742
ELBA — ELBE
the flexural couple across a section normal to either axis has a component about that axis as well as a component about the perpendicular axis. Considering an element ABCD of the section at right angles to the axis of x, contained between two lines near together and perpendicular to the middle plane, the action of the portion of the plate to the right upon the portion to. the left, across the element, gives rise to a couple about the middle line (y) of amount, estimated per unit of length of that line, equal 02w : Gj, say, and to a couple, similarly estimated, fd2w to CI 3ar +<r w,: d2w about the normal (x) of amount - C(1 - cr)^ , =H, say. The
vessels of 209,500 tons in 1891 to 3004 of 568,930 tons in 1899. There is an institute of viticulture. On this and the neighbouring island of Pianosa there are convict prisons, in which from 3000 to 4000 convicts are lodged. The principal towns are Portoferrajo (population, 3 / 3 < in 1881), Marciana (5444), Portolungone (4172), and Rio Marina (2964). Population of the island, about 27,000.
Elbe (the Albis of the Romans), one of the most important rivers of Central Europe. It rises in Bohemia, in the upper gathering-grounds of the Riesengebirge, its chief corresponding couples on an element of a section at right angles feeders being the Weisswasser, which originates near the to the axis of y, estimated per unit of length of the axis of x, are kingly Schneekoppe, at an altitude of about 4600 feet, of amounts - C ^ + °'^)> =Gz ^ and ~H' The resultallt and the Elbseifen, which is formed in the same neighbourS,1 of the shearing stresses on the element ABCD, estimated as hood, but at a little lower elevation. After plunging ° 0Gr 0H and t ie down the 140 feet of the Elbfall, the latter stream unites before, is given by the equation ^ corresponding resultant S2 for an element perpendicular to the with the steep torrential Weisswasser at Madelstegbaude, at an altitude of 2230 feet, and thereafter the united stream axis of y is given by the equation S2= - - g—2. If the plate of the Elbe pursues a southerly course, emerging from the is bent by a pressure p per unit of area, the equation of equili- mountain glens at Hohenelbe (1495 feet), and continuing 0S 0S on at a soberer pace to Pardubitz, where it turns sharply to brium is %da:1 + ^=p, or, in terms of w, 0y the west, and at Kolin (730 feet), some 27 miles farther 4 0 w tiho 0% on, bends gradually towards the north-west. A little 0ad+0y4+ 0a;207/'2 above Brandeis it picks up the Iser, which, like itself, This equation, together with the special conditions at the rim, comes down from the Riesengebirge, and at Melnik it has suffices for the determination of w, and then all the quantities its stream more than doubled in volume by the Moldau, here introduced are determined. Further, the most important of the stress-components are those which act across elements of a river which winds northwards through the heart of normal sections: the tension in direction x, at a distance z from Bohemia in a sinuous, trough-like channel carved through the middle plane measured in the direction of p, is of amount the plateaux. Some miles lower down, at Leitmeritz _3Cz/02mi + <r^Y and there is a corresponding tension in direc- (433 feet), the waters of the Elbe are tinted by the reddish s 1 2h dx oy ) tion y; the shearing stress parallel to y on planes a:=const., 2and Eger, a stream which drains the southern slopes of the . „ , 3C(1 - <r)z d w ‘ Erzgebirge. Thus augumented, and swollen into a stream parallel to x on planes t/ = const., is of amount —— 0^0y ’ 140 yards wide, the Elbe carves a path through the these tensions and shearing stresses are equivalent to two principal basaltic mass of the Mittelgebirge, churning its way tensions, in the directions of the lines of curvature of the surface through a deep, narrow rocky gorge. Then the river into which the middle plane is bent, and they give rise to the winds through the fantastically sculptured sandstone flexural couples. 