by a transcendental equation corresp0nding to the determinantal equation (6).
Numerous examples of this procedure, and of the correspond in treatment of forced oscillations, present themselves in theoretical acoustics. Itmust suffice here to consider the small oscillations of a chain hanging vertically from a fixed extremity. If x be measured upwards from the lower end, the horizontal com onent of the tension P at any point will be Pity/dx, approximately, ifp y denote the lateral displacement. Hence, forming the equation of motion of a mass element, pox, we have
Neglecting the vertical acceleration we have P =gpx, whence 6231 6 Gy
W"§ [, j¢(x'§ °¢>- (39)
Assuming that y varies as ew' we have
5; (x 3) +ky=<>. <4<>>
provided k=a'/g. The solution of (40) which is finite for x=o is readily obtained in the form of a series, thus kx »k'x
y=C 1-;, +, .-5.-...) =CJ@<z>, 4 an
in the notation of Bessel's functions, if z2=4kx. Since y must vanish at the upper end (x =l), the admissible values of a are determined by a'=gz'/41, ]°(z)=o. - (42)
The function L, (z) has been tabulated; its lower roots are given by Z/1f= '7555» 1'757I, 275451 - - -1
approximately, where the numbers tend to the form s-i. The frequency of the gravest mode is to that of a uniform bar in the ratio ~9815. That this ratio should be less than unity agrees with the theory of “ constrained types " already given. In the higher normal modes there are nodes or points of rest (y=o); thus in the second mode there is a node at a distance -1901 from the lower end. Authorities.-For indications as to the earlier history of the subject see W. W. R. Ball, Short Account of the History of Mathematics; M. Cantor, Geschichtc der Mathematik (eipzig, 1880 . . .); J. Cox, Mechanics (Cambridge, 1904); E. Mach, Die Mechanik in ihrer Entwickelung (4th ed., Leipzig, 1901; Eng. trans.). Of the classical treatises which have had a notable influence on the development of the subject, and which may still be consulted with advantage, we may note particularly, Sir I. Newton, Philosophiae natural is Principia Mathematica (1st ed., London, 1687); ]. L. Lagrange, Mécanique analylique (2nd ed., Paris, 1811-1815); P. S. Laplace, tllécanique céleste (Paris, 1799-1825); A. F. Mobius, Lehrbuch der Statik (Leipzig, 1837), and Mechanik des Himmels; L. Poinsot, Elements de statique (Paris, 1804), and Théorie nouvelle de la rotation des corps (Paris, 1834).
Of the more recent general treatises we may mention Sir W. Thomson (Lord Kelvin) and P. G. Tait, Natural Philosophy (2nd ed., Cambridge, 1879-1883); E. J. Routh, Analytical Statics (2nd. ed., Cambridge, 1896), Dynamics of a Particle (Cambridge, 1898), Rigid Dynamics (6th ed., Cambridge 1905); G. Minchin, Statics (4th ed., Oxford, 1888); A. E. H. Love, Theoretical Mechanics (2nd ed., Cambridge, 1909); A. G. Webster, Dynamics of Particles, &c. (1904); E. T. Whittaker, Analytical Dynamics (Cambridge, 1904); L. Arnal, T raité de mécanique (1888-1898); P. Appell, Mécanique rationelle (Paris, vols. i. and ii., 2nd ed., 1902 and 1904; vol. iii., 1st ed., 1896); (Q. Kirchhofi, Vorlesungen 'llbéf Mechanik (Leipzig, 1896); H. Helmholtz, Vorlesungen uber theoretische Physik, vol. i. (Leipzig, 1898); J. Somoff, Theoretische Mechanik (Leipzig, 1878~1879). The literature of graphical statics and its technical applications is very extensive. We may mention K. Culmann, Graphische Statik (2nd ed., Zürich, 1895); A. Fogpl, Technische Mechanik, vol. ii. (Leipzi, 1900); L. Henneberg, Stati des starren Systems (Darmstadt, 1886)-VI. Lévy, La statique graphique (2nd ed., Paris, 1886-1888); li. Müller-Breslau, Graphische Statik (3rd ed., Berlin, 1901). Sir R. S. Ball's highly original investigations in kinematics and dynamics were published in collected form under the title Theory of Screws (Cambridge, 1900).
Detailed accounts of the developments of the various branches of the subject from the beginning of the 19th century to the present time, with full bibliographical references, are given in the ourth volume (edited by Professor F. Klein) of the Encyclopddie der mathemalischen Wissenschaften (Leipzig). There is a French translation of this work. (See also DYNAMICS.) (H. Lu.)
