 differed too widely from the truth to serve as the basis of the Avork, and they Avere 

 provisionally corrected by a series of heliocentric longitudes derived from observa- 

 tions extending from 1781 to 1861. Finally it Avas found th-at the adopted method 

 of computing the perturbations, that of the " variation of elements," though not 

 deserving of the disfavor into Avhich it has fallen of late years, Avas practically 

 inapplicable to the computation of the most difficult terms, namely, those of the 

 second order Avitli respect to the disturbing forces. Indeed, it appeared to the 

 author that the only method of computing those terms which Avas at the same time 

 general, practicable, and fully developed, Avas that of Hansen. But, Avere this 

 method adopted, all that had previously been done Avould have been, useless, even 

 for the purpose of comparison and verification, OAving to the expression of the co- 

 ordinates in terms of a disturbed mean anomaly. It appeared to the author that, 

 although this form of theory led to expressions having fewer terms than the other, 

 it Avas not Avithout its relatiA^e disadvantages. Other considerations being equal, he 

 conceived that astronomers generally Avould greatly prefer to see the perturbations 

 expressed directly in terms of the time, OAving to the case Avith Avhich the results 

 of different investigators could then be compared, and Avith Avhich corrections to 

 the theory may be introduced. 



Under these circumstances the method described in the first chapter of the 

 present paper Avas Avorkcd out. The question hoAV much it contains that is essen- 

 tially ncAv is one that the author has never closely examined: it is, however, certain 



(iii) 



