CALCULATIONS OF THE DISCOVERERS. 



doing, and believing that he stood alone in his adventurous and, 

 as would then have appeared, hopeless attempt. Nevertheless, 

 both not only solved the problem, but did so with a completeness 

 that filled the world with astonishment and admiration, in which 

 none more ardently shared than those who, from their own 

 attainments, were best qualified to appreciate the difficulties of 

 the question. 



The question, as has been observed, belonged to the class of 

 indeterminate problems. An infinite number of different planets 

 might be assigned which would be equally capable of producing 

 the observed disturbances. The solution, therefore, might be 

 theoretically correct, but practically unsuccessful. To strip the 

 question as far as possible of this character, certain conditions 

 were assumed, the existence of which might be regarded as in the 

 highest degree probable. Thus it was assumed that the disturb- 

 ing planet's orbit was in or nearly in the plane of that of Uranus, 

 and therefore in that of the ecliptic ; that its motion in this orbit 

 was in the same direction as that of all the other planets of the 

 system, that is, according to the order of the signs ; that the orbit 

 was an ellipse of very small eccentricity ; and, in fine, that its 

 mean distance from the sun was, in accordance with the general 

 progression of distances noticed by Bode, nearly double the 

 mean distance of Uranus. This last condition, combined with 

 the harmonic law, gave the inquirer the advantage of the 

 knowledge of the period, and therefore of the mean heliocentric 

 motion. 



Assuming all these conditions as provisional data, the problem 

 was reduced to the determination, at least as a first approxima- 

 tion, of the mass of the planet and its place in its orbit at a given 

 epoch, such as would be capable of producing the observed 

 alternate acceleration and retardation of Uranus. 



The determination of the heliocentric * place of the planet at a 

 given epoch would have been materially facilitated if the exact 

 time at which the amount of the advance (L I/) of the observed 

 upon the tabular place of the planet had attained its maximum 

 were known ; but this, unfortunately, did not admit of being 

 ascertained with the necessary precision. When a varying 

 quantity attains its maximum state, and, after increasing, begins 

 to diminish, it is stationary for a short interval ; and it is always 

 a matter of difficulty, and often of much uncertainty, to determine 

 the exact moment at which the increase ceases and the decrease 

 commences. Although, therefore, the heliocentric place of the 



* That is, the place in which, the planet would appear to be, if the 

 observer were at the centre of the system, where the sun is. 



N 2 179 



