

Astronomy. 445 



My method assigns for the limits of this element 3 10° -5 and 335°-6, 

 always taking but 5" for the uncertainty of the modern data. 



The number of Mr. Walker, and my own, are within these limits. 



V. Eccentricity. — I have given 0*1076 as the value of this element, 

 whilst that of Mr. Walker is only 0-0088. 



The discordance may appear considerable to those who know not 

 how ill determined are the eccentricity and major axis of an ellipse 

 from even a considerable portion of the curve known, as will be seen 

 in the two ellipses of gamma Virginis. But all embarrassment van- 

 ishes when we take the trouble to calculate the limits of uncertainty 

 of the eccentricity, corresponding to the limits of uncertainty of the 

 data. 



It is actually found, in admitting only 5" of uncertainty, in the mod- 

 ern data, that the eccentricity of the body causing the irregularities of 

 Uranus may be arbitrarily chosen between 0-2031 and 0*0592; limits 

 already much extended, and which explain why the eccentricity of 

 j Neptune was determined so poorly, by its action on Uranus. But 



J more, we have only to carry the uncertainty of the modern data to 7" 



J or 8" as may properly be done, to see that the inferior limit of eccen- 



] tricity falls nearly to zero, and the superior limit is raised to 0*25, or 



even higher. Thus the eccentricities given by my theory and by the 

 orbit of Mr. Walker come completely within those which are author- 

 ized by the uncertainty of the modern data. 



The eccentricity, I repeat, is but a mathematical auxiliary ; and it 

 is not by such auxiliaries that the degree of precision of a theory must 

 be judged. They may vary very much, as we see, without the final 

 result, which is here the longitude, experiencing any remarkable varia- 

 tions. I have often had occasion to show the Academy, how changes, 

 almost insignificant, in the positions (deduced from direct observations) 

 which have served to determine the elements of a comet or planet, 

 have produced considerable changes in the values of the elements. 



By filly choosing the auxiliaries, and comparing them without know- 

 ing their signification, we could, in appearance, spoil any theory what- 

 ever. What an enormous discordance there seems, at first sight, be- 





tween two theories in which the tangents of the same angle are repre- 

 sented by the numbers 206265 and 344. And yet on referring to the 

 angles themselves, that is to say, to the direct object of the question, 

 it will be found that they differ but ten minutes. 



VI. Semi-axis major. — The semi-axis major gives occasion for the 

 same remarks as the eccentricity. There has been thought to be an 

 insurmountable difficulty, in what I have said, (page 240 of the memoir 



of 1846,) viz., that the semi-axis major was contained between the 

 limits 37-90 and 35-04, while Mr. Walker makes it 80-20, 



I do not believe, to tell the truth, that the direct observations of Nep- 

 tune, which we have at the two ends of an arc one hundred degrees in 

 length, are sufficient for a good determination of the semi-axis major 

 of the orbit. But I have said that I will not contest this point. 



The solution of the pretended difficulty lies wholly in the fact, that 

 the limits 37'90 and 35*04 are not absolute limits, but limits relative to 

 the arbitrary hypothesis that the modern data do not allow of a greater 

 uncertainty than 5". It will not be forgotten that I explained on the 



