Fourier's Double Integrals to Optical Problems. 



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that Fourier's theorem of double integrals enables us to express a 

 wide class of functions in terms of circular functions. In fact, sub- 

 . ject to certain limitations, 



It is proved in the course of the present paper that this process is 

 always legitimate when f(t) is such a function of time as can occur in 

 a physical problem. 



The above theorem reduces f(t) to the limit of a sum of simple 

 circular function of time, the element being du{C cos ut + S sin ut). If 

 we write this du . E cos {ut + ^), a simple vibration of amplitude J&du, 

 period 2tt/u and phase \j/ is suggested. To connect this analysis with 

 the physical analysis of light into a continuous spectrum is tempting. 

 The present essay is an attempt to prove that, in certain very general 

 cases, such a connection exists. The proof depends upon the two 

 principles (i) that we can have no cognizance of instants of time, but 

 can observe only the contents of small intervals of time ; (ii) that, 

 in spectrum analysis, we do not deal with definite wave-lengths, but 

 rather with small ranges of wave-length. 



, The fruitfulness of this calculus is illustrated by several applications. 

 The radiation of an incandescent gas is discussed. The trains of waves 

 emitted by molecules are continually being terminated by collisions. 

 It is held that, in dealing with the limiting widths of spectrum lines, 

 this effect must be included in the same investigation with the Doppler 

 effect first pointed out by Lippich and Lord Eayleigh. 



Other problems shown to be within the grasp of this method are : 

 the connection between Eontgen rays as explained by Professor 

 Thomson, and ordinary light ; and the effect of radiative damping of 

 the molecular vibrations in widening the lines of the spectrum. All 

 these investigations are based upon a theorem for dealing with a radia- 

 tion composed of a vast aggregate of similar pulses distributed at 

 random. The theorem is due to Lord Eayleigh. 



It is usual to examine the theory of dispersion by considering the 

 action of a simple periodic force upon a simple vibrator. Since no 

 light is simply periodic, it is necessary to extend the examination. 

 This is done below. We have also inquired whether fluorescence can 

 be due to natural vibrations of the molecules aroused by the non- 

 periodic quality of light. It is shown that so long as the equation of 

 motion is linear, no such explanation is possible. 



f{t) = (C cos ut + & sin uf)du, 

 Jo 



where 



