66 PROCEEDINGS OF THE AMERICAN ACADEMY 



The extension of the investigations to any other mean distances can be 

 made only by assuming a continuous law to pervade the subject of in- 

 quiry, and that there is no important change in the character of the re- 

 sulting perturbations. Guided by this principle, well established, and 

 legitimate, if confined within proper limits, M. Le Verrier narrowed 

 with consummate skill the field of research, and arrived at two fun- 

 damental propositions, namely, — 



1st. That the mean distance of the planet cannot be less than 35, 

 or more than 37.9. The corresponding limits of the time of sidereal 

 revolution are about 207 and 233 years, 



2d. "That there is only one region in which the disturbing planet 

 can be placed, in order to account for the motions of Uranus ; that the 

 mean longitude of this planet must have been, on January 1st, ISOO, 

 between 243^ and 252°." 



" Neither of these propositions is of itself necessarily opposed to the 

 observations which have been made upon Neptune, but the two com- 

 bined are decidedly inconsistent with observation. It is impossible to 

 find an orbit, which, satisfying the observed distance and motion, is 

 subject at the same time to both of these propositions, or even approx- 

 imately subject to them. If, for instance, a mean longitude and time 

 of revolution are adopted according with the first, the corresponding 

 mean longitude in 1800 must have been at least 40° distant from the 

 limits of the second proposition. And again, if the planet is assumed 

 to have had in 1800 a mean longitude near the limits of the second 

 proposition, the corresponding time of revolution with which its motions 

 satisfy the present observations cannot exceed 170 years, and must 

 therefore be about 40 years less than the limits of the first proposition. 

 Neptune cannot, then, be the planet of M. Le Verrier's theory, and 

 cannot account for the observed perturbations of Uranus under the 

 form of the inequalities involved in his analysis. 



" It is not, however, a necessary conclusion that Neptune will not 

 account for the perturbations of Uranus, for its probable mean distance 

 of about 30 is so much less than the limits of the previous researches, 

 that no inference from them can be safely extended to it. An im- 

 portant change, indeed, in the character of the perturbations takes 

 place near the distance 35.3 ; so that the continuous law by which 

 such inferences are justified is abruptly broken at this point, and it 

 was hence an oversight in M. Le Verrier to extend his inner limit to the 

 distance 35. A planet at the distance 35.3 would revolve about the 



