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HIST0BY OF PHYSICAL ASTBONOMY.
minate error, and then treating the latter as one of the unknown quantitiesof the problem. There were, therefore, five quantities to be determinedrelative to the disturbing planet; namely, the corrections to the mean dis-tance and the epoch, and the absolute values of the mass, the eccentricity,and the longitude of the perihelion. These five quantities, together withthe corrections to the mean distance, the epoch, the eccentricity, and thelongitude of the perihelion of Uranus , formed nine unknown quantities,upon the determination of which the solution of the problem depended.
In the investigation of an approximate value of the epoch, Le Verrier employed as the basis of his reasoning a select number of errors of helio-centric longitude. These were obtained by a comparison of the theorywith observations made when the planet was in opposition. It would havebeen impossible to deduce accurately the errors of heliocentric longitudefrom all the observations, because many of the latter were made when theplanet was near the quadratures, and in that case the process of passingfrom the geocentric to the heliocentric longitude could not be rigorouslyeffected without a knowledge of the error of radius vector ; hut this wasaltogether uncertain. He now had recourse to the errors in geocentric lon-gitude, which could be readily computed without a knowledge of the errorsof either of the heliocentric co-ordinates, and were, therefore, deducible withequal accuracy from all the observations. In the first part of his researcheshe had carefully computed the errors in geocentric longitude correspond-ing to two hundred and seventy-nine observations of the planet. Now theanalytical expression for these errors contained the nine unknown quan-tities of the problem. Putting it, therefore, equal to each numerical errorin succession, Le Verrier formed two hundred and seventy-nine equationsof condition, and by means of these he proposed to obtain the solution ofthe problem. With this view he grouped them into thirty-three meanequations, twenty-six of which depended on the modern observations ofUranus , and the remaining seven on those made previous to its recogni-tion as a planet in 1781. He eliminated without difficulty six of the un-known quantities in terms of the three others, these last being the massof the disturbing planet, and the corrections to the epoch and the meandistance. Pursuing a process which it would be out of place to attemptexplaining here, he formed three final equations, involving these threequantities, and then determined their values by successive approximation.This object being once accomplished, it was easy for him to obtain thevalues of the other six quantities which he had first eliminated. Thefollowing are the results relative to the disturbing planet at which hefinally arrived:—
Semi-axis of the orbit ....
36.154
Sidereal revolution .....
217.387 years
Eccentricity ......
0.10761
284° 45'
Mean longitude, 1st January, 1847
318" 47'
Mass .......
Were
True heliocentric longitude, 1st Jan., 1847
326" 32'
Distance from the sun ....
33.06