(1)
The part of the theorem which is most frequently required, and which also is the easiest to investigate, is the determination of the values of the coefficients A0, Ai, Bi. These are
| A0 = | 1 | ∫p0 φ(ξ)dξ; Ai = | 2 | ∫p0 φ(ξ) cos | 2iπξ | dξ; Bi = | 2 | ∫p0 φ(ξ) sin | 2iπξ | dξ. |
| p | p | p | p | p |
This part of the theorem may be verified at once by multiplying both sides of (1) by dξ, by cos (2iπξ/p)/dξ or by sin (2iπξ/p)/dξ, and in each case integrating from 0 to p.
The series is evidently single-valued for any given value of ξ. It cannot therefore represent a function of ξ which has more than one value, or which becomes imaginary for any value of ξ. It is convergent, approaching to the true value of φ(ξ) for all values of ξ such that if ξ varies infinitesimally the function also varies infinitesimally.
Lord Kelvin, availing himself of the disk, globe and cylinder integrating machine invented by his brother, Professor James Thomson, constructed a machine by which eight of the integrals required for the expression of Fourier’s series can be obtained simultaneously from the recorded trace of any periodically variable quantity, such as the height of the tide, the temperature or pressure of the atmosphere, or the intensity of the different components of terrestrial magnetism. If it were not on account of the waste of time, instead of having a curve drawn by the action of the tide, and the curve afterwards acted on by the machine, the time axis of the machine itself might be driven by a clock, and the tide itself might work the second variable of the machine, but this would involve the constant presence of an expensive machine at every tidal station.
(J. C. M.)
For a discussion of the restrictions under which the expansion of a periodic function of ξ in the form (1) is valid, see [Fourier’s Series]. An account of the contrivances for mechanical calculation of the coefficients Ai, Bi ... is given under [Calculating 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 ƒ(t) is known theoretically to be of the form
ƒ(t) = A0 + A1 cos n1t + B1 sin n1t + A2 cos n2t + B2 sin n2t + ...