Hence, instead of three different waves with different velocities, we have just two,—a wave (like that of sound in air or other elastic fluid) in which the motions are perpendicular to the wave front, and the other (like the waves of light in an isotropic medium) in which the motions are parallel to the wave front.
Waves in an Incompressible Solid (Æolotropic or Isotropic).—If the solid be incompressible, we have A = ∞, and u must be zero.
| Hence |
and by a determinantal quadratic, instead of cubic, we find two wave-velocities and two wave-modes, in each of which the motion is parallel to the wave front. In the case of isotropy the two wave velocities are equal.
It is to be noticed that M1, M2, M3 in the preceding investigation are not generally true "principal moduluses," but special moduluses corresponding to the particular plane chosen for the wave front. In the particular case of isotropy, however, the equal moduluses M2, M3 of (11) are principal moduluses, being each equal to the modulus of rigidity, but M1 is a mixed modulus of compressibility and rigidity not a principal modulus. In the case of incompressibility, the two moduluses found from the determinantal quadratic by the process indicated above are not principal moduluses generally, because the distortions by the differential motions of planes of particles parallel to the wave front must generally give rise to tangential stresses orthogonal to them, which do not influence the wave motion.
([[Author:|W. TH.]])
ELATERIUM, a drug consisting of a sediment deposited by the juice of the fruit of Ecbalium Elaterium, the squirt ing cucumber (see vol. vi. p. 688.) To prepare it, the fruit is sliced lengthwise and slightly pressed; the greenish and slightly turbid juice thus obtained is strained and set aside; and the deposit of elaterium formed after a few hours is collected on a linen filter, rapidly drained, and I dried on porous tiles at a gentle heat. Elaterium is met with in commerce in light, thin, friable, flat or slightly incurved opaque cakes, of a greyish-green colour, bitter taste, and tea-like smell. The best kind is the English, prepared at Hitchin, Market Deeping, Mitcham, and else where; the Maltese is generally very inferior. Elaterium is an exceedingly powerful hydragogue and drastic purgative, and not uufrequently produces vomiting. Its active principal is elaterin, a crystallizable body of the formula C20H28O5.
ELBA, the Αίθαλία of the Greeks, and la the Romans, is an island in the Mediterranean Sea, forming part of the Italian province of Livorno, and lying about G miles from the mainland of Italy, from which it is separated by the channel of Piombino, and about 34 miles E. of Corsica. It has a very irregular coast outline, is 18 miles long and 214 to 1012 miles broad, and has a total area of nearly 90 square miles. It is throughout mountainous, and the highest point, Monte Capanne, is 2925 feet above sea-level. The western portion of the island is granitic, the eastern consists mainly of the sandstone locally known as verrucano, which in some places passes into a talc slate. In the vicinity of Porto Ferrajo the hills are cretaceous. The climate is mild, and, except at some spots on the coast, healthy. Springs are numerous, and the soil is not infertile; but agriculture and cattle-rearing are neglected, and there are no manufactures. Wine, wheat, aloes, dyer's lichen, and olives and other fruits are produced. The sardine and tunny fisheries, and the manufacture of sea-salt are of some importance; but the principal industry is mining. The iron mines are mostly in the vicinity of Rio Inferiere, and yield abundance of ore, chiefly hematite, of excellent quality. On account of the lack of fuel the ore is not smelted on the island, but is shipped direct to Follonica on the neighbouring coast of Italy, and to the ports of France and England. Marble, alabaster, sulphur, and ores of tin, lead, and silver are among the other mineral products, The principal places in Elba are the chief town Porto Ferrajo, with about 5000 inhabitants, the residence of Napoleon from May 4, 1814, to February 26, 1815, Rio Ferrajo, San Pietro, Porto Longone, and the village of Capoliveri. The population of the island in 1871 was 21,755.
The Argonauts, in quest of Cireo are said to have loaded at Portus Argous (Αργῶς λιμήν). now Porto Ferrajo, in Elba. The island was early famous for the richness of its mines, alluded to by Virgil (Æn. x. 173). It was attacked by Phayllus with a Syracusan fleet, 453 B.C., and subsequently by Apelles, who is stated to have subjugated it. In the 10th century it became a possession of the Pisans, from whom it was taken by the Genoese in 1290. It fell eventually into Spanish hands, came in 1736 under the jurisdiction of Naples, and in 1801 was ceded to the king of Etruria by the treaty of Lunéville. It was united to France in 1803, made over to Napoleon by the Treaty of Paris in 1814, restored to Tuscany in the following year, and in 1860 annexed to Italy.
ELBE, the Albis of the Romans and the Labe of the Bohemians, a large river of Germany, with a total length, of 705 miles, and a drainage area of about 55,000 square miles. It rises in Bohemia not far from the frontiers of Silesia, on the southern side of the Riesengebirge or Giants Mountains, in 50° 46′ N. lat. and 15° 32′ E. long. Of the numerous small streams (Seifen or Flessen, as they are named in the district) whose confluent waters compose the infant river, the most important are the Weisswasser, or White Water, and the Elbseifen; the former rises to the S.W. of the Schneekuppe in the W T hite Meadow, and the latter in a stone fountain in the Elb Meadow. Augmented successively by the Adler, the Iser, the Moldau, and the Eger, it cuts its way through the Mittelgebirge of Bohemia, traverses the sandstone mountains of Saxon Switzerland, and with a general JST.W. direction continues to meander through Saxony, Anhalt, and Hanover, until at length it falls into the German Ocean about 53° 5′ N. lat. and 8 50 E. long. The principal towns on its banks are Leitmeritz, Pirna, Dresden, Meissen, Torgau, Wittenberg, Magdeburg, Wittenberge, Harburg, Hamburg, and Altona. A short distance above Hamburg the stream divides into a number of branches, but they all reunite before reaching the ocean. At its source the Elbe is about 4600 feet above the level of the sea; after the first 40 miles of its course it is still 658 feet; .but at Dresden it is only 279, and at Arneburg in Brandenburg only 176. At Königgratz the width is about 100 feet, at the mouth of the Moldau about 300, at Dresden 960, and at Magdeburg over 1000. The tide is perceptible as far up as Geesthacht. Of the fifty and more tributaries belonging to the system the most important are the Moldau, the Eger, the Mulde, and the Saale,—the Moldau having a course of 267 miles, the Eger of 235, the Mulde of 185, and the Saale of 220. Though the channel in some places, and especially in the estuary, is encumbered with sandbanks and shallows, the Elbe is of great importance as a means of communication, steamboats being able to ascend the main stream as far as Melnick, and to reach Prague by means of the Moldau. Some idea of the extent of its traffic may be obtained from the statement that in 1870 at Schandau 489 passenger-steamers and 2658 vessels and barges of various kinds passed up the stream, and 489 passenger steamers, 2865 ships, and 1505 rafts down the stream. By one line of canal it communicates with Liibeck, by another with Bremen, and by others with the great network of Mecklenburg and Brandenburg; and several new lines are projected, by which a direct way will be opened up to Hanover, Leipsic, and various other important cities.—For details see Dr Th. H. Schunke's "Die Schifffahrts-Kanäle im Deutschen Reiche," in Petermann's Mittheil., 1877.
