HISTORY OF PHYSICAL ASTRONOMY.

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and partly to the disturbing action of the unknown planet. Thesetwo causes of error, although totally opposite in their natures, had both acommon origin, and their effects on this account were so thoroughly en-tangled in each other, that it was impossible to investigate either inde-pendently. This circumstance, as may well be supposed, vastly increasedthe difficulty of the problem. If it had been possible to ascertain whatportion of the irregularities of the planet was in each case due to theerrors of the elements, the values of these errors might have been deter-mined by the ordinary methods of astronomy; and when once the elementswere thus corrected, it would have been undoubtedly a less arduous task toascend from them to the elements of the unknown planet. But it clearlyfollows, from the remarks we have already made, that the elements of Uranus cannot be determined without taking into account the whole effects of plane-tary perturbation, and this again implies a knowledge of what is the veryobject of inquiry, namely, the elements and mass of the unknown planet.The problem, then, admits of being completely treated only by a methodwhich shall embrace in one simultaneous investigation the corrections to theelements of Uranus and the elements and mass of the unknown planet. Now,the elements of a planet are six in number. The problem, when consideredin all its generality, will therefore involve thirteen unknown quantities.As, however, it appeared from observation that the perturbations of Uranus in latitude were very small, the two planets might be supposed, for all thepurposes of a first approximation, to revolve in the same plane, and by thissimplification the number of unknown quantities was reduced to nine.These were:—the corrections to the mean distance, the eccentricity, thelongitude of the perihelion, and the epoch or mean longitude of Uranus ;and the absolute values of the same elements relative to the unknownplanet together with its mass. Now, in the case of one planet acting uponanother and disturbing its motion, the theory of perturbation enables thegeometer to express the corrections to the co-ordinates of the disturbedplanet in terms of these nine unknown quantities; and, if the hypothesisof an unknown planet was true in the present case, these corrections oughtto account for the differences between the observed and computed positionsof the planet Uranus . Hence, by comparing the analytical formulae forthe corrections to the co-ordinates of Uranus with the numerical errorsof the tables, a number of equations of condition are formed, and theproblem is finally reduced to the elimination of the unknown quantitiesfrom these equations. It may readily be imagined that this operation isattended with difficulties of no ordinary kind. In fact, only a thoroughacquaintance with the nature of planetary perturbation, and a completecommand of analysis, combined with consummate skill in its application,will suggest to the mathematician such artifices as may lead to the dis-engagement of the unknown quantities from the complicated expressionsin which they are involved.

Nor does the mere elimination of the unknown quantities constitute theonly difficulty attending the solution of the problem. Unless the equa-tions of condition be skilfully combined together, it may happen that thevalues of the unknown quantities, although deduced by a strictly legiti-mate process of reasoning, will prove totally erroneous when consideredas physical results. This will be readily understood by the reader when hetakes into account the mode in which the equations of condition are formed.In each case the error of the planet’s place is expressed algebraically interms of the nine unknown quantities of the problem, and the equation of