HISTOBY OF PHYSICAL ASTBONOMY.

187

the case which we are actually considering, that the equations of con-dition involving the errors of longitude may assign to the elements of thedisturbing planet, and to the corrections of the elements of Uranus , valueswhich shall account for these errors with sufficient accuracy, hut still maydiffer widely from the true values. Now, if any such system of erroneousvalues was employed in computing the errors of radius vector, it manifestlydoes not follow as a necessary consequence that, because the motion inlongitude had been satisfied by a mutual compensation of errors, the resultsin this case also would accord equally well with observation. We aretherefore led to conclude that, even if the existence of an exterior planethad been already placed beyond all doubt, the explanation of the errors ofradius vector would prove extremely valuable in testing the accuracy ofresults derived from the errors of longitude. It may be remarked, however, that the Astronomer Royal, in his letter to M. Le Verrier, expressedin very explicit terms his doubts respecting the accuracy of the final resultsof that geometer, without indeed going so far as to reject in toto the hy-pothesis of a disturbing planet. In his letter to Mr. Adams, on thecontrary, he does not suggest any such doubts, but simply inquires whetherthe errors of radius vector were accounted for by his theory as faithfullyas those of longitude. We may suppose, then, that Mr. Adams, who hadfor many years been strongly impressed with the existence of a Trans-Uranian planet, and who had already been conducted to such satisfactoryresults by his researches on the motion in longitude, may not have dulyappreciated the importance attached by the Astronomer Royal to theexplanation of the errors of radius vector. It is to be regretted, however,that he was so tardy in replying to Mr. Airy, especially as he could nothave experienced any analytical difficulty in complying at once with hisrequest *.

An account of the third part of Le Verrier’s labours on the theory ofUranus appeared in the Comptes Rendus for the 31st August, 1846. Themain object of the second part of his researches, as announced in theComptes Rendus for the 1st June, was to obtain an approximate value ofthe epoch or mean longitude of the hypothetic planet at a given instant.When this was once accomplished, the true value might be investigated byapplying to the approximate value an indeterminate correction, and thendeducing it from the conditions of the problem simultaneously with theother unknown quantities, There was this advantage gained by a firstapproximation to the epoch, that, as the correction to the true value mighthe presumed to he small, it was possible so to conduct the investigationthat it would not be necessary to take into account any terms beyond thoseinvolving the first and second powers of the correction. For a similarreason a more accurate value of the mean distance might be obtained bysupposing the approximate mean distance to be affected with an indeter-

* We have maintained that it does not necessarily follow, because the errors of longi-tude are satisfied, that the errors of radius vector are satisfied also. In the present in-stance, however, it happens that such is really the case. Mr. Adams, in his Memoir onthe Perturbations of Uranus , has given an expression for the correction to the radius vec-tor involving the correction to the mean longitude and its differential, together with theeight unknown quantities of the problem; and he has shown that by far the most consider-able part of the expression is due to the term involving the differential of the correction tothe mean longitude. Hence it manifestly follows that, if the tabular errors of longitude besatisfied, the errors of radius vector will be satisfied also. This result, however, dependsupon the particular values obtained for the unknown quantities, and could not be pre-dicated by any a priori reasoning.