HISTORY OF PHYSICAL ASTRONOMY.

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searches. He now found that, when the epoch was made equal to 2.52°,the eighteen errors of the theory could be so diminished by assigning tothe mass, and the two indeterminate errors of observation, values withintheir prescribed limits, that the residual quantities might reasonably besupposed to arise from errors of observation. As the epoch diverged oneach side of 252°, the errors continued small for the first 9°; but after-wards they began to increase, and became inadmissible before it reached234° or 270°. He therefore concluded that the observations of Uranus might generally be satisfied by the hypothesis of a disturbing planet, whosemean longitude at the commencement of the year 1800 was nearly equal to252°.

It still remained for him to determine the approximate value of the truelongitude, a knowledge of which was indispensable to the discovery of theplanet. For this purpose, it was necessary for him to investigate thevalues of the eccentricity and the perihelion corresponding to the value ofthe epoch adopted by him as the most probable approximation to the truth.Now the expressions for these elements were obtained by eliminating themfrom the mean equations of 1793 and 1828, after substituting for the epochthe numerical value 252°, and consequently each of them contained threeunknown quantities; namely, the mass and the two indeterminate errorsof observation. By a skilful discussion of the errors of the theory corre-sponding to the eighteen fundamental equations of conditions, Le Verrierhad already arrived at approximate values of these three quantities *.He therefore readily determined the numerical values of the eccentricityand the longitude of the perihelion j-; by means of which, and the other twoelements already known, he computed the approximate value of the truelongitude. He finally concluded that the true longitude of the disturbingplanet for the 1st of January, 1847, was 325°, and that the probable errordid not exceed 10°.

Le Vemer’s paper contained the earliest account of researches respect-ing the hypothetical planet that was given to the world, and; as may wellbe conceived, it was read with delight by all the astronomers of Europe .Mr. Airy received the Comptes Renclus for the 1st June, on the 23rd ofthe same month. Finding that the place assigned by Le Verrier to thedisturbing planet did not differ more than a degree from that which Mr.Adams had assigned to it eight months previously, he felt a strong per-suasion of the accuracy of the labours of both mathematicians. He wasstill, however, anxious to ascertain whether the action of the hypotheticplanet would account for the errors in the radius vector of Uranus as satis-factorily as it accounted for the errors in longitude, and he now applied toM. Le Verrier for information on this point, as on a similar occasion he

* Le Verrier, indeed, does not give any account of the method by which heassigned to the errors of observation their most probable values; but it is probable thathe effected this object by a careful comparison of the residual errors of the theory, whichin all cases were expressed in terms of the mass, and the two indeterminate quantities de-noting the errors of observation. It is not impossible also, that, in order to assuregreater accuracy to his results in the present instance, he took into account the effectof the errors of observation for 1810 and 1845; for, although the presumed insignifi-cance of these errors allowed their rejection when the object was merely to ascertain theregion of the ecliptic in which the planet was to be found, still, when the question referredto the determination of its actual position, it is not difficult to perceive the desirablenessof taking them into account. In fact, Le Verrier shews, at page 202 of his Memoir, thatthey exercise a very sensible influence on the ultimate values of the errors of the theory.

t The paper in the Comptes Rendus for the 1st June does not contain any numericalvalues of these elements.