868 Fourier's Theorem as applied to Physical Problems. 



Here, if lib is much less than ir, the corresponding- part of 

 the range of integration is approximately cancelled and 

 features of great length are eliminated. 



There are cases where this operation and that of smoothing 

 may be combined advantageously. Thus if we take 



1 C*+a 



$(jB)dx-^\ <j>{x)dx 



1 C° 7 f sin ua sin ub 1 f ~ ' . 1 ,. ox 



= — I aii \ — 7 — > i ( ;cos ux + fesm?^ > , (la) 



we eliminate at the same time the features whose length is 

 small compared with a and those whose length is large 

 compared with b. The same method may be applied to the 

 other formula (9), (10), (11). 



A related question is one proposed by Stokes *, to which 

 it would be interesting to have had Stokes' own answer. 

 What is in common and what is the difference between 

 C and S in the two cases (i.) where (/>(■«) fluctuates between 

 -co and +co and (ii.) where the fluctuations are nearly 

 the same as in (i.) between finite limits ±a but outside those 

 limits tend to zero? When x 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 <fi(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 



1 nu + a 1 /*«+« 



mJu-a %Ju-a 



in place of and S simply. C then becomes 



c 



+ 00 sinav 



av <p [v) cos uv 



av 



<p(v) being replaced by cj)(v) smav-r-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 4>(x) is thus the same as the old one near 

 the origin but tends to vanish at great distances on either 



* Smith's Prize Examination, Feb. ], 1882; Math, and Phys. Papers, 

 vol. v. p. 367. 



