134 REMARKS CONCERNING FOURIER'S THEOREM AS [36
reducing to (5) again when a is made infinitely small. In comparison wit (7) the higher values of ua are eliminated more rapidly. Other kinds < averaging over a finite range may be proposed. On the same lines as abov the formula next in order is (fig. 3)
. r°° , sin au — aucosau ,~ , a • 17 /-IA\
du---------: 3 3------ {G cos ux + is sin ux\ ax. .. .(10)
" J o -rft U
1 r00 7 sin au — aucosau = —
7rJ0
In the above processes for smoothing the curve representing <£ (as), ordinat( which lie at distances exceeding a from the point under consideration ai without, influence. -This mayor may not "be an advantage. A formula i which the integration extends to infinity is
+ d> (a + £) ff-W** dt; = - I due~u2a^{Ccosux + Ssinux] ..... (11)
"
ft V71" J -» 77" JO
In this case the values of ua which exceed 2 make contributions to tt integral whose importance very rapidly diminishes.
( ! J ! The intention of the operation of smoothing is to remove from the cur\
features whose length is small. For some purposes we may desire on tt contrary to eliminate features of great length, as for example in considerirj the record of an instrument whose zero is liable to slow variation from son extraneous cause. In this case (to take the simplest formula) we may sul tract -from <p (a) — the uncorrected record — the average over a length relatively large, so obtaining
1 f35"1"^ 1 I"0 ( sin ub}
<f> (x) — -j I 6 (x) doc = - du\l- - — — [ {G cos ux + S sin ux}. ...(12)
&0 J x-b TT J 0 ( UO )
Here, if ub is much less than TT, the corresponding part of the range integration is approximately cancelled and features of great length a: eliminated.
There are cases where this operation and that of smoothing may be con bined advantageously. Thus if we take
j rx+a j ra+6
•5-1 $(x)dx- -r $ (X) dx
*& J x-a £0 J x-b
1 f°° , (svaua smub) tri ,v . .
= - | du\- --------- r-MCfcosM/B + /SfsinMaj}. ....(13)
TrJo ( ua ub } J v '
; we eliminate at the same time the features whose length is small compare
with a and those whose length is large compared with 6. The same meth< may be applied to the other formulas (9), (10), (11).
A related question is one proposed by Stokes*, to which it would 1
i interesting to have had Stokes' own answer. What is in common and wh
; * Smith's Prize Examination, Peb. 1, 1882 ; Math, and Pkys. Papers, Vol. v. p. 367.ce of glass is very surprising. I have not again met with this anomaly; but