I' KEF ACE. 



TIIK present work was undertaken as far back as the year 1859. But the labor 

 deMited 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 the 

 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 

 IN; I and IN>.">, 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 work, and they were 

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

 tions extending from 1781 to 1861. Finally it was found that the adopted method 

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

 deserving of the disfavor into which it has fallen of late years, was practically 

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

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

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

 general, practicable, and fully developed, was that of Hansen. But, were this 

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

 for the purpose of comparison and verification, owing 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 was not without its relative disadvantages. Other considerations being equal, he 

 conceived that astronomers generally would greatly prefer to see the perturbations 

 expressed directly in terms of the time, owing to the ease with which the results 

 of different investigators could then be compared, and with which corrections to 

 the theory may be introduced. 



Under these circumstanc.es the method described in the first chapter of the 

 present paper was worked out. The question how much it contains that is essen- 

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



(iii) 



