Spectrum of an Irregular Disturbance. 239 



Equation (1) becomes 



1 r 30 



r^=-~ e~ k2 ^ cos kxdk, (5) 



while for the distribution o£ energy in the spectrum by (3) 



e- 2e2x2 dx=-z \ e- h2 l 2e2 dk. . ... (6) 



J — ao . C JO 



<; If an infinite number of impulses, similar (but not neces- 

 sarily equal) to (4) and of arbitrary sign, be distributed at 

 random over the whole range from — go to + go , the intensity 

 of the resultant for an absolutely definite value of k would be 

 indeterminate. Only the probabilities of various resultants could 

 be assigned. And if the value of k were changed, by however 

 little, the resultant woukPagain be indeterminate. Within 

 the smallest assignable range of k there is room for an infinite 

 number of independent combinations. We are thus con- 

 cerned only with an average, and the intensity of each 

 component may be taken to be proportional to the total 

 number of impulses (if equal) without regard to their phase- 

 relations. In the aggregate vibration, the law according to 

 which the energy is distributed is still for all practical 

 purposes that expressed by (6)." 



The factor e~ c ~ x2 in the impulse was introduced in order to 

 obviate discontinuity. The larger c is supposed to be, the 

 more highly localized is the impulse. If we suppose c to 

 become infinite, the impulse is infinitely narrow, and the dis- 

 turbances at neighbouring points, however close, become inde- 

 pendent of one another. It would seem therefore from (6) 

 that in the spectrum of an absolutely irregular disturbance 

 (where the ordinates of the representative curve are indepen- 

 dent at all points) the energy between k and k + dk is pro- 

 portional to dk simply, or that the energy curve is a straight 

 line ichen k is taken as abscissa. If we take the wave-length 

 X (to which k is reciprocal) as abscissa, the ordinate of the 

 energy curve would be as X -2 . 



The simple manner in whicli dk occurs in Fourier^ theorem 

 has always led me to favour the choice of k, rather than of 

 X, as independent variable. This may be a matter of con- 

 venience or of individual preference ; but something more 

 important is involved in the alternative of whether the energy 

 of absolutely arbitrary disturbance is proportional to dk or 

 to dX. In Prof. Schuster's very important application o' 

 optical methods to the problems of meteorology, which seems 

 to promise a revolution in that and kindred sciences, the latter 