33. In the special example of a circular plate, of radius a, sup- mountains of the “Saxon Switzerland,” washing succesported at the rim, and held bent by a uniform pressure p, the sively the feet of the lofty Lilienstein (932 feet above the value of w at a point distant r from the axis is Elbe), the scene of one of Frederick the Great’s military 1 exploits in the Seven Years’ War, Konigstein (797 feet 64 C above the Elbe), where in times of war Saxony has more and the most important of the stress components is the radial than once stored her national purse for security, and the tension, of which the amount at any point is + o')pz(a2 — r2)/h3; pinnacled rocky wall of the Bastei, towering 650 feet above the surface of the stream. Shortly after crossing the maximum radial tension is about —(a/Zi)2/!, and, when the the Bohemian-Saxon frontier, and whilst still struggling thickness is small compared with the diameter, this is a large through the sandstone defiles, the stream assumes a northwesterly direction, which on the whole it preserves right multiple of^. Authorities.—The analysis requisite to prove most of the results away to the North Sea. At Pirna the Elbe leaves behind stated in this article is given by Love, Mathematical Theory of it the stress and turmoil of the Saxon Switzerland, rolls Elasticity, Cambridge, 1892, 1893. Reference may also be made to through Dresden, with its noble river terraces, and finally, Boussinesq, Application despotcntiels,Taxis,; Clebsch, Theoric beyond Meissen, enters on its long journey across the der Elasticitdt fester K'&rper, Leipzig, 1862 (Saint-Venant’s edition, North German plain, touching Torgau, Wittenberg, MagdeParis, 1883); Thomson and Tait, Natural Philosophy, Cambridge, 1879, 1883 ; Todhunter and Pearson, History of the Theory of burg, Wittenberge, Hamburg, Harburg, and Altona on the Elasticity, Cambridge, 1886-93 ; Pochhammer, Gleichgewicht des way, and gathering into itself the waters of the Mulde and elastischen Stabes, Kiel, 1879 ; Chree, “Changes in the Dimensions Saale from the left, and those of the _ Schwarze Elster, of Elastic Solids due to given Systems of Forces,” Trans. Camb. Phil. Havel, and Elde from the right. Eight miles above Soc. xv., 1892; “On Thin Rotating Isotropic Disks,” Proc. Camb. Phil. Soc. vii., 1891 ; “Long rotating^Circular Cylinders,” Proc. Hamburg the stream divides into the Norder (or Hamburg) Camb. Phil. Soc. vii., 1892; Hertz, “liber die Beruhrung fester Elbe and the Slider (or Harburg) Elbe, which are linked elastischer Kbrper,” Crelle, xcii., 1881 ; Michell, “On the Direct together by several cross-channels, and embrace in their Determination of Stress in an Elastic Solid ...” and other papers in Proc. Land. Math. Soc. xxxi. xxxii., 1899-1900; Pearson, “On arms the large island of Wilhelmsburg and some smaller the Flexure of Heavy Beams subjected to Continuous Systems of ones. But by the time the river reaches Blankenese, Load,” Quart. Journ. Math, xxiv., 1889 ; Pearson andFiLON, same 7 miles below Hamburg, all these anastomosing branches title, Quart. Journ. Math, xxxi., 1899. (a. E. H. L.) have been reunited, and the Elbe, with a noble width of Elba ,■ an island of Italy, belonging to the province of 4 to 9 miles between bank and bank, travels on between Leghorn, 4 miles from the nearest point of the mainland the green marshes of Holstein and Hanover until it becomes (Piombino), and 35 miles south from Leghorn. Iron is merged in the North Sea off Cuxhaven. From Dresden not only mined, but since 1900 smelted, in Elba. Iron to the sea the river has a total fall of only 123 feet, ore, wine, and salt are exported to the average (1891-98) although the distance is about 430 miles. For the 75 value of £197,250 (£244,000 in 1899). The shipping miles between Hamburg and the sea the fall is only. 3j which cleared from the various ports increased from 2692 feet. One consequence of this is that the bed of the river
