HISTORY OF PHYSICAL ASTRONOMY.

179

merits of the orbit which it describes ? ” The problem being thus enun-ciated, Le Vender next proceeded to explain his method of solution. Itwas impossible to consider the disturbing planet without also taking intoaccount the errors in the elements of Uranus ; for the irregularities of thelatter planet, which formed the data of the problem, depended on boththese causes ; nor was it possible to eliminate the effects due to either causewithout a knowledge of those due to the other. The problem, therefore,admitted of solution only by a method which would embrace in onesimultaneous investigation the elements of the disturbing planet, andthe corrections due to the elements of Uranus . It was necessary for thispurpose to compute the heliocentric co-ordinates of the latter planet bymeans of the actual ellipse, and the perturbations of the knowm planets,and then to apply to the results 1°, the expression for the perturbations ofthe unknown planet, in terms of its mass and the elements of its orbit,2°, the expression for the correction due to the false ellipse of Uranus ,in terms of the errors of the elements. Each of the two last-mentionedexpressions contained four unknown quantities, and therefore the com-plete value of any of the heliocentric co-ordinates would contain eight un-known quantities. Equating this result to the corresponding value ofthe co-ordinate derived from observation, an equation of condition wasformed between the eight unknown quantities, and the solution of the prob-lem was reduced to the elimination of these quantities from a system ofsuch equations.

Le Verrier employed as the basis of his researches eighteen errors ofheliocentric longitude *. Three of these errors depended on observationsof the planet made by Flamstead, in 1690, 1712, and 1715. The otherfifteen extended over the period embraced between 1747 and 1845, andwere separated by equal intervals of seven years. By means of theseeighteen errors he formed an equal number of equations of condition.He remarked that the corrections to the elements of Uranus might bereadily eliminated from the equations of condition; and hence the re-sulting equations would contain only four unknown quantities, namely, themass, the epoch, the eccentricity, and the longitude of the perihelion of thedisturbing planet. Now, it was easy to obtain expressions for the mass,the eccentricity, and the longitude of the perihelion, in terms of theepoch. The reader who possesses a knowledge of the elements of mathe-matics will hence perceive that by means of these expressions an equationmight finally be formed involving the epoch alone; and upon its elimi-nation the solution of the problem would manifestly depend. This element,however, presented itself in such a complicated form, that Le Verrier did not consider it advisable to attempt the solution of the problem bymeans of such an equation. He proposed to attain the same end by

* He had previously employed these errors in demonstrating the incompatibility of theactual theory of Uranus with the results of observation ; but, in order to avoid unneces-sary complication, we did not make any allusion to them on that occasion. As the errorswere all subject to the condition of being equidistant, it was impossible to determine themgenerally except by a process of interpolation ; and, as the ancient observations were sepa-rated from each other by considerable intervals, it is manifest that in some instances thenumerical values of the errors could not be relied upon for extreme accuracy. Besides,the probable effect of the error of radius vector was not taken into account. Le Verrier clearly saw that the errors arising from both these causes were too insignificant to de-serve notice when the object merely was to obtain an approximate value of the mean longi-tude of the disturbing planet. In his final solution he completely eluded both sources oferror by employing as the basis of his investigation the geocentric longitudes of Uranus .