DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
345 
explain Rontgen rays. In the former, we have the magnetic force great and nega¬ 
tive throughout the pulse ; in the latter, positive and negative are to be so balanced 
that the force integrated through the thickness of the pulse shall vanish. On this 
property, together with that of the thinness of the pulses, Sir George Stokes* bases 
his proof that there will he no sensible diffraction. 
The mathematical consequence of this property will be zero amplitude for the 
infinite wave-length. Practically this means that the energy in the visible spectrum 
is very much smaller than for the Thomson pulse. It will be of order E 2 d . (d/A 0 ) 3 , 
instead of E -cl. d/\ ; a proportion of 10 -9 of the whole, instead of 10~ 3 . Now diffrac¬ 
tion depends chiefly on waves whose lengths are of this order ; very much shorter 
waves will not be diffracted, but will penetrate matter; and in any case would give 
much smaller diffraction patterns. Pulses of both the proposed forms will he sensibly 
free from diffractive properties; those of Stokes in a much higher degree than those 
of Thomson. 
Radiation of an Incandescent Gas, 
§ 37. As an example of the composition of a large number of independent pulses 
of uniform type, we will take the case of radiation from an incandescent gas. We 
will suppose the mass of gas to be at a great distance, and to have no visible 
diameter; we shall thus be enabled to consider the radiation as composed of plane- 
waves travelling at right angles to their own wave fronts. Furthermore, the amount 
of gas is to be so small that the emission is not sensibly affected by absorption. The 
gas is to consist of molecules all having the same period of free vibration. 
§ 38. The light received by the spectator will not be homogeneous. One reason 
for this is the Doppler effect.! The velocities of the molecules in the line of sight will 
alter the period of the light received. Another cause will doubtless be the altered 
vibrations of two molecules when very near to one another. This will perhaps become 
important at high pressures, but we will not further consider it at present. 
§ 39. Lastly, we have to take into account the fact that the train of single waves 
emitted by each vibrating molecule is not infinite in length, but has a definite 
beginning and ending. The effect of this cause is investigated below. The Doppler 
effect is included in the same piece of analysis. We shall arrive at the remarkable 
result that the limiting width of the spectrum line when the pressure is indefinitely 
diminished is less by some 10 per cent, than the width calculated by Lord Rayleigh, 
who took into account nothing but the Doppler effect. 
§40. The vibration of a molecule will be altered by collision with another. The 
velocity will also be altered, in both magnitude and direction. The vibration 
received by the spectator from this molecule will therefore be suddenly and 
fortuitously altered in period, amplitude and phase. The total radiation received 
* Wilde Lecture, ‘Proc. Manchester Phil. Soc.,’ 1897. 
t Lord Rayleigh, ‘ Phil. Mag.,’ vol. 27, April, 1889. 
VOL. CXCV.-A. 2 Y 
