(The negative sign is used since w represents the elevation, positive or negative, above the undisturbed surface, and if u, and v increase with and, in the preceding discussion, there will be a fall of tide.) Referring back now to equation (3), Mr. Airy has shown (par. 85, 86, "Tides and Waves") that each term multiplied by S may be put under the general form cos (it+kw) in which is a function of alone: and also that "the equation between w, u, and v; those between p'", u and v; and that between w, p'', and the terms arising from the disturbing force, being all linear, we may take the terms arising from the disturbing force separately, and, finding the solution for each term, we may add all together. It will be sufficient, therefore, to proceed with the solution of the equation" (instead of equation (3)) 0= cos (it+kw)-gw+p''' and combining this with equations (4), (5) and (6), we have Laplace's differential equations of tidal motions (as given by Mr. Airy). "A general solution of these equations is scarcely to be hoped. for; it is a matter of difficulty to find, in a very limited case, a particular integral which will satisfy them."-(Airy, Tides and Waves.") And the particular integral essayed by Laplace and Mr. Airy is of the following form: w= a cos (it + kw) น - b cos (it + kw) in which a, b, c, a"" are functions of only. It is not my purpose to follow the investigation any further, which is purely analytical and consists in determining for each term of equation (3), the values of these quantities, and, thence, of w, u, v, p'", by which we get the elevation, velocity, direction of motion, &c. of the tides arising from the disturbing forces expressed by the particular term. I will only remark that the compulsory resort to this particular integral fixes the original assumption of an ocean covering the whole surface of the globe, irrevocably,-these values of w, u, v, and p", being simple perturbating functions, whose perturbations in time correspond exactly with those of the forces, while they extend in space through the whole circumference of the earth, without the possibility of limitation in that direction. In short, it is the particular integral which expresses that particular tide belonging to an ocean continuous in longitude. Excepting the arbitrary restriction applied to the variation in depth, the differential equations (3), (4), (5) and (6), are perfectly general, and could a general integral be obtained, a limitation of the ocean's area (approximating feebly to the continental barriers) could be established-and, thence, results which might be considered approximations to the actual phenomena. Such an integral, however, is not likely to be obtained, neither Laplace nor Mr. Airy having cared to attempt it. "As it is, Laplace's theory fails totally in application, from the impossibility of introducing in it the consideration of the boundaries of the sea." "If we look to the results of the theory, it will be found that they are rather of a negative than of a positive kind. They show that, without a far more complete knowledge of the form of the bottom of the sea than we can hope to possess, it will be impossible, even with more powerful mathematics, to calculate tides à priori. They show that the calculations founded on the equilibrium-theory cannot be good for anything. In proving that (with sea at least of a certain shallowness) the part of the equator next to the moon would be a place of low water, they destroy all hope of using an equilibrium-theory, even as an approximation. In establishing the remarkable result as to the non-existence of diurnal tide in height when the depth is uniform, they show that no inference can be drawn from the mere magnitude of a force as to the magnitude of its effects.”—(Airy, "Tides and Waves.") It does not seem to me that so difficult and profound a course of analysis was at all necessary to arrive at everything in these negative results at all important. The remark in the beginning of this paper that "the actual shape of the ocean's bottom is the very foundation of a dynamic theory of the tides," seems almost self-evident. But our mathematics, thus far, has failed to grasp even the simplest approximation to shore-outline, and if we had the most perfect knowledge of the form of the bottom of the sea, it would be far beyond the powers of analysis to introduce, with any accuracy, its consideration into the problem. Remarks of a similar nature might be made as to the conclusions concerning the equilibriumtheory, which, it seems to me, no philosopher could ever have regarded as a solution. Laplace having failed to show what the effect of the continental barriers is, it cannot be considered as proved, that the equilibrium-theory is not as near an "approximation" as anything else we have or are likely to have. My object in this paper has been to show by what simple considerations and processes the differential equations of Laplace's theory may be arrived at. In doing so I am perfectly well aware that the finding a short path to a known result is quite a different affair from the original discovery, and I must also remark that the considerations from which equations (4) and (5) are deduced are pointed out (after he has arrived at the equations by long and tedious processes) by Mr. Airy himself. Mr. Airy concludes his able work on "Tides and Waves," by a "Theory of Waves in Canals," and which, as embracing the subject of the tides, applies to cases such as rivers and arms of the sea, to which neither the equilibrium nor dynamic theory would (if applicable elsewhere) apply, and to "cases of open seas, where the whole may be conceived divided into parallel canals in which the circumstances are nearly similar." The "theory" is a very beautiful one and a very valuable contribution to physical science; more valuable, I think however, for its thorough discussion of waves in all (or nearly all) the aspects in which they present themselves to the navigator, naval constructor, or engineer, than for its application to the tides. Though it doubtless comes much nearer an approximation to the circumstances under which the tides actually flow in rivers or arms of the sea, than the dynamic or equilibrium theories do to the tides of the ocean, yet the vast difference between the actual configuration of shores and beds of such canals, and the simple assumptions the theory is confined to, will probably render this, like all other theories, useless, or nearly so, in practice. The subject of the tides of the ocean, though perhaps as intelligible as a physical phenomenon, as most others in astronomy, is in its actual manifestations, entirely beyond the grasp of our mathematics, beyond any reasonable conception we can form as to the powers of the human mind to grasp, through any supposed improvement in the means of mathematical analysis. It is probable that, for the aid of the investigator, the "equilibrium theory" has done as much as any theory can be expected to do. ART. XXXIX.