358 
MR. C. GODFREY ON THE APPLICATION OF FOURIER’S 
Finally, the motion which we have analysed as 
cos pt j R cos (ut + xjj)du — sin pt j R sin (ut -|- \jj)du 
may be described as a simple vibration whose amplitude and ‘phase vary slowly. 
We might equally well say that the amplitude and period vary slowly; the latter 
within narrow limits. In passing it is interesting to note that all the effects of white 
light would be produced by an approximately simple vibration whose period varies 
slowly, but within wide limits ; the whole range of variation being traversed a great 
number of times in the course of the shortest observable interval of time The great 
gap between 10 -15 second (the period of vibration) and 10“' second (the shortest 
observable time) will give room for a rate of variation small compared with the one 
measure, and great compared with the other. 
§ 61. Returning to the nearly homogeneous light, let is be the effective range of 
integration in 
| R cos (ut + *p)du, | R sin (ut 4- xp)du. 
This of course means that R becomes small outside the limits is. The width of 
the spectrum line will be of order s. Now the two integrals just quoted will give 
irregularly sinuous time-curves, the average extent of a sinuosity being of order 1/s. 
Thus, the varying simple vibration 
p+s 
R cos (ut i r ji)du 
J p — s 
will have entirely changed in amplitude and phase after a time of order 1/s. It 
will therefore be impossible to produce sensible interference with time-differences of 
more than 1/s. This is another aspect of the fact that for lines of width s (measured 
in frequency), the maximum path-difference for interference is of order 1/s. 
§ 62. We can look at the same matter from yet another point of view. We may 
go back to the composition of the radiation from a gas. This we have seen to be 
built up of finite trains of waves. For the moment, let us omit the Doppler effect 
and take all the trains to be of the same period. In an interval during which only a 
small proportion of molecules collide, the amplitude and phase of the composite 
vibrations is but little altered. But after an interval comparable with the mean free 
time of molecules, most of the molecules will have obtained new and independent 
vibrations; the composite motion will be entirely altered in amplitude and phase. 
For the discharge tubes used by Michelson this time is of order 10 5 periods. 
Again, isolating the Doppler effect, we deal with the superposition of infinite 
simple trains. We have already seen that the composite vibration will have a 
materially altered amplitude and phase after a time equal to the reciprocal of the 
range of frequency in the component trains. For Michelson’s experiments it 
happens that this time is again some 10 5 periods. 
