APPENDIX.

C05

astronomers, by applying to the elliptic motion of Uranus the perturba-tions produced by Neptune as represented by the formula; of analysis, havesucceeded in satisfying the observation of 1690 with almost perfect accuracy,the outstanding error being less than 1". The question appears to admit ofa definitive solution by adopting the following mode of procedure :—Sincethe action of Neptune upon Uranus continued insensible from 1070 to1800, it necessarily follows that the motion of Uranus , after subductingfrom it the effects produced by the disturbing action of the other planets,was purely elliptic during the whole of the interval of time includedbetween these two epochs. Hence it is obvious, that if the elements ofUranus be deduced from a sufficient number of observations made withinthe included interval, the motion of the planet, when calculated fromsuch elements, ought to satisfy the totality of the observations, extendingfrom 1690, the year of Flamsteed’s earliest observation, down to 1800,or even a few years later.

(75.) The elements of Neptune being considerably different from thoseof the hypothetical planets of Le Verrier and Adams, and its mean motionbeing nearly commensurable with the mean motion of Uranus , the theoryof its action upon the latter planet presents a wide discordance, when com-pared with the theory of either of the geometers just mentioned. Itis to be borne in mind, however, that this circumstance is immaterial,when the question relates merely to the perturbations produced in themotion of Uranus , on the occasion of one conjunction with Neptune .Prof. Peirce, however, took a different view' of the subject. He con-tended, on the ground of the discordance above referred to, that Neptune was not the planet designated by geometry, and that, in fact, its discoverymust be regarded as a happy accident. “ The solutions of Adams andLe Verrier, ” says he, “ are perfectly correct for the assumption to whichthey are limited, and must be classed with the boldest and most brilliantattempts at analytical investigation, richly entitling their authors to allthe eclat which has been lavished upon them on account of the singularsuccess with which they are thought to have been crowned. But theirinvestigations are nevertheless wholly inapplicable to the theory of themutual perturbations of Uranus and Neptune . The successive periodsof conjunction and opposition, occurring at intervals of eighty-four years,that is, in about the time of a revolution of Uranus , this planet is alwaysat the same part of its orbit when it is most affected by the action ofNeptune . The action of Neptune consequently assumes a fixed, perma-nent undisturbed character, so that it can hardly be recognised as pertur-bation by the practical observer. It is far otherwise with the ordinaryclass of perturbations, where the place of greatest disturbance varies frompoint to point of the orbit: thus the place of greatest disturbance, inthe case of the theoretical planet, would not have remained stationary,but have varied 80° upon the orbit of Uranus at each successive conjunc-tion and opposition; so that the disturbance could not in this case bedisguised to any great extent under the fixed laws of ordinary ellipticmotion. In the case of Neptune , its action on Uranus is to be detectedin the comparatively small differences between its character and that ofan elliptic motion, and the difference between the influence at oppositionand that at conjunction.”*

(76.) The assertion of Prof. Peirce—that the investigations of Adams andLe Verrier are inapplicable to the theory of the mutual action of Uranus * Proc. Arner. Acad, of Arts and Sciences, vol. i., p. 341.