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HISTORY OF PHYSICAL ASTRONOMY.

assigning to the epoch a succession of numerical values, and, having de-termined by means of it the corresponding values of the other elements,then ascertaining which system of values agreed best with the obser-vations.

Le Vender eliminated the corrections to the elements of Uranus from theequations of 1715, 1775, 1810, and 1845; and, setting aside those of 1690and 1715, he grouped the remaining twelve into three mean equations cor-responding to the years 1758, 1793, and 1828. Each of these equationscontained the mass, the eccentricity, the longitude of the perihelion, and theepoch of the disturbing planet. Now, by assigning any particular valueto the epoch, it was easy to determine the values of the other three un-known quantities. These values might then be employed in computingthe place of Uranus corresponding to any given observation; and, by acomparison of the observed and computed places, the error of the theorydepending on the assumed value of the epoch might be ascertained. LeVender proposed, by this means, to compare his theory with the observa-tions of 1690 and 1747 for a great number of values of the epoch, withthe view of discovering whether the errors in any case were so small thatit might be fairly presumed they were due to errors of observation. Thiswas the criterion by which he resolved to test the legitimacy of histheory; and its suitableness for this purpose may be readily understood,for it is. manifest that any errors committed in the calculation of the ele-ments of the disturbing planet by means of the equations of condition,founded mainly on the modern observations, could not fail to producesensible effects when the theory was compared with the more remote ob-servations of 1690 and 1747*. In order to confine his labours wdthin asnarrow a sphere as possible, he proceeded to inquire what values of theepoch were really admissible, for it was useless to employ any values thatdid not tally with the conditions of the problem. Now, it was clear thatthose values of the epoch which assigned negative values to the mass oughtto be rejected, for all such values implied that the disturbing planet exertedon Uranus a pushing force, and not an attractive force, conformably to thetheory of gravitation. Le Verrier found by an elaborate and skilful scrutinyof the form of the algebraic expression for the mass that it had a positive signi-fication for all those values of the epoch comprised between 96° 40' and189° 55', and also for those between 263° 8' and 358° 4U; and that it wasnegative for all the remaining values comprised within the circuit of theecliptic. Again, it was evident that those values of the epoch which made themass immoderately large were inadmissible ; for, in all such cases, the planetwould produce sensible perturbations in the motion of Saturn ; but this con-clusion was at variance with observation. Rejecting all such values, Le Verrier succeeded in bringing still nearer to each other the limits he hadpreviously found. The arc which had 96° 40' and 189° 55' for its limitsnow extended only between 108° and 162°; while the arc which was

* Some of our readers may be disposed to conclude, by similar reasoning, that itwould be more advantageous to test the theory by the equations of 1712 and 1690 thanby those of 1747 and 1690. This is, no doubt, true in an absolute sense; but it mustbe borne in mind that the equation of 1715 was employed for the purpose of eliminatingthe corrections to the elements of Uranus , and was on this account invariably equal tozero. It is easy, then, to perceive, when we take into account the close proximity of theobservations of 1712 and 1715, that the equation of 1712 would, in all cases, be exceed-ingly small, and consequently it could be of little value in testing the accuracy of thetheory in any particular case.