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 ma}' 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 tin- 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 

 t would then be 312 years, and conjunctions with 

 Uranus would o< < ur nearly at intervals of 115 years. 



(2.) The eccentticily of the disturbing planet. — Even with 

 the large residuals of 1'ianus employed in the investi] 

 tions of I.e Verriei and .Adams, the derived value of the 

 eccentricity of Neptune was entirely illusory. The several 

 values of • of Neptune resulting from their in- 



vestigations aie as follows : 



Adams {first hypothesis) 0.16103 



Le Vt rrier 0.10761 



Adams (second hypothesis) 0.120615 



The eccentricity given by investigation of the orbit of 

 Neptune from observations of the planet was : 



New comb (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° 57' 



Le Verrier 284 45' 



Adams (second hypothesis) _ 299 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 the 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 ejrVy 



Le Verrier 0.0001075 ttAto 



Adams (second hypothesis) 0.00015003 Krfirir 

 While the most reliable mass of Neptune from observation 

 was : 



Newcomb (motion of the satellite) 0.00005160 79^0 

 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 scries of residuals ; the mass of Neptune finally 

 adopted in the tables, tF7"tttt, 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 nritfff. Our nrst 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- 

 liable 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- 

 \ iewed 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. 

 (which I finally considered inductively the most probable) 

 I set down the longitude of the exterior planet equal to 



162' i 6°. This result was reached on the evening of the 

 14th ol 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 iSo toaoo", respect- 

 ively. 1 therefore, on some day in the latter pail of 



