(2)
where the periods 2π/n1, 2π/n2, ... of the various simple-harmonic constituents are already 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 A0, A1, B1, A2, B2, ... by means of a series of recorded values of the function ƒ(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 2π/n1, 2π/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 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 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 diffraction-grating 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 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 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 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 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
| ρ | ∂²y | = T | ∂²y | , |
| ∂t² | ∂x² |
(3)
where T is the tension, and ρ the line-density. In a “normal mode” of vibration y will vary as eint, so that
| ∂²y | + k²y = 0, |
| ∂x² |
(4)
where
k² = n²ρ/T.