-dot force does not act. Now it is evident, if the part of this equation under the sign S be equal to the differential of the remainder of the first part, taken supposing a variable, we shall have rido? + drz + =0, in which a is variable as well v andr. And 2dt 2a da dR M to determine da we have dR dr=dR, as R is a 2 a? dv dr 2a? dR function of r and . Hence da= fe This is the celebrated formula by which it is proved that, regarding only the first power of the disturbing force, the greater axes of the orbits of the planets are subject to no secular variations. dR contains only periodical terms. The invariability of the greater axes of the planetary orbits is one of the most interesting results that has been obtained in this science. As far as regards the first power of the disturbing force the proof is sufficiently easy. When we consider the second power, the process becomes very difficult, but it has been executed with great ability by M. Poisson.* Who will prove it generally? That the eccentricities of the orbits are also only subject to variations included within narrow limits, is another conclusion that bas been deduced. This is conceived to follow from an equation first discovered, we believe, by Laplace. This equation, given by Mr. Woodhouse, p. 460, is mvā e’ +ni' Vaé? +m"da" e"? + &c.=K, in which K is a constant quantity, and very small on account of the magnitude of e, é, é", &c., and it is thence inferred that all the values of e, é', &c. must always continue very small. But it is obvious that such reasoning cannot apply if one of the bodies, m', be much smaller than another mi", because it may require a great change in é' to compensate for a small change in e". This escaped the notice of several writers on physical astronomy. Lagrange first appears to have remarked it, and it has been recently noticed by Mr. Bowditch, in the fourth volume of the American Transactions. Lagrange states the objection very clearly: * Il suit de la que si les excentricités des orbites qui appartiennent à des masses très grandes sont une fois très-petites, elles le seront toujours, ce qui est le cas de Jupiter et Saturne; mais celles qui appartiennent à des masses fort petites pourront croître jusqu'à l'unité et au-delà, et on ne pourra déterminer leurs véritables limites que par l’intégration des équations différentielles, comme on le verra ci-après.'Méc. Analytique, tom. ii. p. 147.+ • Journal de l'Ecole Polytech. tom. viii. K2 I If the solution of the problem of the three bodies be not yet complete in all its parts, it we cannot by approximations shew the places of the bodies of our system during an indefinite period, we can do that which, if it be well considered, must place this science among the first that do honour to the industry and ingenuity of man--we can calculate the motions and predict with unerring certainty the relative positions of these vast bodies for thousands of years to come-by applying the laws of matter and motion to the result of modern observation. These laws we collect from a few simple phenomena principally terrestrial. With respect to the astronomical tables as belonging to the present age, nothing can be conceived more accurate than the results deduced from the most intricate calculations :-even the tables of the new planets have already arrived at a great degree of exactness. We may instance those for the planet Vesta. The perturbations of this planet by Jupiter, Saturn and Mars have been computed by M. Daussy: the perturbations by the earth are too small to require notice. Those by Mars only become sensible from the relation of the mean distances of Mars and Vesta ; as the relation of the distances of Jupiter and Saturn produced the great equations of these planets. The results of the great labour of M. Daussy were published in 1814. Thirty different equations or corrections arising from the perturbations are required for deducing the longitude of the planet, some of them amounting to several minutes; yet such is their exactness, that in May, 1818, as we happen to know, the computed differed from the observed place by less than one ninute. It is unnecessary to add any thing further to shew the estimation in which we hold Mr. Woodhouse's work. We hope that the student will not be deterred from entering on it by a notion that the books which he has been accustomed to read, at Oxford or Cambridge, are not sufficient to enable him to become, readily, master of its contents; and that those who have not had the advantage of an university education will not suppose for a moment that the elementary treatises on mechanics, and flusions, in English, are not tully competent to assist them in understanding this volume. To the university student we would certainly recommend that he should defer the study of Mr. Woodhouse's book till he has made himself acquainted with the Principia’ of Newton; if this be too much, he may, if he please, enter on it after he has mastered the doctrine of motion as given in Wood's Principles of Mechanics, and with whatever knowledge of Auxions he may have obtained froin Vivce's or Dealtry's treatises. He may, perhaps, suppose that it is required he should know the method of partial diljerences and the calculus of variations. We intreat bini not to be be frightened at mere sounds. Let him examine for himself. He will find that in the application of these branches of mathematics in Mr. Woodhouse's book, he will have little more to do than to understand a simple notation. With regard to the calculus of variations, he will easily comprehend that if V be a function of a, b, c, x, y, 2, dr, dy, &c., and that any circumstances should make it necessary to tind the fluxions of V arising from a and y having varied, it is convenient, in order to distinguish the new from the former fluxions, to denote them by eu, dy; and if, in consequence of the variations of these quantities, e, x, and dr should also have changed, that their new differentials are to be expressed by če, dr, adr. The rules of the fluxional calculus obtain in finding the fluxions of V; also odr=dôr, which is easily proved. This contains almost all of the calculus of variations that is applied to physical astronomy. The notation of