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MACHINES

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Originally appearing in Volume V12, Page 958 of the 1911 Encyclopedia Britannica.
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MACHINES . A more See also:

general See also:form of the problem of See also:harmonic See also:analysis presents itself in See also:astronomy, in the theory of the tides, and in various magnetic and meteorological investigations. It may happen, for instance, that a variable quantity f(t) is known theoretically to be of the form f(t)=Ao+A1cos nit+Bisin nit +A2cos n2t+B2sin nit+ . . . (2) 957 (3) In a " normal mode " (4) (5) where the periods 22r/n1, 2w/n2, . . . of the various See also:simple-harmonic constituents are alzeady known with sufficient accuracy, although they may have no very simple relations to one another. The problem of determining the most probable values of the constants Ao, A1, B1, See also:A2, B2, . . . by means of a See also:series of recorded values of the See also:function f(t) is then in principle a fairly simple one, although the actual numerical See also:work may be laborious (see See also:TIDE). A much more difficult and delicate question arises when, as in various questions of See also:meteorology and terrestrial See also:magnetism, the periods 2a/ni, 2g/n2, . . . are themselves unknown to begin with, or are at most conjectural. Thus, it may be desired to ascertain whether the magnetic See also:declination contains a periodic See also:element synchronous with the See also:sun's rotation on its See also:axis, whether any periodicities can be detected in the records of the prevalence of sun-spots, and so on. From a strictly mathematical standpoint the problem is, indeed, indeterminate, for when all the symbols are at our disposal, the See also:representation of the observed values of a function, over a finite range of See also:time, by means of a series of the type (2), can be effected in an See also:infinite variety of ways.

Plausible inferences can, however, be See also:

drawn, provided the proper precautions are observed. This question has been treated most systematically by See also:Professor A. Schuster, who has devised a remarkable mathematical method, in which the See also:action of a diffraction-grating in sorting out the various periodic constituents of a heterogeneous See also:beam of See also:light is closely imitated. He has further applied the method to the study of the See also:variations of the magnetic declination, and of sun-spot records. The question so far chiefly considered has been that of the representation of an arbitrary function of the time in terms of functions of a See also:special type, viz. the circular functions See also:cos nt, See also:sin nt. This is important on dynamical grounds; but when we proceed to consider the problem of expressing an arbitrary function of space-co-ordinates in terms of functions of specified types, it appears that the preceding is only one out of an infinite variety of modes of representation which are equally entitled to See also:consideration. Every problem of mathematical physics which leads to a linear See also:differential See also:equation supplies an instance. For purposes of See also:illustration we will here take the simplest of all, viz. that of the transversal vibrations of a tense See also:string. The equation of See also:motion is of the form Pay—Tax2 where T is the tension, and p the See also:line-See also:density. of vibration y will vary as ei' ", so that ax +See also:key=o, k2 =n2p/T. where If p, and therefore k, is See also:constant, the See also:solution of (4) subject to the See also:condition that y=o for x=o and x=l is y = B sin kx (6) provided kl =See also:sir, [s =I, 2, 3, ...]. (7) This determines the various normal modes of See also:free vibration, the corresponding periods (grin) being given by (5) and (7). By See also:analogy with the theory of the free vibrations of a See also:system of finite freedom it is inferred that the most general free motions of the string can be obtained by superposition of the various normal modes, with suitable amplitudes and phases; and in particular that any arbitrary initial form of the string, say y=f(x), can be reproduced by a series of the type f(x) = Bisin7+B2sin2 x+B3sin3ix+...

(8) So far, this is merely a restatement, in mathematical See also:

language, of an See also:argument given in the first See also:part of this See also:article. The series (8) may, moreover, be arrived at otherwise, as a particular See also:case of See also:Fourier's theorem. But if we no longer assume the density p of the string to be See also:uniform, we obtain an endless variety of new expansions, corresponding to the various See also:laws of density which may be pre-scribed. The normal modes are in any case of the type y= Cu(x)eini where u is a solution of the equation d2u n2p dx2+ TT u=o• (to) The condition that u(x) is to vanish for'x=o and x=l leads to a transcendental equation in n (corresponding to sin kl=o in the previous case). If the forms of u(x) which correspond to the various roots of this be distinguished by suffixes, we infer, on See also:physical grounds alone, the possibility of the expansion of an arbitrary initial form of the string in a series f(x) =C1u1(x)+C2U2(x)+C3u3(x)+ . . . (II) It may be shown further that if r and s are different we have the conjugate or orthogonal relation flpu,.(x)ue(x)dx=0. 0 (9) This enables us to determine the coefficients, thus C,= f 1pf(x)u,(x)dx4- f 1p{u,(x)}2dx. (13) The See also:extension to spaces of two or three dimensions, or to cases where there is more than one dependent variable, must be passed over. The mathematical theories of See also:acoustics, See also:heat-See also:conduction, See also:elasticity, See also:induction of electric currents, and so on, furnish an in-definite See also:supply of examples, and have suggested in some cases methods which have a very wide application. Thus the transverse vibrations of a circular membrane See also:lead to the theory of See also:Bessel's Functions; the oscillations of a spherical See also:sheet of See also:air suggest the theory of expansions in spherical harmonics, and so forth. The physical, or intuitional, theory of such methods has naturally always been in advance of the mathematical.

From the latter point of view only a few isolated questions of the See also:

kind had, until quite recently, been treated in a rigorous and satisfactory manner. A more general and comprehensive method, which seems to derive some of its See also:inspiration from physical considerations, has, however, at length been inaugurated, and has been vigorously cultivated in See also:recent years by D. Hilbert, H. See also:Poincare, I. Fredholm, E. See also:Picard and others. (H.

End of Article: MACHINES

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MACINTOSH, CHARLES (1766-1843)

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