APPENDIX.

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theoretical researches of these geometers and the actual position of theplanet Neptune , must be regarded as wholly fortuitous. Let us examinesome of the arguments which he adduces in support of this peculiar viewof the subject.

(92.) It has been stated (p. 185) that Mr. Adams, in his final communi-cation to the Astronomer Royal, remarked that the observations of Uranus would in all probability be satisfied best by adopting for the hypotheticalplanet a mean distance equal to 83.6. He was led to entertain this opinionby a comparison of the errors of his theory for the three oppositions of Uranus in 1843-44-45, resulting from the two hypotheses of the mean distancewhich he bad already employed in his researches. It is manifest, whenthis inference is viewed in connexion with his previous results, thatAdams had renounced all faith in even an approximation to Bode’s lawof the distances of the planets, and that the current of his researches w'asrapidly conducting him to a mean distance of the hypothetical planetagreeing with the actual mean distance of the planet Neptune . Mr.Gould seeks to depreciate the merit of this sagacious conclusion by con-tending that a mean distance equal to 33.6 would give erroneous results.“ Le Verrier, ” says he, “ has shewn that the assumption of even 35 asthe mean distance would lead to intolerable discordances. Peirce hasfurther proved that an important change in the character of the perturba-tions takes place near the distance 35.3. It is therefore evident that noclaims can be based upon the rough inference alluded to.”

It has been mentioned that Peirce objected to the reasoning bywhich Le Vender established the inferior limit of the mean distance of thehypothetical planet (35.04), on the ground of the continuity of the investi-gation being broken at the distance 35.3 by the commensurability of themean motions of the two planets. It is plain, therefore, that the pro-positions announced by these two geometers are mutually incompatible ;and yet Mr. Gould adduces them as confirmatory of each other! Wehave already had occasion to remark that there do not exist grounds forsupposing that either of them is entitled to any confidence.

(93.) Mr. Gould admits that Le Vender may be considered the discovererof the planet Neptune , in so far as he proved not only that it was impossibleto represent the motions of Uranus without the assumption of some un-known disturbing body, but that the perturbations were of that analyticalform which belongs to an exterior planet.

Now it appears to me, that the latter assertion is at direct variance withthe actual state of the question. Le Verrier demonstrated, by his re-searches, that the perturbations were such as would be produced by thedirect action of an exterior planet during the interval of time over whichthe observations extended ; but, with respect to the analytical form of theseperturbations, it depended on the elements of the disturbing planet, whichwere beyond the scope of investigation, and in fact turned out to beentirely different from those deduced by Le Verrier.

(94.) Mr. Gould further remarks that Le Verrier omitted the considera-tion of the terms depending on a near approach to commensurability; butthat this, although certain !y a defect, cannot be considered as an error inthe theory, since within the limits where he had reason to suppose thatthe orbit was situated, these terms are almost uniformly negligible.

With reference to this point it maybe remarked, that the irregularitiesin the motion of Uranus depended on the direct action of Neptune duringthe period of last conjunction, and not on the analytical theory of that