partial differences, all that is used by Mr. Woodhouse, is so easy that it almost explains itself. He refers, (but the reference is almost unnecessary,) for the explanation of the term partial differential co-efficient, to his Principles of Analytical Calculation, published in 1803. The integration of equations to partial differences is indeed a difficult subject; but the knowledge of this is scarcely required even for reading the Mécanique Céleste. The integration of only two or three (we believe) equations of partial differences occur in that work; from none of which the student will find any difficulty in bis progress. In giving this volume to the public, Mr. Woodhouse has preferred a new claim, we think, and that no small one, to the gratitude of his country. We owe him much for bis former works, and we trust that the estimation in which the present must be held will insure the production of another volume on those subjects which he has led us to hope will ó furnish matter for further speculations.' Art. VII.- Eastern Sketches, in Verse. By Henry Gally Knight, Esq. Second Edition. 1819. London. pp. 207. MR. R. Gally Knight is a traveller as well as a poet; and seeking to combine utility with pleasure, he has employed his poetry as a vehicle for imparting to his readers whatever is most striking in the customs and manners of the countries which he has visited. His stories,' he says, ' are not merely fables; they are intended to be portraits faithfully representing the features of the respective countries in which the scene of each is laid.' This plan, though perhaps attended with an increase of difficulty to the writer, is productive, we think, of considerable advantage to the reader. T'he accuracy of an observant traveller, and K 3 the the ardent imagination of the bard, are qualities of so distinct a nature, that, in general, the absence of the one is almost a proof that the other will not be found wanting. The reader, therefore, who is disappointed in the poetry, may throw himself boldly on the instruction which it is intended to convey; and with a fair chance of being entertained as well as informed, he, or his author, must be singularly unfortunate if neither should take effect in the course of their acquaintance. Mr. Knight appears to us to have attained considerable success in both departments. The general tone of his poetry, and more especially the succinct and picturesque notices collected in the preface, (which is drawn up with singular neatness and elegance,) bear testimony to his merits as an observer of national manners; and, if we are not mistaken, he performed his travels at an age when discrimination and judgment could have derived but little assistance from experience. His Muse also, though not of that daring description which penetrates to the sources of our feelings, and rouses and hurries along our imagination at the very moment when most repudiated by our taste, has many attractions of the milder and more amiable kind. Her song gives no indications of a spirit disturbed by moody passions, or scarred and scathed by painful recollections, which take their gloomy tinge rather from conscience than from misfortune. When she deals with vice or suffering, with the wicked or the unhappy, the distressing images naturally raised by such pictures are pleasingly softened by the glimpses of a gentle and benevolent turn of mind, which, like the moon-beams in Abdallagh's bower, (p. 3.) continually steal through the darker colouring, and produce an agreeable effect of contrast. The countries in which Mr. Knight has, respectively, laid the scenes of his manners-painting' fables, are Syria, Greece and Arabia, all of which, as he justly remarks, are abundantly stored with objects that speak to the imagination, and are calculated to make up by their poetical fertility for that inherent liberty which the Muse, in confining herself to certain localities, has for a season voluntarily surrendered. • The Muse,' says our author,' is, perhaps, never so much at her ease as when she sports in the regions of pure fancy; but in touching upon the countries of the east, truth and poetry may still be united. There is indeed a strength of genius capable of extracting honey from every weed—a power, of which the potent distillations effected by Mr. Crabbe's minute and searching pen may be adduced as lively examples. But it is true, at the same time, that there is a line of beauty for the imagination no less than for the eye, and that class of common undistinguishable objects, which lie confounded in the same plane, and do not stand stand forward, as it were, to the senses, are no less unaccommodating in poetry than the straight, the formal, and the angular in painting or sculpture. The bard who sits and thinks with his Muse' on the banks of a Dutch canal must import his whole stock of imagery, and conceal the native poverty of the spot with exotic foliage; but to name Syria or Greece is to awaken the choicest and most agreeable ideas that nature, whether living or inanimate, is capable of suggesting. The opening of Ilderim,' the first tale in the series, affords a favourable specimen of Mr. Knight's poetical powers. He is describing the luxurious garden of Abdallagh, a Syrian chief, residing at Balbec, the scene of his guilty, and as yet triumphant usurpation. · There eastern art display'd Yet, matchless Nature, in that scene so fair, Or purple clusters woo'd from overhead, Or yellow cassia bloom'd, and spicy fragrance shed.' We have not room (nor indeed is it necessary) to give the details of the story, in which there is not much of novelty; but we will present our readers with another extract from it of considerable force and beauty. We have marked one or two careless expressions which might easily have been avoided. The passage is towards the close of the poem, where the ladies of Abdallagh's seraglio, on the point of being murdered in pursuance of a precautionary order given by the tyrant to his Moorish slaves before the engagement in which he lost his life, are unexpectedly rescued by the hero (Ilderim) and his followers, who suddenly burst into the Harem by a secret way. • Their chieftain held the torch ; long aisles of gloom, That, |