MACHINES.
A more general form of the problem of harmonic analysis presents itself in 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 simpleharmonic 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, A2, B2, . . . by means of a series of recorded values of the function f(t) is then in principle a fairly simple one, although the actual numerical work may be laborious (see TIDE). A much more difficult and delicate question arises when, as in various questions of meteorology and terrestrial 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 declination contains a periodic element synchronous with the sun's rotation on its axis, whether any periodicities can be detected in the records of the prevalence of sunspots, and so on. From a strictly mathematical standpoint the problem is, indeed, indeterminate, for when all the symbols are at our disposal, the representation of the observed values of a function, over a finite range of time, by means of a series of the type (2), can be effected in an infinite variety of ways. Plausible inferences can, however, be drawn, provided the proper precautions are observed. This question has been treated most systematically by Professor A. Schuster, who has devised a remarkable mathematical method, in which the action of a diffractiongrating in sorting out the various periodic constituents of a heterogeneous beam of light is closely imitated. He has further applied the method to the study of the variations of the magnetic declination, and of sunspot 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 special type, viz. the circular functions cos nt, sin nt. This is important on dynamical grounds; but when we proceed to consider the problem of expressing an arbitrary function of spacecoordinates 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 consideration. Every problem of mathematical physics which leads to a linear differential equation supplies an instance. For purposes of illustration we will here take the simplest of all, viz. that of the transversal vibrations of a tense string. The equation of motion is of the form
Pay—Tax2
where T is the tension, and p the linedensity. of vibration y will vary as ei' ", so that
ax +key=o, k2 =n2p/T.
where
If p, and therefore k, is constant, the solution of (4) subject to the condition that y=o for x=o and x=l is
y = B sin kx (6)
provided kl =sir, [s =I, 2, 3, ...]. (7) This determines the various normal modes of free vibration, the corresponding periods (grin) being given by (5) and (7). By analogy with the theory of the free vibrations of a 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 language, of an argument given in the first part of this article. The series (8) may, moreover, be arrived at otherwise, as a particular case of Fourier's theorem. But if we no longer assume the density p of the string to be uniform, we obtain an endless variety of new expansions, corresponding to the various laws of density which may be prescribed. 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 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 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 acoustics, heatconduction, elasticity, induction of electric currents, and so on, furnish an indefinite supply of examples, and have suggested in some cases methods which have a very wide application. Thus the transverse vibrations of a circular membrane lead to the theory of Bessel's Functions; the oscillations of a spherical sheet of 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 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 inspiration from physical considerations, has, however, at length been inaugurated, and has been vigorously cultivated in recent years by D. Hilbert, H. Poincare, I. Fredholm, E. Picard and others.
(H. LB.)
End of Article: MACHINES 

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