Fourier s Theorem as applied to Physical Problems. 867 

 From (5) we find that (8) is equivalent to 



2 f j l — cosua . 



^ 1 rt * fc — ^2 — \ C cos if# + fe sin im?} , . 



7TJ Q u a 



(9) 



, sin a% — aw cos ar« , ~ ~ . . , ,„_,, 



an 3 3 { (J cos ux 4- b sin ux\dx. (10) 



reducing to (5) again when a is made infinitely small. In 

 comparison with (7) the higher values of ua are eliminated 

 more rapidly. Other kinds o£ averaging over a finite range 

 may be proposed. On the same lines as above the formula 

 next in order is (fig. 3) 



In the above processes for smoothing the curve representing 

 (j>{x), ordinates which lie at distances exceeding a from the 

 point under consideration are without influence. This may 

 or may not be an advantage. A formula in which the 

 integration extends to infinity is 



1 c °° 



= ~ \ du e~ u2(l2/4 { cos ux + S sin ux] . . . (11) 



In this case the values of ua which exceed 2 make con- 

 tributions to the integral whose importance very rapidly 

 diminishes. 



The intention of the operation of smoothing is to remove 

 from the curve features whose length is small. For some 

 purposes we may desire on the contrary to eliminate 

 features of great length, as for example in considering 

 the record of an instrument whose zero is liable to slow 

 variation from some extraneous cause. In this case (to 

 take the simplest formula) we may subtract from cf>(x) — the 

 uncorrected record — the average over a length b relatively 

 large, so obtaining 



=- - r du { 1 - S -^ } { C cos ux + S sin ux J . (12) 



