B.C. 721, March 19, as related by Ptolemy, confirms the acceleration, as, for this eclipse, the true place of the moon precedes the compared place by a quantity equivalent to fifty minutes of time. From a careful discussion of these eclipses, Mr. Dunthorne thinks that the coefficient of the acceleration is not greatly different from 10".*

The paper of Mr. Dunthorne here quoted is, at the present time, valuable only as showing that the lunar tables, even so long ago as the middle of the preceding century, were sufficiently precise to put in evidence the fact of the acceleration, and to enable its amount to be assigned with tolerable accuracy.

Recently Mr. Airy has had occasion to discuss three of the most remarkable eclipses of antiquity, namely, the eclipse of Agathocles, the eclipse of Larissa, and the eclipse of Thales,† with the advantage of the use of very much better solar and lunar tables than any which were previously employed, namely, the solar tables of Hansen and Olufsen, and the lunar tables of Hansen. This paper is so well known that any remarks upon it are at present superfluous, and we are only concerned with the results as bearing on the amount of the lunar acceleration.

These results the Astronomer Royal states as follows:— 1. The eclipse at Larissa, 556, May 19, is established as a real eclipse at a well-defined point, and may be adopted for critical reference in deciding on the value of lunar tables, as applicable to distant places of the moon.

2. Professor Hansen's Tables very well represent the phenomena of the three eclipses of Agathocles, Larissa, and Thales, &c.

3. If any change is permitted in the two elements, of secular acceleration of longitude, and change of the argument of latitude, it must be of the nature of increasing the acceleration, and increasing the argument of latitude in the distant ages.

In an addendum to this paper it is announced that the eclipse of Stiklastad confirms the preceding result, namely, that if any correction is made to the coefficient of the acceleration, it must be by way of increase, and not of decrease.

It may, then, be taken for granted that the coefficient of the acceleration must be at least as great as that used by Professor Hansen in his Tables, namely, 12" 12.

We now come to the consideration of the controversy

* In addition to this paper by Dunthorne on the existence of the secular acceleration of the moon's mean motion, as exhibited by ancient eclipses, a very interesting paper by La Place, printed in the Connaissance des Tems for 1800,"On the Secular Equations of the Motions of the Apogee and Nodes of the Lunar Orbit," may be consulted. In this paper La Place announces his discovery of the secular equation of the lunar apogee, and gives the results of comparison with twenty-seven ancient eclipses, compared with the tables, at his request, by M. Bouvard.

"On the Eclipse of Agathocles, the Eclipse at Larissa, and the Eclipse of Thales, &c." By G. B. Airy, Esq., Astronomer Royal. (Mem. Ast. Soc., vol. xxvi.)

respecting the amount of this coefficient, as derived from theory, a controversy vitally interesting and important; both as regards the great names that are ranged on different sides, that is, in support of the commonly received coefficient on the one side, and of that recently insisted on by Mr. Adams on the other; and as regards the accuracy of those analytical processes employed in the different investigations which have given rise to the present dispute. In support of the old coefficient are the great names of Plana and Pontécoulant, who deduce a coefficient very nearly equal to that required to satisfy the ancient eclipses; while Professor Hansen, by using an essentially different method, arrives at nearly the same result. On the other side are Adams and Delaunay, who, pursuing processes totally different, and carrying their expansions to a degree never before attempted, arrive at a result smaller by one-half than that commonly received and contended for by their opponents. That which renders the present controversy still more interesting is the fact that the astronomers named above are, with very few exceptions, the only savans in the world at the present time who have such complete mastery of the complicated and difficult analysis necessary for the solution of the problem as to enable them to judge of the merits or defects of the various processes employed, which, from the discordant results, cannot all be

correct.

In endeavouring to lay before the Society the present state of the question, it is necessary to refer to two former short articles on this subject, written by me, and contained in the sixteenth volume of the Monthly Notices of this Society, and a very brief recapitulation of their contents may be useful. M. Plana had been induced by Mr. Adams's paper, published in the Phil. Trans. for 1853,* to review his theory of the secular acceleration given in the first volume of the Théorie du Mouvement de la Lune, and imagined that he had completely reconciled his own value of the coefficient with that given by Mr. Adams, by the detection of an error at pages 60 and 61 of the Théorie. He investigates at great length the terms which had been omitted, and arrives at a result identical with that of Mr. Adams, as far as the latter had carried his development. He also intimated his intention of carrying the expansion to terms of the seventh order. This memoir is dated April 16, 1856. In June 1856, however, M. Plana published another memoir, Sur l'Equation Séculaire du moyen Mouvement de la Lune,t in which he gives the additions to be made to his formula for the acceleration given in the first volume of the Théorie on account of the error before adverted to, but in which he seems to deny, what he had before admitted, the

*On the Secular Variation of the Moon's Mean Motion."

J. C. Adams, Esq., M.A., F.R.A.S.

† Both papers of Plana are extracted from the Turin Memoirs.

By

correctness of the reasonings and the results of Mr. Adams. This whole paper is so obscure, and the references to his own former researches and to the results of Mr. Adams are so indistinct, that nothing can be gained from it as to the probable correctness of his own decision; and all that is learnt from the paper is, that he assumes that Mr. Adams has taken into his computations some terms which do not properly belong to the secular equation of acceleration. It was imagined by astronomers at the time of the writing of this memoir, that is, in the year 1856, that M. Plana would attempt to confirm his views by an elaborate discussion of the whole subject; for, from some hints which occur in his previous memoirs, it was plainly his intention to carry the approximation to terms of the seventh order. Nothing, however, has since appeared from his pen (at least I am not aware of any publication) bearing on this subject; and it only remains to show what has been the nature of the researches and the results of his opponents, Adams and Delaunay.

The paper in which Mr. Adams first announced his discovery that the usually received coefficient of the acceleration was incorrect, has been before quoted; and it is due to its author to state, in proof of the clearness of his views, that he indicates in a geometrical and very simple form the mistake which he assumes to exist in the ordinary theory. This theory, though it has been much developed by Plana, is that of La Place without essential alteration;* and therefore, whatever objection is fatal to La Place's theory, is fatal to the correctness of the deductions of that astronomer, to whatever degree of development they may be carried.

For the sake of such members of the Society as may not be familiar with the cause of the acceleration of the moon's mean motion, it may be worth while to state the leading features of La Place's theory.

The time of revolution of the moon round the earth depends upon the amount of her gravitation towards the earth; and, if her orbit were not disturbed by the sun and planets, it would be absolutely constant. The sun's disturbing force, however, on the whole, during a revolution, diminishes the central or radial force, and her orbit is, therefore, larger and her period greater than it would be if she were undisturbed. The period will, therefore, depend upon the sun's distance from the earth, and will be different for the different months of the year, but will have a mean value which will be pretty nearly constant for each year. On account, however, of the decrease of the excentricity of the earth's orbit, the sun's average distance in any year is slightly greater than it was in the preceding, and his disturbing power diminishing the moon's gravity towards the earth is less, and therefore the moon's

* See Adams's Paper on the Secular Acceleration, referred to above, Phil. Trans. for 1853, p. 398.

gravitation towards the earth is greater, and her motion becomes accelerated.

La Place, with his usual sagacity and power of analysis, deduced the effect of this secular disturbance in the mean motion; but he concluded, Mr. Adams thinks erroneously,* that the radial force only is concerned in producing this effect, and that the equal description of areas would not be affected by it, the tangential force producing no effect. It is a curious fact that in the infancy of the lunar theory a similar mistake was made with respect to the motion of the moon's apogee. It was assumed correctly that only the radial disturbing force was concerned directly in producing the motion of the apse, but it was not taken into account that the tangential force, by diminishing or increasing the velocity, caused the position of the moon to be at a given time different from what it would otherwise have been, and therefore indirectly altered the radial force, and tended to produce a motion of the apse: and it is remarkable that this indirect effect of the tangential disturbing force was in this case of the motion of the apogee equal to the direct effect of the radial disturbing force; and that Clairaut by this discovery relieved men's minds from one of those periodical panics, which, ever since the publication of the Principia, have at intervals seized the minds of astronomers with regard to the absolute truth of the theory of gravitation.

It is then with this radical defect in omitting the effect of the action of the tangential part of the secular disturbing force arising from the diminution of the earth's excentricity, that Mr. Adams charges the theory of La Place, and therefore the processes of Plana and Pontécoulant, and he proceeds in a very simple and satisfactory manner to show how he introduces into his analysis the additional terms which will thence arise.

By a correspondence with the Astronomer Royal, to which I have had access, it appears that ever since the publication of his paper he has at intervals been steadily employed in developing his results to terms of higher order, and he has recently published the result of his researches by giving the value of his coefficient of acceleration correctly, as he believes, to terms of the seventh order. This result was first announced in a letter to M. Delaunay, which is printed in the Comptes Rendus (vol. xlviii. p. 247).

He there gives for his expression, from which the coefficient of acceleration is deduced:

* It was evidently the intention of La Place to carry this development of the function expressing the acceleration only to the second power of m, within which limit the equality of the areal velocity is not affected; and, therefore, it cannot be assumed that any mistake was committed by him in his investigation; but the same apology will not apply to his followers.

and hence, he says, the coefficient of the acceleration is 10"-66-2" 341"580"71-0" 25 5"-78.

=

On this result Mr. Adams remarks that "the convergence of this series, though slow up to this point, converges more rapidly afterwards, and there is no fear that the sum of the terms neglected should exceed o"08."

This remark is very important, because, if the convergence of the series be admitted, there can be very little doubt of the correctness of the result, since, as will shortly be seen, Delaunay's expression is identical with that of Adams, and Professor Hansen can imagine only the failure of convergence as producing what he calls, perhaps prematurely, the error of Delaunay.

Before proceeding to the consideration of Delaunay's method and results, it will be well to give the substance of a letter from Mr. Adams to the Astronomer Royal, which contains one of the most curious and interesting specimens extant of the nature of the errors which will, when all pains have been used to avoid them, creep into computations of this kind, and will at the same time show the absolute identity of the results of the two astronomers.

On receiving from M. Delaunay his expression for the secular acceleration, Mr. Adams found a discrepancy between it and his own value in the coefficient of m7, of which he was enabled to trace the origin, and which we give in his own words, premising that M. Delaunay's term was

17103741

576

"The simplicity of the difference," he says, "between

these two quantities, when put under the form

100000

1152

shows

that it is caused by a mistake in a figure in the numerator of one of the fractions which go to form the coefficient. Accordingly, yesterday I looked over my old calculations, and soon found that there was an error of this nature, and that when this was corrected, my result perfectly agreed with M. Delaunay's. The error took place, as it not unfrequently happens, in one of the simplest parts of the calculation, namely, in dividing by 2 - 2 m, in order to obtain the coefficient of the term in the moon's longitude involving

nd t

é' d c' cos 2 (Pontécoulant's notation). The last term in this. 3372845 coefficient I had inadvertently written as m5 instead of 10368 3572845 m5. The correction to be applied is, therefore, 10368

100000

m5;

5184

and the corresponding correction to be applied to the multiplier

de' dt

dn nd t

of e' in the value of is the above correction multiplied

by mt. Hence this correction

2

100000, which is just the dif

1152

ference between M. Delaunay's result and mine. This agree