APPENDIX.

617

theory and limits of Le Verrier , and corresponds with Adams’s solution ;the other is the orbit of Neptune .” *

With respect to the existence of two mean distances of least possibleerror, with an interval included between them, any mean distance corre-sponding to which is incapable of satisfying the observations with sufficientaccuracy, it seems to be in the highest degree improbable. This will bereadily seen by reference to the theoretical researches of Le Verrier andAdams. The elements of the first and second planet of Adams, andthose which Le Verrier deduced from his final investigation, exhibit asuccessive diminution of the mean distance. Now, in each of these threecases, the mean distance was greater than the true value; but this defectwas remedied by increasing the eccentricity in a corresponding degree, andplacing the perihelion near the point of conjunction of N eptune and Uranus .By this means the distances of the disturbing body were rendered in eachcase very nearly equal to the true distances in the part of the orbit whereconsiderable precision was indispensable; and the effect of the error inthe mean distance was thrown upon the opposite portion of the orbitextending on each side of the aphelion, where it was incapable of exer-cising any influence. The following table will exhibit this view of thesubject in a clearer light:—

Hypothetical Planet.

Mean

Distance.

Perihelion

Distance.

Aphelion

Distance.

Longitude ofPerihelion.

Adams, Hyp. I.

38.400

32.216

44.584

315° 57'

„ Hyp. II. .

37.478

32.958

41.998

299 11

Le Verrier . . .

36.154

32.264

40.044

284 45

It appears from these results, that the perihelion distance is almost thesame for each of the three planets, and that the longitude of the periheliondoes not in any case differ materially from the longitude of Uranus andNeptune (273°) on the occasion of their last conjunction, about the begin-ning of the year 1822. On the other hand, the aphelion distance variesnearly at a rate corresponding with the variation of the mean distance.Now if we suppose the mean distance of the hypothetical planet to bediminished below the value assigned by Le Verrier , so as to approachnearer the mean distance of Neptune , have we not strong reason to believethat, by similarly throwing the effect of the change mainly upon theaphelion distance, where it would be altogether uninfluential, the observa-tions of Uranus would be satisfied with the same degree of precision asin the foregoing cases? Indeed it seems very probable that this objectmight be accomplished by employing any mean distance within a rangeextending considerably both above and below the mean distance of Nep­ tune , the perihelion being turned towards the point of conjunction whenthe mean distance was greater, and the aphelion being turned towards thesame point when the mean distance was less, than the true value.

(98.) We may recapitulate the conclusions suggested by the discovery ofthe planet Neptune in the following terms :—Two contemporary geometers,

* Report, p. <55.