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HISTORY OF PHYSICAL ASTRONOMY.

heliocentric co-ordinates of the planet, the previous omission of which,he shewed, was alone sufficient to occasion sensible errors in the tables.

An account of the second part of Le Vender's researches appeared inthe Comptes Rendus, for the 1st of June, 1846. After having correctedthe theory of the planet, he next proceeded to compare it with the obser-vations, in order to ascertain whether the latter might be satisfied byapplying suitable corrections to the elements, or whether it was impossibleto account for them by any ellipse whatever and the perturbations ofthe known planets. He selected, for this purpose, nineteen ancient ob-servations of the planet, and 260 observations made at the Observatories ofGreenwich and Paris , during the period comprised between the discovery ofthe planet in 1781 and the month of September, 1845. In order ef-fectually to guard against every contingency of error, he undertook thelaborious task of reducing anew all these observations. Having employedthe results in computing the geocentric co-ordinates of the planet for eachobservation, he then proceeded to deduce from theory the correspondingvalues of the co-ordinates. It was necessary for this purpose to determine,in the first instance, the heliocentric co-ordinates of the planet. When theseare known, and also the place of the earth in her orbit, a simple calculation,the converse, in fact, of that to which we have already alluded*, will sufficefor passing from the heliocentric to the geocentric co-ordinates. Le Verrier therefore calculated the heliocentric longitude, latitude, and radius vectorof the planet; and then, by means of these data and the places of the earthin her orbit, derived from the solar tables, he determined the values of thegeocentric co-ordinates. Finally, subtracting the value of each co-ordinatefrom the corresponding value assigned by observation, he obtained theerrors of his theory. The question which now offered itself was this : Canthese errors be destroyed by applying suitable corrections to the ellipticelements of the planet? Now, it is easy to obtain the analytical expressionfor the error in geocentric longitude, in terms of the errors of the elementsof the planet, these errors being exhibited in indeterminate forms. Inthe present case, the only elements which it is necessary to take intoaccount are the mean distance, the epoch, the eccentricity, and thelongitude of the perihelion ; for the effects due to errors in the other twoelements are of an inferior order of magnitude, and, in consequence, maybe neglected. By putting the algebraic expression for the error of thetheory equal to each of the numerical values which he had previouslyobtained by a comparison of the observed and calculated geocentric longi-tudes, Le Verrier formed a series of equations of condition, which oughtto be satisfied if the discordances between theory and observationwere really due to errors in the elements. Every observation gavehim a corresponding equation of condition; so that the total numberof equations thus formed by him amounted to two hundred andseventy-nine. Setting aside the equations of condition depending onthe ancient observations, he formed four final equations, by a suit-able combination of all the others, and from them deduced the cor-rections to the four elements. The motion of the planet, whencalculated for a great number of places by means of the new ellipse,resulting from the application of these corrections to the tabular elements,still exhibited discordances with the actual motion, which were generallyby far too great to be accounted for by any probable errors of observation.

* See page 171.