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History of physical astronomy.

1793 and 1828, and then to eliminate from them the eccentricity and tholongitude of the perihelion of the disturbing planet. From what wc havealready said, relative to the composition of these equations, it is manifestthat each of the eliminated elements would be expressed in terms of threeunknown quantities, namely, the mass, and the two indeterminate errorsof observation. He now proposed, by means of them, to determine thoerrors of the theory in 1690, 1747, and 1758, in order to ascertain whetherthey might be made sufficiently small for any value of the epoch. Theseerrors no longer appeared in the form of simple numerical results, as theydid by his first method. They now contained the mass, and the twoerrors of observation in indeterminate forms; and it was only by a discus-sion founded on the limits, within which each of these three quantitiesmight possibly vary, that the numerical value of the error of the theorycould be in any case arrived at.

Le Verrier proceeded to apply this method of investigation, purposingto extend it to the whole of the ecliptic. For this purpose he madethe epoch of the disturbing planet vary for every 9° from zero to 360°,and computed the corresponding errors of the theory in 1690, 1747, and1758. Examining the expressions for these errors, lie found that, for allvalues of the epoch from zero to 189°, no admissible values of the massand the other two indeterminate quantities could render the results in allthe three cases so small that they might reasonably be attributed to errorsof observation. As the epoch increased from 189°, the errors in each ofthe three cases began to diminish, and they all became very small, whenit attained a value equal to 243° or 252°, or any of the intermediatevalues. From 252° the errors began to increase and soon becameinadmissible; nor did they present any diminution throughout the remain-ing part of the circumference of the ecliptic. Le Verrier therefore cameto the conclusion that there is only one region of the ecliptic in which it ispossible to place the disturbing planet so as to account for the movements ofUranus , and that the mean longitude of this planet, on the 1st of January,1800, must be included between 243° and 252”.

Tho next point to ascertain was, whether a planet actually situated inthe region indicated by his researches would account generally for theirregularities of Uranus ; for, although he had found that a planet whosemean longitude was somewhere about 252°, and whose eccentricity andlongitude of perihelion were determined by the equations of 1793 and1828, would also satisfy the equations of 1690, 1747* and 1758, it did notfollow as a necessary consequence that all the equations of condition depend-ing severally upon the individual observations of the planet would also besatisfied I 13 7 the same supposition. In order to establish this point, heresolved to compute the errors of the theory corresponding to the eighteenindividual observations which formed the basis of his researches. Therewas this advantage, however, gained by his previous labours, that it was nolonger necessary to extend his investigation to the whole circuit of theecliptic, it being merely sufficient to examine the region in which he hadalready found that the planet, if really existing, must be situated, at thecommencement of the year 1800. With this view he formed eighteenexpressions for the errors of the theory for five equidistant values of theepoch comprised between 234° and 270°, the corresponding values ofthe eccentricity and the longitude of the perihelion being derived, asbefore, from the two mean equations of 1793 and 1828. The result ofthis investigation afforded a complete confirmation of his previous re-