72 THEORBITOFNEPTUNE. 



Residuals of equations. ^"'"^^ ""'^" resicluals or ap- 



'■ parent errors oi theory. 



/-I-2.8 — 1.2fi 3 + o'q.S — 0."40^ 



8th series, )— 2.4 —O.dfx 7 —0.34 —Q.Ufi 



1863-1864,-0.3 +0.7^ 7 —0.04 +0.10^ 



\_2.0 +0.9^ 6 —0.33 +0.15^ 



§ 32. The coefficients of fi, taken negatively, represent the changes which would 

 be produced in the residuals if we suppose the mass of Uranus to be nothing. It 

 will be seen that these coefficients are generally smaller than the residuals them- 

 selves, and that their actual effect on the modern residuals never amounts to 

 ' more than four-tenths of a second. Supposing that the modern observations 

 cannot be relied on within this limit of error, we should arrive at this remarkable 

 result, — that if the planet Uranus were unknown, its existence could scarcely be 

 inferred from all the observations hitherto made on Neptune, unless these were 

 combined in such a way as to show the systematic error of the theoretical radius 

 vector. In fact, the orbit of Neptune, computed without regard to the perturb- 

 ations of Uranus, would only exhibit an error of 9" when compared with Lalande's 

 position ; and a discussion of the modern observations would exhibit no sensible 

 error in the heliocentric longitudes. Jhis circumstance furnishes a very good 

 illustration of the propriety of developing the long-period perturbations, the co- 

 efficients of which amount to whole minutes, as perturbations of the elements 

 which shall vanish at the epoch 1850. 



Under these circumstances, no reliable correction of the mass of Uranus can be 

 concluded from the motions of Neptune. The solution of the preceding residuals 

 does, indeed, indicate an increase of this mass by one-third, which seems altogether 

 inadmissible, and is certainly very unreliable. Of the twenty-nine residuals, 

 fifteen indicate an increase of the -mass, thirteen a diminution, and for one the 

 coefficient of fi vanishes : so that the increase of the mass of Uranus is indicated 

 only by the fact that the residuals which favor it are generally a little larger than 

 those which do not. 



§ 33. If Uranus could scarcely be detected from the motions of Neptune, much 

 less can an extra-Neptunian planet, unless it happened to be nearly in conjunction 

 with Neptune at the present time, and to have a much greater mass than Uranus, 

 — a highly improbable combination of circumstances. That there is no present 

 indication of any such action is shown by the smallness of the apparent mean 

 errors of theory in heliocentric longitude and radius vector during the whole 

 period from 1846 to 1864. The following table shows the mean value of these 

 errors during each of the seven series of modern observations, and the error of 

 the geocentric longitude of the Lalande observations, putting ^=zO. The error 

 of radius vector is expressed as error of annual parallax. It will be remembered 

 that the first of the four equations of each series arise from observations made 

 about half-way between the first quadrature and the opposition, the second at 

 opposition, the third between opposition and last quadrature, and the fourth near 

 the last quadrature. Each series, therefore, gives four equations of the first 

 degree between the errors of heliocentric longitude ^>:, and annual parallax S^). 