II.-APPLIED MECHANICS 1
§ 1. The practical application of mechanics may be divided into two classes, according as the assemblages of material 1 in view of the great authority of the author, the late Professor Macquorn Rankine, it has been thought desirable to retain the greater 4 part of this article as it appeared in the 9th edition of the Encyclopaedia Britannica. Considerable additions, however, have been introduced in order to indicate subsequent developments of the subject; the new sections are numbered continuously with the old, objects to which they relate are intended to' rernainfixed or to move relatively to 'each other-the former class being comprehended under"the term “Theory of Structures ” and the latter under the term “ Theory of Machines.” . PART I.-OUTLINE OF THE THEORY OF STRUCTURES
§ 2. Support of Structures.-Every structure, as a whole, is maintained in equilibriiim by the joint action of its own weight, of the external load or pressure applied to it from without and tending to displace it, and of the resistance of the material which supports it. A structure is supported either by resting on the solid crust of the earth, as buildings do, or by floating in a fluid, as ships do in water and balloons in air. The principles of the support of a floating structure form an important part of Hydromechanics (q.v.). The principles of the support, as a whole, of a structure resting on the land, are so far identical with those which regulate the equilibrium and stability of the several parts of that structure that the onl principle which seems to require special mention here is one which comprehends in one statement the power both of liquids and of loose earth to support structures. This was first demonstrated in a paper “ On the Stability of Loose Earth, " read to the Royal Society on the 19th of June 18z56 (Phil. Trans. 1856), as follows:- Let E re resent the weight o the portion of a horizontal stratum of earth which is displaced by the foundation of a structure, S the utmost weight of .that structure consistently with the power of the earth to resist displacement, ¢> the angle of repose of the earth; then S: I +Sin 2
A ET l"'S1I1 ¢
To apply this to li uids ¢» must be made zero, and then S/E=1, as is well known. For a proof of this expression see Rankine's Applied Mechanics, 17th ed., p. 219.
§ . 3. Composition of a Structure, and Connexion of its Pieces.-A structure is composed, of pieces, -such as the stones of a building in masonry, the beams of a timber frame-work, the bars, plates and bolts of an iron bridge. Those pieces are connected at their joints or surfaces of mutual contact, either by simple pressure and friction (as in masonry with moist mortar or without mortar), by pressure and adhesion (as in masonry with cement or with hardened mortar, and timber with glue), or by the resistance of fastenings of different kinds, whether made by means of the form of the joint (as dovetails, notches, mortices and tenons) or by separate fastening pieces (as trenails, pins, spikes, nails, hold fasts, screws, bolts, rivets, hoops, straps and sockets.
§ 4. Stability, Stifness and Strength.-A structure may be damaged or destroyed in three ways:-first, by displacement of its pieces from their proper positions relatively to each other or to the earth; secondly by disfigurement of one or more of those pieces, owing to their being unable to preserve their proper shapes under the pressures to which they are subjected; thirdly, by breaking of one or more of those pieces. The power of resisting displacement constitutes stability, the power of each piece to resist dishlgurement is its stifness; and its power to resist breaking, its strengt § 5. Conditions of Stability.-The principles of the stability of a structure can be to a certain extent investigated independently of the stiffness and strength, by assuming, in the first instance, that each piece has strength sufficient to be safe against being broken, and stiffness sufficient to prevent its being disfigured to an extent inconsistent with the purposes of the structure, by the greatest forces which are to be applied to it. The condition that each piece of the structure is to be maintained in equilibrium by having its gross load, consisting of its own weight and of the external pressure applied to it, balanced by the resistances or pressures exerted between it and the contiguous pieces, furnishes the means of determining the magnitude, position and direction of the resistances required at each joint in order to produce equilibrium; and the conditions of stability are, first, that the position, and, secondly, that the direction, of the resistance re uired at each joint shall, under all the variations to which the load is subject, be such as the joint is capable of exerting conditions which are fulfilled by suitably adjusting the figures and positions of the joints, and the ratios of the gross loads of the ieces. As for the magnitude of the resistance, it is limited by conditions, not of stability, but of strength and stiffness., § 6. Principleof Least Resistance.-Where more than one system of resistances are alike capable of balancing the same system of loads applied to a given structure, the smallest o those alternative systems, as was demonstrated by the Rev. Henry Moseley in his Mechanics of Engineering and Architecture, is that which will actually be exerted but are distinguished by an asterisk. Also, two short chapters which concluded the original article have been omitted—ch. iii., “ On Purposes and Effects of Machines, " which was really a classification of machines, because the classification of Franz Reuleaux is now usually followed, and ch. iv., “ Applied Energetics, or Theory of Prime Movers, " because its subject matter is now treated in various special articles, e.g. HYDRAULICS, STEAM ENGINE, GAS ENGINE, OIL ENGINE, and 'fully developed in Rankine's The Steam Engine and Other Prime Movers (London, 1902). (ED. E.B.)