DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
341 
\jj — ut u lie at random all round an origin. The phase of the resultant will be 
arbitrary, while the mean value of its modulus is 
rd . cf)(u ) * 
Now this is the amplitude of the compound harmonic motion A. 
If we pass to a frequency u -j- da, where T da is small, the new phases will differ 
but slightly from the old ; but if T da is finite or great, the new phases will differ 
finitely from the old, and the resultant for u -j- du will have no apparent connection 
as regards phase with the resultant for v. 
Let us consider what happens when we observe this radiation. First, we can only 
observe the content of a certain interval of time, which we have taken to Ire T ; we 
receive into our apparatus the total energy of all the pulses in T. Now Schuster’s 
theorem expresses the connection between the total energy of a radiation, and its ex¬ 
pression as a Fourier integral (see p. 335). If the Fourier expression for the resultant 
of the n pulses in the present case is 
co 
| It cos (ut 4- 0)du, 
0 
the total energy is 
co 
7r I B?du. 
Jo 
We have just seen that Ft is not a definite function of u, but partakes of the 
random character of the sequence of pulses. At this point we make use of 
Rayleigh’s principle ; that we are not concerned with particular wave-lengths, but 
rather with the average energy over small ranges of wave-length. Bearing in mind 
the average value of R, we see that, for practical purposes, we have a spectrum whose 
intensity of energy for period 2 tt/u is 
§ 25. The fluctuations in the energy-wave-length curve will be less rapid as we 
descend to the longer waves of the spectrum. As we have just seen, when the time 
of vibration is small compared with T, the fluctuations are so crowded as to be 
indistinguishable ; the eye, or sensitive plate, will take the meancurve n<f) z (u). But if 
the time of vibration is large compared with T the range of angle included in the set 
\Jl — UT X , Xp — Ur. 2 , . . . \b — UT n 
will be but small; the resultant will possess a phase intermediate between the 
extreme values, and a modulus of almost ncf>(u). As we pass continuously to quicker 
vibrations, the modulus will diminish from n<p(u ) to zero, and so on ; the divergences 
of the energy curve from <f>~(u) being no longer rapid but on a broad and theoretically 
distinguishable scale. 
It will be found, however, that for short pulses the amount of energy in the slow 
* Lord Rayleigh, ‘Phil. Mag.,’ Aug., 1880 ; or ‘ Theory of Sound,’ ed. 1894, p. 40. 
