1 68 
SCIENCE. 
telescopic search to determine whether that search was 
worth undertaking, and if so, at what point approximately 
it was best to begin. 
I. — We now consider, seriatim , the errors of the elements 
of the perturbed planet — errors which the very hypothesis 
of a disturbing body introduces, and which must have 
entered into the tables of the inferior planet, as constructed 
independently of unknown exterior perturbation. We con- 
sider what the effect of these errors may be, and how far it 
may be eliminated or subtracted from the residuals of the 
actual theory of the planet. These residuals are, of course, 
first corrected for any known error of theory or tables, or 
erroneous masses of known perturbing planets. 
(i) The error of mean distance of the perturbed planet. — Any 
error of radius vector enters very largely into the residuals 
of heliocentric longitude, if the observations are made at 
any considerable intervals from the planet’s opposition. If 
it is suspected that the error of radius vector will vitiate 
the residuals of longitude, we may avoid its effect by pass- 
ing to residuals of geocentric longitude. Or we may con- 
fine our research to the mean residuals of observations near 
the opposition points, and symmetrically placed with refer- 
ence thereto. The effect of erroneous radius vector is 
thereby eliminated. 
(2.) The error of periodic time of the perturbed planet. — If 
the residuals are examined graphically, the eye will readily 
detect whether any correction to the periodic time is advis- 
able. 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 co-efficient of which is indicated by the divergence of 
the two residual-lines. 
I had considered the problem only thus far when it oc- 
curred to me to apply the method, only partially developed, 
to the determination of an approximate position of Nep- 
tune from the residuals of Bouvard’s Tables of Uranus, 
published in 1821. Taking also the residuals from obser- 
vations up to 1824, and not permitting myself a knowledge 
of the longitude of Neptune at any epoch, a very little labor 
gave me an approximate position of the disturbing planet 
from which, it now appears, Neptune might easily have been 
found some twenty years in advance of its actual discovery. 
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 farther 
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 forth elements of the perturbed 
planet — the error of eccentricity, and the error of longitude 
of perihelion. 
(3.) "The error of eccentricity of the perturbed planet. — (See 
Sir John Herschel’s Outlines of Astronomy , page 536. 
(4.) The error of longitude of perihelion of the perturbed 
planet. — (Ibid., page 537.) 
When the longitude-residuals have been corrected in this 
manner, we proceed on the assumption that any outstanding 
residuals are due to unexplained exterior perturbation. 
II. — 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 themselves. 
(1.) The mean distance of the disturbing planet. — Regard- 
ing the next order of distance beyond Neptune as occupied 
by the planet for which we are searching, 1 assumed, as a 
first value of mean distance, 0=46.0 : this value seemed to 
be indicated by a fair induction. The periodic time of the 
planet would then be 312 years, and conjunctions with 
Uranus would occur nearly at intervals of 115 years. 
(2.) The eccentricity of the disturbing planet. — Even with 
the large residuals of Uranus employed in the investiga- 
tions of Le Verrier and Adams, the derived value of the 
eccentricity of Neptune was entirely illusory. The several 
values of eccentricity of Neptune resulting from their in- 
vestigations are as follows : 
Adams (first hypothesis) — 0.16103 
Le Verrier 0.10761 
Adams ( second hypothesis) 0.120615 
The eccentricity given by investigation of the orbit of 
Neptune from observations of the planet was : 
Newcomb ( Tables of Neptune) 0.0089903 
We should, therefore, expect nothing of any attempt to 
arrive at the eccentricity of an orbit exterior to that of Nep- 
tune. 
(3.) The longitude of perihelion of the disturbing planet . — 
Much the same remark obtains in reference to this element. 
The several values of longitude of perihelion of Neptune, 
resulting from the researches in perturbations of Uranus, 
are as follows : 
Adams ( first hypothesis) 315 0 57' 
Le Verrier 284° 45' 
Adams {second hypothesis) 299 0 11' 
The longitude of perihelion given by observations of the 
planet is : 
Newcomb ( Tables of Neptune) 46° 6' 39" .7 
Evidently it would not be wise to include this element in 
the investigation. 
(4.) The epoch of the disturbing planet. — If we can ob- 
tain even a rough approximation to die value of this ele- 
ment, the end of the investigation is fully attained. An 
inspection of the outstanding residuals, graphically exhib- 
ited, will show, without further labor, the epochs of maxi- 
mum disturbance. We may prepare an approximate pertur- 
bative curve, the epochs of maximum disturbance of which 
shall be in harmony with the assumption of mean distance 
of the exterior planet. By applying this to the plot of out- 
standing residuals, we may decide at what points the appli- 
cation of the perturbative curve best accounts for them. 
The amount of excursion in its several sinuses we need not, 
for this purpose, attend to with any great care : this will de- 
pend upon the mass and distance of the disturbing planet ; 
and, that it will be unavailing to attempt any determination 
of the mass in the present case will be evident from the fact 
that the mass of Neptune, from the theoretical investiga- 
tions of LeVerrier and Adams, was widely discrepant: 
Adams ( first hypothesis ) 0.0001656 njfa? 
Le Verrier 0.0001075 915V0 
Adams (second hypothesis') 0.00015003 
While the most reliable mass of Neptune from observation 
was : 
Newcomb {motion of the satellite') 0.00005160 75a so 
We thus have the inverse problem of perturbation re- 
duced to a very simple rational form. The residuals of 
longitude of Uranus were next treated in accordance with 
this method. 
In his Investigation of the Orbit of Uranus , Newcomb pre- 
sents three series of residuals ; the mass of Neptune finally 
adopted in the tables, Tg-hiTT, corresponds very nearly to the 
mean of the first and third series. But the mass of Nep- 
tune which was employed in this investigation is that given 
by Newcomb’s discussion of the motion of the satellite of 
Neptune, and is -j-g^-g-o-. Our first step, then, was to cor- 
rect these mean residuals into accordance with this adopted 
mass. 
Afterward, examining these corrected residuals accord- 
ing to the method just related, in reference to unexplained 
perturbing action, I concluded that Uranus was in conjunc- 
tion with an exterior perturbing body between the years 
1780 and 1795, and that another conjunction would take 
place at some time before the close of the present century. 
The most probable position of the exterior planet I there- 
fore considered to be about 170° of longitude; the pro- 
bable error of the position I considered roughly io°. This 
result was reached on the morning of the 10th of October, 
1877. During the few days immediately following I re- 
viewed this examination, as much as possible independ- 
ently of the previous result, and at the same time varying 
the assumed mean distance. With a value of a = 52.0 
(which I finally considered inductively the most probable) 
I set down the longitude of the exterior planet equal to 
162° ± 6°. This result was reached on the evening of the 
14th of October. I now turned my attention toward a 
similar treatment of the residuals of Neptune, with a slight 
hope of getting a confirmatory result. Two suppositions 
agreed in fixing the longitude at about 180 0 to 200°, respect- 
ively. I therefore, on some day in the latter part of 
