610

APPENDIX.

means varies proportionally to the uncertainty of the data. It varies muchmore rapidly. Thus while for a probable error of 5" in the data, we find2.86 for the interval between the limits of the mean distance, this intervalso diminishes with the probable error, that when the latter is reduced tohalf this quantity we no longer find any value of the mean distance whichcan satisfy the question. And on the contrary, when the probable errorof the modern observations is extended beyond 5", the inferior andsuperior limits of the mean distance will be found to vary with rapidity,and to leave the greatest latitude in the choice of that auxiliary.” *These remarks are not borne out by the results contained in the foregoingtable. The action of the planet Neptune accounts for the irregularitiesof Uranus , without even supposing so large an error as 5" in the modernobservations, and yet its mean distance is 30.0363, which is far belowLe Verrier ’s inferior limit of the mean distance. The modern observa-tions of Uranus are satisfied to within 3" by the action of Neptune ,whereas Le Verrier found that if the probable error of these observationswas reduced to 2".5, there is no mean distance of the hypothetical planetwhich could satisfy the observations. The trifling difference between3" and 2".5 can hardly serve to account for the discrepancy which herepresents itself between the results of Le Verrier ’s theoretical researches,and those relative to the perturbations actually produced by Neptune .

It is manifest, without pursuing the subject any further, that the limitsassigned by Le Verrier to the mean distance of the hypothetical planet,are at direct variance with the mean distance of Neptune , as deducedfrom actual observation. Nor is it a matter of any importance, whetherthe elements employed in calculating the action of Neptune do or do notrepresent the true orbit of that planet. It is sufficient that an orbit canbe assigned, the mean distance of which lies far below the inferior limitassigned by Le Verrier , in which if a planet of a given mass be supposedto revolve, its action will be capable of completely accounting for theirregularities of Uranus .

(83.) The question then arises, where are we to look for the origin of thisdiscordance between theory and observation ? In order to arrive at someconclusion upon this point, it is necessary to direct attention to the methodof investigation by which Le Verrier deduced his final results respectingthe hypothetical planet. The groundwork of these results consisted ofthirty-three equations of condition. Le Verrier concluded from his pre-vious researches that the ratio of the mean distance of Uranus toa!

that of the hypothetical planet, must be very nearly equal to .51, and thate, its mean longitude at the beginning of the year 1800, must be con-tained somewhere between 234° and 270°. He, therefore, in his final in-vestigation assumed — r — 51 + 02y, and e = 252° -f IS 0 ?, y being a

quantity which he imagined would not differ much from unity, while Q wassupposed by him to lie somewhere between + 1 and — 1. Besidesthese quantities each of the equations contained other seven unknownquantities, namely, the corrections of the four elements of Uranus , andthe eccentricity, longitude of perihelion, and mass of the disturbingplanet. The corrections y and S enter into the equations in a very com-

* Comptes Rendus, tome xxvii., p. 330.