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1912]                               APPLIED  TO  PHYSICAL  PROBLEMS                                135
is the difference between C and S in the two cases (i) where <> (//;) fluctuates between  oo and + co and (ii) where the fluctuations are nearly the same as in (i) between finite limits  a but outside those limits tends to zero ? When oo is numerically great, cos ux and sin ux fluctuate rapidly with u; and inspection of (5) shows that $ (x) is then small, unless C or S are themselves rapidly variable as functions of u. Case (i) therefore involves an approach to discontinuity in the forms of C or S. If we eliminate these discontinuities, or rapid variations, by a smoothing process, we shall annul 4* (x) at great distances and at the same time retain the former values near the origin. The smoothing may be effected (as before) by taking
~\     ru+a                    "I     ru+a
-         Cdu,     -~         Sdu Zaj-u-a,             2aju-a
in place of G and S simply.    G then becomes
r+0 ,   . , N            sin av av <p (v) cos uv--------,
/                                    nit
J -co                                           MM
<j> () being replaced by (/> (v) sin av + av. The effect of the added factor disappears when av is small, but .when av is large, it tends to annul the corresponding part of the integral. The new form for (f> (x) is thus the same as the old one near the origin but tends to vanish at great distances on either side. Case (ii) is thus deducible from case (i) by the application of a smoothing process to C and S, whereby fluctuations of small length are removed.
We may sum up by saying that a smoothing of <p (ae) annuls C and S for large values of u, while a smoothing of 0 and S (as functions of u) annuls </> (an) for values of as which are numerically great., if ub is much less than TT, the corresponding part of the range integration is  approximately  cancelled  and features  of great length a: eliminated.