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[Proceedings of the Royal Society, A, Vol. xc. pp. 318323, 1914.]
ACCORDING to Fourier's theorem a curve whose ordinate is arbitrary over the whole range of abscissae from # =  oo to x = + <x> can be compounded of harmonic curves of various wave-lengths.. If the original curve contain a discontinuity, infinitely small wave-lengths must be included, but if the discontinuity be eased off, infinitely small wave-lengths may not be necessary. In order to illustrate this question I commenced several years ago calculations relating to a very simple case. These I have recently resumed, and although the results include no novelty of principle they may be worth putting upon record.
The case is that where the ordinate is constant (TT) between the limits + 1 for ,-/; and outside those limits vanishes.
In general
                                 i r    r^00
<jf>($) = -|   die I     dv <p (v) cos k (v  x)................(1)
/* H-QQ                                                                                       rx                                                       m Ti 0
I     dv 6 (v) cos k (v - x} = ZTT cos kx    dv cos kv = 2?r cos kx r-^
J-oo                                                                        JO                                                  *
= j (sin k (x + 1) - sin k (x  1)},
and                         a=anfc0 + l)-fflnfc(0-l)] ................ (2)
As is well known, each of the integrals in (2) is equal to + |TT ; so that, as was required, < (x) vanishes outside the limits  1 and between those limits takes the value TT. It is proposed to consider what values are assumed by 0 (x) when in (2) we omit that part of the range of integration in which k
exceeds a finite value kit
152en x is moderate, we may use.. (12)