Spectrum of an Irregular Disturbance. 211 



disappears if we take the mean with respect to f . We may 

 conclude that in the mean the energy of: every mode is the 

 same ; and since the modes are uniformly spaced with respect 

 to their frequency (proportional to s) and not with respect to 

 their period or wave-length, this result corresponds with 

 a constant ordinate of the energy curve when k is taken as 

 abscissa. 



It is to be noted that the above corresponds to an arbitrary 

 localized velocity. We shall obtain a higher and perhaps 

 objectionable degree of discontinuity, if we make a similar 

 supposition with respect to the displacement. Setting in (9) 

 ^ = throughout and yo^O except in the neighbourhood of 

 f, we get B s = and 



A^Jsin^Yi, (11) 



where Y 1 = fy ^<3?. By (8) the mean energy in the various 

 modes is now proportional to 1/t/ or to s 2 . When I is made 

 infinite, so that t s may be treated as continuous, we have an 

 energy curve in which the ordinate is proportional to s 2 or k 2 , 

 k being abscissa. 



We may sum up by saying that if the velocity curve is 

 arbitrary at every point the energy between k and k + dk 

 varies as dk, but if the displacement be arbitrary the energy 

 over the same range varies as k 2 dk. 



In Schuster's Periodogram, as applied to meteorology, the 

 conception of energy does not necessarily enter, and the 

 definitions may be made at pleasure. But unless some strong- 

 argument should appear to the contrary, it would be well to 

 follow optical (or rather mechanical) analogy, and this, 

 if I understand him, Schuster professes to do. If the energy 

 associated with the curve 4>(x) to be analysed is represented 

 by \{</>(.v) ) 2 dx, <f)(x) must be assimilated to the velocity and 

 not to the displacement of a stretched string. 



We have seen that when cp(x) is arbitrary at all points the 



ordinate of the energy curve is independent of k. In the 



curves with which we are concerned in meteoroloo-v the values 



. *■' 

 of <j)(x) at neighbouring points are related, being influenced by 



the same accidental causes. But at sufficiently distant points 



the values of <f>{x) will be independent. Equation ((.J) suggests 



that in such cases the ordinate of the energy curve (k being 



abscissa) will tend to become constant when k is small 



enough. 



Another illustration of the application of Fourier's theorem 



to the analysis of irregular curves may be drawn from the 

 Phil. Mag. S. 6. Vol. 5. No. 2v. Feb. 1903. U 



