228 DD. P. Todd—Search for a Trans-neptunian Planet. 
(2) The error of periodic time of the perturbed planet.—f 
the residuals are examined graphically, the eye will readily 
detect whether any correction to the aes time is advisable. 
If, in general, the mean line of the residuals is nearly a right 
line, and makes a given angle with the line of zero-residual, it 
may fairly be concluded that the residuals need a correction 
depending directly on the time, the magnitude of the coéfficient 
f which is indicated by the divergence of the two residual- 
lin 
TI had considered the problem only thus far when it occurred 
to me to apply the method, only partially developed, to the 
determination of an 2a position of Neptune from the 
residuals of Bouvard’s f Uranus, published in 1821. 
Taking also the aideati: pare observations up to 1824, and 
not permitting myself a knowledge of the longitude of Nep- 
tune at any epoch, a very little labor gave me an approximate 
position of the distarbing planet from which, it now appears, 
Neptune might easily have been found some twenty years in 
advance of its actual discover 
When my work had advanced to this stage, a mere chance 
threw in my way a copy of Sir John Herschel’s Outlines of 
Astronomy, (which I had never before examined): I at once 
observed that my treatment of the residuals of Uranus with 
reference to a planet exterior to Neptune was quite similar to 
his “dynamical” exposition of the perturbations of Uranus 
arising from Neptune itself. And I was further gratified to 
' find that he had given a very full and lucid statement of the 
effect upon the longitude-residuals caused by errors of the 
third and fourth elements of the perturbed planet—the error of 
eccentricity, and the error of longitude of perihelion. I there- 
fore adopted, without hesitation, the continuance of the graph- 
ical method therein detailed ; and shall do no more here than 
to refer to the pages o Herschel’s treatise where these ele- 
ments are = with. 
3) or of eccentricity of the perturbed planet.—(See Sir 
7 Herichal's Outlines of Astronomy, page 536. 
id, The he of longitude of perthelion of the perturbed planet.— 
peice page 537.) 
hen the longitude-residuals have been corrected in this 
manner, we proceed on the assumption that any outstanding 
residuals are due to unexplained exterior perturbation. 
IL.—Of the seven elements of the disturbing planet, we must 
assume a value of one: the values of three others, together 
with the mass of the disturbing planet, we may consider as 
theoretically determinable from the longitude-residuals them- 
selves 
(1) ‘The mean distance of the disturbing planet.—Regarding 
