614

APPENDIX.

by his previous researches, that the observations would be very nearlysatisfied by supposing y = 0, G = 0. It might, therefore, be inferredthat the equations would assign to y and G values of inconsiderable magni-tude, which would probably satisfy the observations to a still greaterdegree of accuracy. But if the algebraic function of the second degree,involving y, G, and m', did not represent with sufficient fidelity the equationsresulting from the elimination of the other unknown quantities of theproblem, it would manifestly follow that any values of y and G, considerablyremoved from the particular values by means of which Le Verrier con-structed the algebraic equations, would fail, in a corresponding degree, tosatisfy these equations, even although such values should satisfy the ob-servations of the planet as well as any others.

(89.) According to this view of the subject, the extreme values of the meandistance resulting from Le Vender's researches ought to be considered asindicating, not the limits within which the observations of Uranus mightbe satisfied, by attributing to each of them a probable error of a certainmagnitude, but those within which an algebraic function of the seconddegree, involving y, G, and m', constructed from certain particular valuesof the two former quantities, is capable of representing the equations re-sulting from the elimination of the other unknown quantities of theproblem. Admitting this to be true, it would follow that if Le Verrier had assumed 30.0363 as his approximate mean distance, and constructedhis equations by means of a series of particular values of y and G, cor-responding to mean distances in the immediate vicinity of the approximatevalue, he would have obtained for the latter merely a slight correction,while at the same time he w'ould have deduced limiting values whichwould have totally excluded the results of his actual researches.

(90.) The conclusion definitively suggested by the foregoing remarks is,that Le Verrier ’s theory of limits is rather specious than real, in so far asit relates to the possibility of representing the observations of Uranus within a certain range of mean distance upon the supposition ofeach of them containing a probable error of a certain magnitude.It is not pretended, however, that these remarks involve the trueexplanation of the remarkable discordance which prevails between theresults of Le Verrier ’s researches and those deducible from observa-tions of the planet Neptune . They are merely put forth with the view ofindicating a probable mode of accounting for the discordance, in theabsence of a thorough investigation of the subject. Until this be accom-plished there is full warrant for suspecting the legitimacy of the methodby which Le Verrier arrived at his results. This is a point, however, ofvery trivial importance, which does not in the remotest degree affect themerits of that illustrious geometer, in so far as his theoretical discoveryof the planet Neptune is concerned.

(91.) We shall conclude with a few remarks on a recent publica-tion entitled “ Report to the Smithsonian Institution on the History of theDiscovery of Neptune." *

The author of this Report, Mr. Gould, of Cambridge, U.S. , whilegiving full credit to both Le Verrier and Adams for the accuracy of theirrespective solutions of the inverse problem of perturbation, stoutly main-tains the opinion expressed by his countryman, Prof. Peirce, to the effectthat the remarkable connexion which was found to subsist between the

* Published at Washington, 1850.