8G6 Lord Rayleigh : Remarks concerning 



extends to a large range of x ; otherwise the interference of 

 components with somewhat different values of u may limit 

 the periodicity to a comparatively small range. Conversely, 

 a prolonged periodicity is associated with an approach to 

 discontinuity in the values of C or S. 



The complete curve representing <j>(oc) will in general 

 include features of various lengths reckoned along *, and a 

 feature of any particular length is associated with values of 

 u grouped round a corresponding centre. For some purposes 

 we may wish to smooth the curve by eliminating small 

 features. One way of effecting this is to substitute every- 

 where for <£(#) the mean of the values of <j>(x) in the neigh- 

 bourhood of #, viz. 



1 fx+a 



Ya) #*)'«**> ( 6 > 



tJx-a 



the range (2a) of integration being chosen suitably. With 

 use of (5) we find for (6) 



x- I <b(x)dx = — I du {C cos ux + S sin hoc], (7) 



la Jx-a ^Jo UCl 



differing from the right-hand member of (5) merely by the 

 introduction of the factor sin ua--ua. The effect of this 

 factor under the integral sign is to diminish the importance 

 of values of u which exceed ir/a and gradually to annul the 

 influence of still larger values. If we are content to speak 

 very roughly, we may say that the process of averaging on 

 the left is equivalent to the omission in Fourier's integral 

 of the values of u which exceed irj2a. 



We may imagine the process of averaging to be repeated 

 once or more times upon (6). At each step a new factor 

 sinwa-i-wa is introduced under the integral sign. After 

 a number of such operations the integral becomes practically 

 independent of all values of u for which ua is not small. 



In (6) the average is taken in the simplest way with 

 respect to x, so that every part of the range 2a contributes 

 equally (fig. 1). Other and perhaps better methods of 



Fig. I. Fig. 2. Fig 3. 



smoothing may be proposed in which a preponderance is 

 given to the central parts. For example we may take 

 (% 2) 



^£(«-?){<K*+?) +<#>-?) Mf • • (8) 



