The surface equations of equilibrium can also be written down at once by substituting the values of stresses in terms of strains in the expressions for F, G, H given above.
The Problem of Equilibrium.—To find the strain under given forces we have to solve the body and surface equations of equilibrium, when X, Y, Z and F, G, H are given. This problem has not been completely solved except in a few cases. It was solved by Lamé and Lord Kelvin for a solid or hollow sphere; it has also been solved for an infinite solid bounded by two parallel planes, or by a circular cylinder. Many particular solutions, however, are known for bodies of other shapes. Some of these solutions are of great practical value, e.g. St. Venant's solutions for the torsion and flexure of prisms. For bodies in which one or two dimensions are small, i.e. for thin plates and shells, and for thin rods, approximate theories have been given, which are partly deduced from the above exact equations, and partly from plausible hypotheses, a complete treatment based on the exact equations being in most cases impracticable.
The Problem of Vibrations.—When a body is vibrating, the mass acceleration parallel to Ox of the particle at (x, y, z) is ρ dx dy dz d2u/dt2. The equations of vibration are therefore found
by writing - d2u/dt2 instead of X, in the first body equation of equilibrium, and similarly with the others. The surface conditions will usually be that F, G, H are zero. The problem has been completely solved by H. Lamb for a solid or hollow sphere. For the elastic solid theory of the luminiferous ether, see Ether; for some practical solutions of the general equations of equilibrium, see Strength of Materials.—Bibliography: A. E. H. Love, Mathematical Theory of Elasticity; Lord Kelvin and Tait, Natural Philosophy; Todhunter and Pearson, History of Elasticity and Strength of Materials.
El´ateridæ, the name of a family of beetles, remarkable for their ability to throw themselves to a considerable height in the air, when placed on their back, by a vigorous muscular movement. Hence their names of springing-beetles, click-beetles, skip-jacks, &c. When alarmed, the elater counterfeits death. Flowers, grass, and decaying wood are the habitations of these animals, which are almost always found singly. The larvæ are often very injurious to vegetation, especially those which devour the roots of herbaceous plants (as in the genus Agriōtes), and are known from then slenderness and hardness as wire-worms. The fireflies of America belong to the family. In these a pair of luminous organs is found on the thorax, while there is a third on the under side of the base of the abdomen. The Pyrophŏrus noctilūcus, called cucujos in Brazil, is used as a personal ornament by ladies. The largest species of the genus Elater, the Elater flabellicornis, is 2½ inches in length.
Elate´rium, a substance obtained from the fruit of the squirting or wild cucumber (Ecballium agreste). The juice of the unripe fruit, when expressed and allowed to stand, deposits elaterium as a green sediment with an acrid taste, a faint odour, and powerful cathartic properties. It is a violent purgative, and is poisonous, but its action is not constant. The active principle in it is called elaterin.
Elba (Lat. Ilva), a small island in the Mediterranean, in the province of Livorno (Leghorn), Italy, separated from the mainland by the Strait of Piombino, about 6 miles wide. The island is 18 miles long and from 2½ to 10½ miles broad, and is traversed by mountains rising to a height of over 3000 feet. It is rich in iron, marble, granite, salt, &c.; and iron ore is exported. Excellent wine and fruits are produced. It has two seaports—Porto-Ferrajo (the capital) and Porto-Longone. The Treaty of Paris in 1814 erected Elba into a sovereignty for Napoleon, who resided in it from 4th May, 1814, to 26th Feb., 1815, when he escaped and landed at Cannes on 1st March. After Napoleon's departure the island was restored to Tuscany, which became part of Italy in 1860. Pop. 30,450.
Elbe (elb; Ger., pronounced el´be; Lat. Albis; Bohem. Labe), an important river in Central Europe. It rises on the south-west slopes of the Schneekoppe or Snowcap, one of the Riesengebirge, between Bohemia and Silesia. From this point it flows nearly due south into Bohemia for about 50 miles, when it turns to the west, and after about 40 miles takes a general north-north-west direction till it falls into the North Sea, intersecting Saxony and a considerable portion of Prussia. The finest scenery of its valley is in the Saxon Switzerland. Its length is 725 miles; drainage area, 56,865 sq. miles. The principal affluents are: on the right, the Iser, Schwarz-Elster, and Havel; on the left, the Alder, Moldau, Eger, Mulde, and Saale. In the lower part of its course the river divides into several arms, which unite again about 5 miles below Hamburg. It is more or less navigable for about 525 miles, but its estuary is much encumbered with sand-banks. In 1870 its navigation was declared free from Hamburg to Melnik in Bohemia. The North Sea and Baltic ship canal connects its estuary with Kiel Bay, and there are other important connected canals. It is well stocked with fish.
Elberfeld (el´bėr-felt), a town of Rhenish Prussia, in the government of and 15 miles east of Düsseldorf, on both sides of the Wupper, enclosed by lofty hills. Taken with Barmen it stretches along the Wupper valley for about 7 miles. The old streets are narrow and irregular, but the newer quarters are well built. It is a great seat of manufacturing industry, among its leading products being cottons, woollens, silks, velvet, mixed textile goods, buttons, ribbons, lace, yarns, thread, carpets, aniline dyes, iron and steel, machinery, pianofortes, and paper. Calico-printing, dyeing, and bleaching are very extensively carried on. It has given its name to a system of poor relief, combining organized voluntary effort and individual treatment. Pop. 170,195.
Elbeuf (el-beuf), a town of France, department of Seine-Inférieure, 11 miles S.S.W. of Rouen, in a valley on the left bank of the Seine, connected by two bridges with St. Aubin on the opposite side of the river. It is an important centre for the production of woollen manufactures, chiefly of lighter cloths and fancy goods, and is also an entrepôt for the finer and heavier cloths of Louviers and Sedan. It communicates by steamers with Paris, Rouen, and Havre. Pop. 19,240.