-On some Fossil Plants of Recent Formations; by LEO LESQUEREUX. THE fossil plants of our recent formations have until now attracted little attention. The difficulty of identifying species of dicotyledonous plants from fragments of leaves only, is perhaps the cause of this neglect. Nevertheless the plants of the tertiary and quaternary strata will likely give a solution to some important problems in natural history. Botanists are now intently looking at the flora of those formations, not only to satisfy their minds in regard to the distribution of species of plants in the different strata, but to trace to its farthest limits the history of our present vegetation. They wish to find the origin of some genera and species now living on our earth, to trace their geographical distribution by recording their appearance and destruction at certain places and at a precise time, collecting thus, if possible, some facts that may help to unravel the causes which have changed and may still modify the march of vegetation. It is besides well known and easily understood that plants are more easily influenced by atmospherical changes than animals, at least than testaceous animals, which are those most commonly preserved in the geological strata, these only showing the changes in the sea. Even as characteristic of alluvial or fresh water formations, plants are more reliable than the remains of terrestrial animals, exposed to accidental and unaccountable migrations. The leaf of a palm tree found in the quaternary strata of Northern Russia could never have excited such discussions as did the remains of the elephant found there imbedded in the ice. We may therefore expect to obtain from botanical paleontology more precise indications about the succession of certain geological strata than from shells and animal remains only. This expectation is confirmed by the flora of the different strata of the coalmeasures which is evidently different, at least as regards some of the species of plants, for each bed of coal. Among the collections of fossil plants that have lately come under my examination, the most interesting, by far, is the one made by Dr. John Evans in his U. S. Surveying expedition of Oregon territory, Vancouver Island, &c. A description of these fossil plants appears just now to be a valuable contribution to science, and with the approval of the Secretary of the Interior, I have been advised by Dr. Evans to publish my remarks on those plants in advance of the publication of his report which will contain a full description of the fossil leaves with correct figures. It will be interesting to mention and compare at the same time some species of fossil plants found by Prof. Jas. M. Safford in the Pliocene of Tennessee, and some others collected by Dr. D. Dale Owen and myself in the chalk banks or Pleistocene of the Mississippi. Species of Fossil Plants collected by Dr. John Evans at Nanaimo (Vancouver Island) and at Bellingham bay, Washington Territory. 1. Populus rhomboidea (Lsqx.). Leaves rhomboidal, with the margins irregularly toothed above, and entire near the slightly decurrent base. Lateral primary nerves diverging at an acute angle like the secondary ones, and ascending to both corners of the rhomb of the leaf, all strongly marked with scarcely visible percurrent veinlets. It is much like Populus repando-crenata of Heer, differing only by the leaves somewhat broader and by the undulations and teeth a little deeper. The Populus mutabilis with its numerous varieties is a characteristic plant of the upper Molasse or Miocene of Europe, especially found in the upper strata of Oeningen. (Nanaimo). 2. Salix Islandicus (Lsqx.). Leaves large, lanceolate, pointed, serrulate, rounded at the base. Secondary nerves in acute angles with the medial nerve, nearly straight and numerous. Subdivisions of the nerves invisible. A willow with very large leaves, apparently identical with Salix macrophylla (Heer) of the Miocene of Europe. (Bellingham bay.) 3. Quercus Benzoin (Lsqx.). Leaves shining, oval, with undulate and entire margins decurrent on the petiole. Basilar secondary nerves opposite and emerging in an acute angle above the margin and ascending to the third of the length of the leaves. Upper secondary nerves more open and diverging. The kind of nervation of this leaf is peculiar to a few species of oaks, and has also some likeness to that of the genus Benzoin. This species is distantly related to Quercus Charpentieri (Heer), common in the Miocene of Switzerland. (Nanaimo.) 4. Quercus multinervis (Lsqx.). Leaves apparently shining and oval like the former; but differing much in the numerous, deeply marked, secondary nerves all parallel, emerging in an obtuse angle from the medial nerve, and slightly arched. It is related to Quercus neriifolia (Braun), a species plentifully found at Oeningen. (Nanaimo.) 5. Quercus Evansii (Lsqx.). Leaves thick, coriaceous, half a foot long or more, elliptical, with wavy and entire margins. Primary and secondary nerves deep and broad, apparently keeled. Secondary nerves oblique, curved along the margin of the leaves. This species has the same form and nervation as Quercus undulata, integrifolia, ovalis, and platyphylla of Göppert, all species which may be referred to the same and found in abundance at Shossnitz. The size of our species is twice larger. (Bellingham bay.) 6. Quercus Gaudini (Lsqx.). Leaves oval-lanceolate in general outline, narrowed and somewhat decurrent at the base (sometimes rounded), sinuate, dentate above, entire below, pointed. Nerves deeply marked like the former. Apparently a very variable species, which but for the size of the leaves could be referred to the former. Among our living species, its nearest relative is Quercus densiflora, a species of California. (Bellingham bay.) |