Biography of Percival Lowell

by A. Lawrence Lowell

ῶ—The place of its perihelion, that is the position of its nearest approach to the Sun.

(These last two determine the shape of the ellipse, and the direction of its longer axis with respect to that of the other planet.)

m—Its mass.

Now formulas, or series of equations, that express the perturbations caused by one planet in the orbit of another must contain all these elements, because all of them affect the result. But there are too many of them for a direct solution. Therefore Leverrier assumed a distance of the unknown planet from the Sun, and with it the mean motion which is proportional to that distance; worked out from the residuals of Uranus at various dates a series of equations in terms of the place of the unknown in its orbit; and then found what place therein at a given time would give results reducing the residuals to a minimum—that is, would come nearest to accounting for them. In fact, supposing that the unknown planet would be about the distance from the Sun indicated by Bode’s law, the limits within which he assumed trial distances were narrow, and, as it proved, wholly beyond the place where it was found. This method, which in its general outline Percival followed, consisted therefore of a process of trial and error for the distance (with the mean motion) and for the place of X in its orbit (ε). For the other three elements (e, ῶ and m) he used in the various solutions 24 to 37 equations drawn from the residuals of Uranus at different dates, and expressed in terms of ε. He did this in order to have several corroborative calculations, and to discover which of them accorded most closely with the perturbations observed.

We have seen that in 1908-09 Percival was inquiring about the exact residuals of Uranus, and he must have been at work on them soon afterwards, for on December 1, 1910, he writes to Mr. Lampland that Miss Williams, his head computer, and he have been puzzling away over that trans-Neptunian planet, have constructed the curve of perturbations, but find some strange things, looking as if Leverrier’s later theory of Uranus were not exact. This work had been done by Leverrier’s methods “but with extensions in the number and character of the terms calculated in the perturbation in order to render it more complete.” Though uncertain of his results, he asks Mr. Lampland, in April 1911, to look for the planet. But he was by no means himself convinced that his data were accurate, and he computed all over again with the residuals given by Gaillot, which he considered more accurate than Leverrier’s in regard to the masses, and therefore the attractions, of the known planets concerned. Incidentally he remarks at this point in his Memoir,[45] in speaking of works on celestial mechanics, that “after excellent analytical solutions, values of the quantities involved are introduced on the basis apparently of the respect due to age. Nautical Almanacs abet the practice by never publishing, consciously, contemporary values of astronomic constants; thus avoiding committal to doubtful results by the simple expedient of not printing anything not known to be wrong.” His result for X, as he called the planet he was seeking, computed by Gaillot’s residuals, differed from that found in using Leverrier’s figures by some forty degrees to the East, and on July 8 he telegraphs Mr. Lampland to look there.

These telegrams to Mr. Lampland continue at short intervals for a long time with constant revisions and extensions in the calculations; and, as he notes, every new move takes weeks in the doing; but all without finding planet X. Perhaps it was this disappointment that led him to make the even more gigantic calculation printed in the Memoir, where he says: “In the present case, it seemed advisable to pursue the subject in a different way, longer and more laborious than these earlier methods, but also more certain and exact: that by a true least-square method throughout. When this was done, a result substantially differing from the preliminary one was the outcome. It both shifted the minimum and bettered the solution. In consequence, the whole work was done de novo in this more rigorous way, with results which proved its value.”

Then follow many pages of transformations which, as the guide books say of mountain climbing, no one should undertake unless he is sure of his feet and has a perfectly steady head. But anyone can see that, even in the same plane, the aggregate attractions of one planet on another, pulling eventually from all possible relative positions in their respective elliptical orbits with a force inversely as the square of the ever-changing distance, must form a highly complex problem. Nor, when for one of them the distance, velocity, mass, position and shape of orbit are wholly unknown, so that all these things must be represented by symbols, will anyone be surprised if the relations of the two bodies are expressed by lines of these, following one another by regiments over the pages. In fact the Memoir is printed for those who are thoroughly familiar with this kind of solitaire.

For the first trial and error Percival assumed the distance of X from the Sun to be 47.5 planetary units (the distance of the Earth from the Sun being the unit), as that seemed on analogy a probable, though by no means a certain, distance. With this as a basis, and with the actual observations of Uranus brought to the nearest accuracy by the method of least-squares of errors, he finds the eccentricity, the place of the perihelion and the mass of X in terms of its position in its orbit. Then he computes the results for about every ten degrees all the way round the orbit, and finds two positions, almost opposite, near 0° and near 180°, which reduce the residuals to a minimum—that is which most nearly account for the perturbations. Each of these thirty tried positions involved a vast amount of computation, but more still was to come.

Finally, to be sure that he had covered the ground and left no loophole for X to escape, he tried, beside the 47.5 he had already used, a series of other possible distances from the Sun,—40.5, 42.5, 45, 51.25 units,—each of them requiring every computation to be done over again. But the result was satisfactory, for it showed that the residuals were most nearly accounted for by a distance not far from 45 units (or a little less if the planet was at the opposite side of its orbit), and that the residuals increased for a distance greater or less than this. But still he was not satisfied, and for greater security he took up terms of the second and third order—very difficult to deal with—but found that they made no substantial difference in the result.

So much for the longitude of X (that is its orbit and position in the plane of the ecliptic) but that was not all, for its orbit might not lie in that plane but might be inclined to it, and like all the other planets he supposed it more or less so—more he surmised. Although he made some calculations on the subject he did not feel that any result obtained would be reliable, and if the longitude were near enough he thought the planet could be found. He says: