nary, considering the assiduity and signal success with which the other departments of the science are cultivated. For these departments, however, a very slight acquaintance with mathematical science suffices: indeed, if we except the planetary perturbations, for the computation of which even the most refined analysis is scarcely sufficient-the only mathematical knowledge necessary to the practice of astronomy is confined to a few of the elementary theorems of Geometry, the simple rules of Algebra, and Spherical Trigonometry. We speak not of the observation of nebulæ, and double stars, or other phenomena: for this purpose, it is sufficient to be able to read angles; and hence, no doubt, one cause of the general interest taken in such obser

vations.

Although the Memoirs are thus strikingly deficient in contributions, which profess a profound knowledge of the higher geometry, it would be extremely unjust to ascribe to the Council of the Astronomical Society any indifference towards the noblest and most important department of their science. On the contrary, they seem, in their official capacity, to have used every means in their power to rouse the dormant energies of their mathematical members. They have repeatedly proffered their gold medal as a reward for interesting analytical discussions; they have successively invited attention to the theory of the satellites of Saturn,-of the four new planets,-of the lunar motions; and, finally, they have left to their geometers the choice of their own subject, requiring only some extension or improvement of methods or formulae already in use. The Council have therefore, in this respect, shown a laudable regard for the interests of Physical Astronomy; and if their efforts have hitherto been unattended with success, their zeal at least is deserving of every praise. They also remark, justly, that they are only the officers of the Society, and can merely select and arrange such materials as are put into their hands. It belongs, therefore, to the individual members to uphold, or rather create, the mathematical reputation of the Society, and maintain that connexion between theory and observation which, since the days of Apollonius of Perga, has been productive of the greatest advantages to astronomy.

It may be remarked, as accounting in some measure for the paucity of physico-mathematical papers in the Memoirs, that the Society is only one of many associations now existing in the country, within the range of whose enquiries such subjects are included. Other societies will divide with it the labours of the small number of mathematicians who are really capable of applying the integral calculus with effect to the developement of

the more abstruse consequences of the theory of gravitation; and the Royal Society, notwithstanding all the sins it has to answer for on this head, will probably continue to receive the most valuable contributions. It has been through this channel, that Mr Ivory has given to the world his important researches respecting the Attraction of Spheroids, Atmospherical Refractions, and other subjects connected with Physical Astronomy;-researches which vindicate the honour of British science, and which, in the history of mathematical and physical investigation, assure to the name of their illustrious author a place beside those of Laplace and Lagrange.

The Memoir to which we have above alluded, and which, on account of its singularity, is deserving of particular consideration, is the production of M. Plana, of the Turin Observatory; and its object is the discussion of some of the methods of analysis developed by Laplace in the Mécanique Céleste, for the computation of the planetary perturbations. The value of a discussion of this nature ought not to be measured merely by the correction of numerical results; in able hands it seldom fails to be accompanied by some extension or improvement of analysis, and it is even of importance to possess different modes of investigating the same subjects. The results of the theory of the world evolved by analysis are indeed so remote from the first principles, that, as Laplace remarks, astronomy can hardly receive a greater benefit than in the verification of their exactitude. It was not, however, the hope of finding any thing essential to correct in the immortal work of Laplace, that induced M. Plana to undertake the very laborious investigation which in this paper he has presented to the Astronomical Society, but to show that the improvements recently made in the methods of computing the perturbations rendered many questions susceptible of an easier solution than the want of symmetry in the ancient methods permitted: and, as he remarks in his introduction to the subject, it will not appear surprising, if, aided by the recent discoveries of geometers who have extended the boundaries of their science, he should be conducted to results more accurate than any previously publish

ed.

M. Plana commences his Memoir by examining a particular artifice of analysis, employed by Laplace for the purpose of transforming the elements of the fictitious orbit assigned by theory, when the mutual action of the planets is neglected, into those of the real orbits determined by observation. Laplace's method for this purpose-and the same had been previously employed by Lagrange-consists in the introduction, into one of the integrated formulæ of motion, of a supernumerary arbitrary constant, assu

med in such a manner that the term which in the resulting series expresses the mean motion, shall be equal to zero; thus leaving only the deviations produced by the action of the disturbing forces. M. Plana accomplishes the required transformation without having recourse to any indirect assumption, and his method has the advantage of keeping separate, through the whole course of the analysis, the terms on which the perturbing forces have no influence. The results of this direct method confirm the accuracy of the conclusions of Laplace.

In the succeeding part of his Memoir, M. Plana examines the method employed in the Mécanique Céleste, of determining the amount of the action of the fixed stars on the secular variations of the planetary system. Although the phenomena of sidereal astronomy leave scarcely any room to doubt that the stars are centres of attraction like the sun, yet on account of their vast distances they can hardly be supposed to exercise any influence on our system; Laplace, however, in order to leave nothing to a priori reasoning which could be reached by the calculus, thought it necessary to submit this point to analytical investigation; especially as it is not impossible that forces in themselves infinitely small, may produce sensible effects by being exerted through an infinity of time. In this case, M. Plana's results differ somewhat from those of Laplace in numerical value; but it is demonstrated by both authors, that the action of the stars is so feeble that it must accumulate during many hundreds of centuries before it can sensibly affect the secular inequalities of the planets. There is, however, a curious remark to be made on this subject. The two equations which are generally supposed to assure the stability of the planetary system,—namely, that which exists between the squares of the eccentricities, the masses, and the square roots of the axes; and the analogous one between the squares of the tangents of the inclinations, the masses, and the square roots of the axes,-are, by reason of the action of the stars, not mathematically exact, and can only be regarded as demonstrated for the simple powers of the perturbing forces; for the action of the stars is of an order equal to the squares or cubes of those forces. The causes here assumed to be in operation cannot, indeed, become sensible till after an almost inconceivable length of time; yet one almost regrets the possibility of the existence of any cause by which the stability of the system can be affected; or that beautiful consequence of its arrangement rendered nugatory, through which it is internally secured from any unlimited departure from its actual conditions.

One of the means provided in the economy of nature for the protection of the system from any permanent derangement, con

sists in the incommensurability of the mean motions of the planets. This circumstance, in general, renders it unnecessary to have regard to the squares or higher powers of the perturbing forces; but in certain cases, as when the mean motions of two planets are nearly commensurable, those powers cannot be safely neglected in the computation of their mutual action. Thus the mean motion of the Earth exceeds only by a small quantity four times the mean motion of Mercury; and in the analytical expression of the perturbation of the longitude of that planet, there are certain terms divided by the square of the difference between the mean motion of the Earth and four times the mean motion of Mercury, to which it is indispensable to have regard, because, however small the expression of the force itself may be, it may acquire, through the smallness of its divisor, a very sensible value. The direct computation of those terms is an operation of overwhelming labour; to avoid which, Laplace had recourse to artifices which give an approximate value of them with comparative facility. M. Plana, distrusting such approximations, has gone through the whole process of computing, by the direct method, the perturbations of Mercury by the Earth, and has obtained numerical values of the coefficients of the equations differing considerably from those of Laplace. As, however, the values of the coefficients in question are exceedingly small, the investigation is of more importance to theory than to the tables.

Another instance of the same kind occurs in the case of Jupiter and Saturn. The irregularities of these two planets are so considerable, that they had been remarked by Halley and Lambert, before they had undergone any theoretical discussion; and the discovery of their laws, periods, and physical causes, forms one of the most brilliant triumphs of the genius of Laplace. When the calculus is not pushed beyond the simple powers of the perturbing forces, it was found that the inequalities of the mean motions of the two planets have opposite signs, and are connected together by the following very simple relation. Denoting the mass of Jupiter bym, and his mean motion by a, and taking m' and a' to represent the analogous quantities with regard to Saturn; then the great inequality of Jupiter is to that of Saturn as ma is to m' a', so that when the mean motion of the one is accelerated, that of the other is retarded in this ratio. It is easy to see how greatly the numerical computation of the inequalities of the two planets must be facilitated by the above relation; and it is, consequently, of importance to determine how far it is accurate. Laplace had considered that it holds good even when regard is had to the square of the perturbing force; but M. Plana endeavours to prove that it only subsists in re

spect of the simple powers of that force; and, having computed the perturbations of each planet separately, by a method which he has developed with the most minute details, he found, to his surprise, that his final result had a contrary sign to that obtained by Laplace, and amounted only to a third part of the value. This conclusion was much too important, in a theoretical point of view, to be admitted without farther investigation. Accordingly the subject was resumed by Laplace himself; and in a short notice, which was published in the Connoisance des Tems for 1829, he announced that his new researches had led him to a relation which the results of M. Plana were far from satisfying, and which differed little from that given in the Mécanique Céleste. M. Poisson has also undertaken the investigation of this subject, and confirmed the conclusions of Laplace. He acknowledges, indeed, that he had not verified the long computations of M. Plana-having no reason to suspect their accuracy; but on making a complete enumeration of all the combinations afforded by the argument of the inequalities in question, (five times the mean motion of Saturn minus twice that of Jupiter,) he found that M. Plana was far from having exhausted them, and that he had neglected terms of a magnitude comparable to that of others which he had included in his analysis. Hence it may be inferred that the discrepancy between the results of the Italian geometer and those of the Mécanique Céleste is owing to the magnitude of some of the neglected terms.

M. Plana, in a Memoir lately presented to the Turin Academy, has replied to the objections started by Laplace and Poisson, and still considers the results published in the Memoirs as accurate; observing, not without some appearance of reason, that the most effectual way of putting a stop to the controversy, or at least of rendering it subservient to the amelioration of the tables of Jupiter and Saturn, would be to point out some error in his calculations, instead of trusting to approximations purely theoretical, and of which the advantages are frequently found to be illusory. It is to be feared that few will be inclined to accept this challenge, or to engage in a process of numerical computation, the very aspect of which is appalling; but M. Plana, in starting the subject, and treating it in so detailed a manner, has done a real service to science-by provoking farther discussion on one of the most interesting and important points of the Mécanique Céleste.

Among the contributions to the Memoirs most deserving of notice, are some accounts of observations made at Paramatta, in New South Wales, under the direction of Sir Thomas Brisbane, the late governor of that colony. Sir Thomas Brisbane,