HISTORY OF PHYSICAL ASTRONOMY.

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included, allowing each of the observations to be affected with an error asgreat as could possibly be due to it. He found that the mean distancecould not he greater than 37.90, nor less than 35.04, and hence he con-cluded that the limits of the periodic time were 207 years and 233 years.Adopting a given value of the mean distance, and supposing all the otherelements to vary, he found that the recorded positions of Uranus mightstill be represented within the limits assigned by the errors of observation.At length the error in one of the positions having become equal to thegreatest possible value assigned to the error of observation, the values ofthe elements were henceforth restricted by the condition that the error inthis position should retain its maximum value, and the locus of thedisturbing planet for the 1st January, 1847, corresponding to the differ-ent systems of elements derived from this hypothesis, was a continuouscurve. By continuing the variation of the elements, the error inanother of the positions became as great as possible, and, the valuesof the elements being henceforth restricted by it, the locus of theplanet was transformed into a different curve. Proceeding in this manner,Le Verrier found that the space within which the planet must be included,corresponding to the assumed value of the mean distance, was a curvilinearpolygon whose sides were discontinuous, and for different values of themean distance he obtained different polygons. The amplitude of thepolygons diminished as the value of the mean distance approached its ex-treme limits, and when it actually attained either of them the polygon be-came a point. This circumstance indicated that there was only one posi-tion of the planet that could satisfy the observations. When a sufficientnumber of such polygons was constructed, they might then be all cir-cumscribed by one continuous curve, and tangents drawn to it wouldnecessarily include the disturbing planet. Le Verrier found that thelongitude of the tangent, drawn from the sun to the east side of thebounding curve, was 321°, whence, as he had already obtained 326° 32 /for the most precise value of the planet’s longitude, it followed that thespace to be explored in this direction extended only to about 5|°. Thewestern limit was much more remote from the precise longitude, but heremarked that its amplitude might be reduced considerably by assigninga probable limit to the eccentricity. Assuming that the value of theeccentricity did not exceed .125, he obtained 335° for the limiting longi-tude in this direction. He also found that the value of the mass couldnot be greater than nor less than Xihsii'

We have mentioned that Le Verrier assigned 326° 32' as the true helio-centric longitude of the disturbing planet on the 1st January, 1847. Thisdetermination, to use his own words, placed the planet about 5° to theeast of the star I of Capricorn. Only twelve days had elapsed since itwas in opposition, and on this account strong hopes were entertained byhim that astronomers might succeed in discovering it before it plungedagain into the rays of the sun. With the view of directing attention morestrongly to this point, he made some interesting remarks relative to theapparent magnitude and visibility of the planet. The apparent magnitudedepends on the volume of the body and its distance from the sun. Thevolume, again, is deducible from the mass and density. Now, he hadalready found that the mass of the planet was two and a half times greaterthan that of Uranus . There remained, therefore, only the density to beascertained. He had no established principles to guide him in coming toa conclusion on this point, and therefore he was compelled to have re-