D. P. Todd—Search for a Trans-neptunian Planet. 227 
the investigation are not sufficiently well marked to justify the 
execution of so laborious a research, especially if it be found 
that a simple, rational treatment, unencumbered with the re- 
finements of analysis, may be fairly interpreted as indicating 
the position of an exterior perturbing body with merely a 
rough approximation. 
(2) Even in the case of Uranus, and the theoretic search for 
Neptune, where the residuals of longitude were very strongly 
marked, many of the elements pertaining to the disturbing 
planet which Adams and LeVerrier sought to determine theo- 
retically, turned out afterward, when their real values became 
known, to have been indicated with only meagre precision. 
Much less should we now expect these elements to be given 
with any certainty in the case of a planet exterior to Neptune. 
I was also much impressed by a remark in Sir George Airy’s 
Account of some Circumstances historically connected with the 
Discovery of the Planet exterior to Uranus—“T have always 
considered the correctness of a distant mathematical result to 
e a subject rather of moral than of mathematical evidence.”* 
This provisional treatment of the residuals of Uranus was 
undertaken, then, as a preliminary to the proposed telescopic 
search, to determine whether that search was worth undertak- 
ing; and, if so, at what point, approximately, it was best to 
begin. I should remark, also, that this portion of the work, 
as an investigation to these ends, was never undertaken with 
reference to publication. 
—Let us now consider, seriatim, the errors of the elements 
1) 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 con- 
siderable interval from the planet’s opposition. If it 1s sus- 
pected that the error of radius vector will vitiate the residuals 
of longitude, we may avoid its effect by passing to residuals of 
geocentric longitude. Or, we may confine our research to the 
mean residuals of observations near the opposition-points, and 
symmetrically placed with reference thereto. e effect of 
erroneous radius vector is thereby eliminated. 
* Memoirs Royal Astronomical Society, vol. xvi, p. 398. 
