l'excentricité donnee par M. Bouvard doit être augmentée, et qu'il en est de même de la longitude du perihele; deux causes qui contribuent, à cause de la position actuelle de la planète dans son orbite, à augmenter le rayon vecteur. Je ne transcris pas ici les valeurs de ees sccroissements, parceque je ne les ai pas encore avec toute la rieur precise, mais je les aurai recute avant un mois, et je me ferai un devoir, Monsieur, de vous les transmettre aussitôt, si cela vous est agreable.

Je me bornerai à ajouter que la position en quadrature, deduite en 1944 des deux oppositions qui la comprennent, au moyen de mes formalis, se difère de la position observée que de 6; ce qui prozne que l'erreur du rayon entièrement dis

** C'est même une des considérations qui derront donner plus de probabilné à la verité de mes rescitats, qu'ils rendent un compte scrupulent de toutes les circonstances da probleme. Ainsi, bien que je n'are Sait usage dans mes premières recherches que des oppostions, les quadratures n'ont pas laissé de se trouver calculées STEE toute l'exactitade possible. Le TAYOR Vecteur s'est trouvé rectifié de lui-même, sans que l'on l'eat pris en considération d'une manere directe. Excusez-moi, Monsieur, d'insister sur ce point. C'est une suite du désir que j'ai d'obtenir votre suffrage.

“Je recevrai avec bien du plaisir les observations que vous voulez bien m'annoncer. Malheureuse ment le temps presse; l'opposition approche; il fact de toute necessité que j'ar époque.

cette

it pas

comprendre ces observations dans mon travail. Mais elles me seront très-utiles pour me servir de vérifications; et c'est ce à quoi je les employerai certainement."

It is impossible, I think, to read this letter without being struck with its clearness of explanation, with the writer's extraordinary command, not only of the physical theories of perturbation, but also of the geometrical theories of the deduction of orbits from observation, and with his perception that his theory ought to explain all the phenomena, and his firm belief that it had done so. I had now no longer any doubt upon the reality and general exactness of the prediction of the planet's place.

On the 13th of July, I transmitted to Professor Challis “Suggestions for the Examination of a Portion of the Heavens in search of the external Planet which is presumed to exist and to produce disturbance in the motion of Uramus," and I accompanied them with the following letter:—

G. B. Airy to Professor Challis.

"Royal Observatory, Greenwich, July, 13, 1846.

"I have drawn up the inclosed paper, in order to give you a notion of the extent of work incidental to a sweep for the possible planet.

"I only add at present that, in my opinion, the importance of this inquiry exceeds that of any current work, which is of such a nature as not to be totally lost by delay."

On August the 7th, Professor Challis, writing to my confidential assistant Mr. Main) in my supposed absence, said—

Professor Challis to the Rev.

R. Main. [EXTRACT.]

“Cambridge Observatory, August 7, 1846.

"I have undertaken to search for the supposed new planet more distant than Uranus. Already I have made trial of two different methods of observing. In one method, recommended by Mr. Airy

* I met with a difficulty which I had anticipated. ** I adopted a second method.'

From a subsequent letter (to be cited hereafter), it appears that Professor Challis had commenced the search on July the 29th, and had actually observed the planet on August the 4th, 1846.

At Wiesbaden (which place I left on September the 7th), I received the following letter from Professor Challis :

dressed the following very important letter to Greenwich :

J. C. Adams Esq., to G. B. Airy.

"St. John's College, Cambridge, September 2, 1846.

[This letter, which contains some very important and minute calculations, is too long for insertion, but it contains the following observations :-]

"I am at present employed in discussing the errors in latitude, with the view of obtaining an approximate value of the inclination and position of the node of the new planet's orbit; but the perturbations in latitude are so very small that I am afraid the result will not have great weight. According to a rough calculation made some time since, the inclination appeared to be rather large, and the longitude of the ascend

am now treating the subject much more completely, and hope to obtain the result in a few days.

"I have been thinking of drawing up a brief account of my investigation to present to the British Association.

On the 31st of August, M. Le Verrier's second place of the disturbing planet paper on the (the third paper on, the motion of Uranus) was communicated to the French Academy. I place the notice of this paper after those of September 2, &c. because, in the usual course of transmission to this country, the No. of the Comptes Rendus containing this paper would not arrive here, at the earliest, before the third or fourth week in September; and it does not appear that any earlier notice

of its contents was received in England.

