Spectrum of an Irregular Disturbance. 345 



Telocity-curve or the displacement-curve is taken to be 

 arbitrary. Here k is the frequency, or the inverse of the 

 period. I arrived at my own conclusion by translating the 

 results of an investigation, in which Fourier's analysis was 

 employed, into what I thought to be its complete optical 

 analogy. The. definitions that I gave of what I called the 

 Intensity of the Periodogram are, I think, quite clear and 

 definite, but the optical and mechanical analogy is not as 

 good as I thought. This depends on the fact that I took the 

 periodic time to be the independent variable instead of the 

 frequency. It does not matter which we take, if we deal 

 only with a number of separate periods^ each having finite 

 amplitude, the ordinates in that case measuring energy, but 

 if the periods approach each other indefinitely and the energy 

 between any two of them is to be represented by the area 

 included between the corresponding ordinates, the curve and 

 the axis of abscissa, the abscissa must represent frequencies 

 unless some correction is made to the ordinates. 



I was therefore wrong, when translating my results into 

 optical language, to compare the ordinate of the periodogram, 

 as I had defined it, with the intensity of a luminous disturb- 

 ance, and Lord Eayleiglr's criticism is quite justified. 



It may be desirable to alter my definition so as to make 

 the optical analogy complete, though practical considerations 

 have to be taken into account. 



Whether frequencies or periods are best plotted depends 

 on individual cases, but if we take the period we may still 

 preserve the optical analogy which in that case would be 

 with the spectrum of a diffraction-grating. It would only be 

 necessary for this purpose to divide the ordinates by the 

 squares of the periods. This would not involve any additional 

 arithmetical labour, unless the Fourier coefficients are ob- 

 tained by mechanical means. As Lord Rayleigh points out, 

 meteorological irregularities should be made to correspond to 

 velocities in the mechanical analogy, and this I had realized. 



I should like, in conclusion, to refer to a subject intimately 

 connected with the matter under discussion. We constantly 

 meet with the assertion that Eontgen radiations are due to 

 impulses and not to regular oscillations. I have no objection 

 to this statement if its meaning is clearly understood, but 

 generally it is put forward in such a way as to imply that an 

 impulse is something smaller, or even something of a different 

 order of magnitude, than a periodicity of short wave-length. 

 The statement that Eontgen rays are impulses differs from 

 the statement that Eontgen rays are short waves, only in so 

 far that the impulse theory asserts that there are long waves 



