170

HISTORY OF PHYSICAL ASTRONOMY.

condition is formed by putting this expression equal to the actual error asdetermined by a comparison of the observed and computed places of theplanet. Every tabular error furnishes a corresponding equation, and thusthe number of equations of condition is limited only by the number of in-dependent observations. Now, if the latter were mathematically accurate,the equations of condition would all be consistent with each other, so thatany nine of them would suffice for the determination of the nine unknownquantities of the problem, and the values hence deduced would rigorouslysatisfy all the other equations. But, as all observations are necessarilymore or less erroneous, the errors resulting from a comparison of the ob-served and calculated places of the planet will not be wholly dependenton the unknown quantities of the problem; and the equations of conditionbeing vitiated in consequence, will prove to a certain extent incom-patible with each other, so that every nine equations selected for the de-termination of the unknown quantities will conduct to different results.With a view to elude this source of error, the number of equations ofcondition is generally made to exceed in a great degree the number ofunknown quantities, and the grand object, then, is to combine them to-gether, so that the errors of observation may destroy each other, and asystem of equations may finally be arrived at which will assign the truevalues of the unknown quantities.

Mr. Adams first proceeded to examine the perturbations producedin the motion of Uranus by the other planets, in order to assure himselfbeyond doubt that the errors of Bouvard’s tables did not proceed froman erroneous application of the existing theory*. For this purpose herecomputed the principal perturbations due to Jupiter and Saturn,and introduced some new inequalities which had been first pointedout by Hansenf. He also took into account the correction to Jupiter ’smass, to which recent researches had conducted Astronomers. Not-withstanding these improvements, the theory still failed to representthe motion of the planet. Two important advantages were, however,gained by these preliminary labours. In the first place, it was clearlyestablished that the cause of the irregularities must be sought elsewherethan in the development of the actual theory. In the second place, theapplication of the improvements had the effect of exhibiting the errors ofthe tables as residual facts wholly dependent on some extraneous influence,and consequently they now assumed a more precise and definite characterthan they had previously done. The first point to be decided in thisinquiry was, the most suitable mode of exhibiting the irregularities inthe motion of the planet. We have mentioned already, that both theheliocentric longitude and radius vector had been discovered to besensibly in error. Now it may be remarked, that the problem admitsof solution by employing as the basis of investigation an adequate numberof the errors of either of these co-ordinates, independently of those of theother. In point of fact, however, the errors of the radius vector weretoo inconsiderable to be employed with safety in such an inquiry, althoughthey might be subsequently used with advantage in verifying resultsotherwise derived. Mr. Adams, therefore, had recourse to the errors ofheliocentric longitude, which in some instances amounted to 2' or 3', andextended over a period embracing more than a revolution and a half of

* Mem. Ast. Soc., vol. xvi.; Naut. Aim., 1851.

t These inequalities are of the order of the squares of the disturbing forces, and are byfar the most considerable of the class to which they belong.