It is not my design here to give a complete analysis of this remarkable paper; but I may advert to some of its principal points. M. Le Verrier states that, considering the extreme difficulty of attempting to solve the problem in all its generality, and considering that the mean distance and the epoch of the disturbing planet were determined approximately by his former investigations, he adopted the corrections to these elements as two of the unknown quantities to be investigated. Besides these, there are the planet's mass, and two quantities from which the eccentricity and the longitude of perihelion may be inferred; making, in all, five unknown quantities depending solely on the orbit and mass of the disturbing planet. Then there are the possible corrections to the mean distance of Uranus, to its epoch of longitude, to its longitude of perihelion, and to its eccentricity; making, in sil, nine unknown quantities. To obtain these, M. Le Verrier groups all the observations into thirtythree equations. He then explains the peculiar method by which he derives the values of the unknown quantities from these equations. The elements obtained are,— Semi-ax's Major ..... $6.154

It is interesting to compare these elements with those obtained by Mr. Adams. The difference between each of these and the corresponding element obtained by Mr. Adams in his second hypothesis is, in every instance, of that kind which corresponds to the further change in the assumed mean distance recommended by Mr. Adams. The agreement with observations does not appear to be better than that obtained from Mr. Adams's elements, with the exception of Flamsteed's first observation of 1690, for which (contrary to Mr. Adams's expectation) the discordance is considerably diminished.

M. Le Verrier then enters into a most ingenious computation of the limits between which the planet must be sought. The principle is this: assuming a time of revolution, all the other unknown quantities may be varied in such a manner, that though the observations will not be so well represented as before, yet the errors of observation will be tolerable. At last, on continuing the variation of elements, one error of observation will be intolerably great. Then, by varying the elements in another way, we may at length make another error of observation intolerably great; and so on. If we compute, for all these different varieties of elements, the place of the planet for 1847, its locus will evidently be a discontinuous curve or curvilinear polygon. If we do the same thing with different periodic times, we shall get different polygons; and the extreme periodic times that can be allowed will be indicated by the polygons becoming points. These extreme periodic times are 207 and 233 years. If now we draw one grand

curve, circumscribing all the polygons, it is certain that the planet must be within that curve. In one direction, M. Le Verrier found no difficulty in assigning a limit; in the other he was obliged to restrict it, by assuming a limit to the eccentricity. Thus he found that the longitude of the planet was certainly not less than 321°, and not greater than 335° or 345°, according as we limit the eccentricity to 0.125 or 0.2. And if we adopt 0.125 as the limit, then the mass will be included between the limits 0.00007 and 0.00021; either of which exceeds that of Uranus. From this circumstance, combined with a probable hypothesis as to the density, M. Le Verrier concluded that the planet would have a visible disk, and sufficient light to make it conspicuous in ordinary tele

scopes.

M. Le Verrier then remarks, as one of the strong proofs of the correctness of the general theory, that the error of radius vector is explained as accurately as the error of longitude. And finally, he gives his opinion that the latitude of the disturbing planet must be small.

My analysis of this paper has necessarily been exceedingly imperfect, as regards the astronomical and mathematical parts of it; but I am sensible that, in regard to another part, it fails totally. I cannot attempt to convey to you the impression which was made on me by the author's undoubting confidence in the general truth of his theory, by the calmness and clearness with which he limited the field of observation, and by the firmness with which he pro

claimed to observing astronomers, "Look in the place which I have indicated, and you will see the planet well." Since Copernicus declared that, when means should be discovered for improving the vision, it would be found that Venus had phases like the moon, nothing (in my opinion) so bold, and so justifiably bold, has been uttered in astronomical prediction. It is here, if I mistake not, that we see a character far superior to that of the able, or enterprising, or industrious mathematician; it is here that we see the philosopher. The mathematical investigations will doubtless be published in detail; and they will, as mathematical studies, be highly instructive but no details published after the planet's discovery can ever have for me the charm which I have found in this abstract which preceded the discovery.

I understand that M. Le Verrier communicated his principal conclusions to the astronomers of the Berlin Observatory, on the 23rd of September, and that, guided by them, and comparing their observations with a star-map, they found the planet on the same evening. And I am warranted, by the verbal assurances of Professor Challis, in stating that, having received the paper on the 29th of September, he was SO much impressed with the sagacity and clearness of M. Le Verrier's limitations of the field of observation, that he instantly changed his plan of observing, and noted the planet, as an object having a visible disk, on the evening of the same day.