DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
355 
proportional to r, not to r 3 . In fact, the taller the component curves are, the 
narrower they are. This consideration is overlooked if we allow ourselves to substitute 
the maxima of the curves for the curves themselves when these become narrow. 
§ 53. This latter source of error, by which r~ is substituted for r, will not affect the 
shape of the resultant energy-curve if the average length of train is taken to be 
independent of the velocity. But we have seen that greater velocities give shorter 
trains. And this error tends in the same direction as the other ; for it gives too great 
prominence to the longer trains, i.e., to the smaller velocities, which velocities send 
light to the middle of the spectrum line. Hence the effect of the error is to make the 
resultant curve too steep. 
Accordingly, the only accurate way of investigating the limiting width for zero 
pressure is to form the general energy function as on p. 349, and then to proceed to the 
limit by diminishing the number of molecules in unit volume. 
Effect oj Damping in the Widths of Spectrum Lines. 
§ 54. It has been urged by LommeiA that, whereas the vibrations of an atom are 
undoubtedly damped by radiation, the light emitted by a simple gas will be to some 
extent continuous. The same idea has been recently developed by Jaumann. f 
Both of these writers rely on a Fourier analysis of the vibration 
e~ Kt sm(pt + xfj) . . .(x.) 
Their results have not been accepted generally ; as has been pointed out in the 
course of the present paper, their procedure will have no physical meaning for the 
single train of waves with which they deal. 
But we have also seen that a vast aggregate of independent emissions of this form 
will really give the result which Lommel suggests. 
§ 55. The trains of waves emitted by the molecules of a gas cannot strictly be 
represented by (x.). As a matter of fact, such a motion as (x.) will never be allowed 
to go on indefinitely ; it will always be checked at a certain stage by a new collision. 
Nevertheless, if the radiation is so rapid that the vibration of a molecule has generally 
become insignificant before the next collision occurs, we shall not be making a 
serious error by allowing (x.) to represent the train of waves. We will proceed to 
investigate the effect on the assumption that the damping is so rapid as to allow this 
procedure. 
§ 56. Both Lommel and Jaumann make an erroneous application of Fourier’s 
theorem. The analysis should be as follows :— 
* Lommel, ‘ Wied. Ann.,’ vol. 3, p. 251, 1878. 
t Jaumann, 1 Wied. Ann.,’ vols. 53 and 54. 
2 z 2 
