DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
353 
Lord Rayleigh’s expression for the half-width is 
u = \ . 
h/2 
V 
v 
Now fi — 10/9; hence the theory developed in the present paper necessitates a 
further addition of about 10 per cent, to the values calculated from Lord Rayleigh’s 
formula. But the effect of this is quite obscured by the above-mentioned necessity 
of doubling the figures which Michelson deduces from theory. The conclusion of 
his paper should be that Lord Rayleigh’s theory accounts for a certain fraction of 
the observed ividths of spectral lines. The fraction varies from f to ^ for the different 
substances examined. 
If the distribution of energy in the spectral line be given by 
(f>(x) = 2 ‘ r!;V = e 
ip 
« 0 *> 
then 
cos 27 tux clx 
7T% 
dl ' 
_7J®!4*S 0 * 
e ~t~ 
and 
Y 
l 
„ ttVS 
= 2 o 
If we construct the curve representing the energy of the spectrum on a scale of 
reciprocal wave-lengths, the “ half-width ” in this curve will be S 0 . Lienee the “ half¬ 
widths” in the energy-curve and the visibility-curve are connected by the relation 
n 
o 
I 1 
IT ‘ S 0 ’ 
§ 48. Our result that 10 per cent, ought to be added to the visibility half-width 
means that the theoretical width of the spectrum line should be diminished by a 
similar percentage. 
It might have been expected that, with pressure small and collisions comparatively 
few, the modifying effect of the curtailment of free trains would disappear, the free 
paths being now on the average long. We might have expected that the formula for 
the width of the lines would converge to that given by Lord Rayleigh, in which the 
Doppler effect alone is considered. If the above reasoning is valid, there is no such 
convergence of the two theories when the pressure is indefinitely reduced; the 
results derived from them differing by some 10 per cent. It is noteworthy that the 
present theory leads us to expect narrower lines than does Lord Rayleigh. This 
result is certainly paradoxical, and calls for further justification. 
The modification in theory has been to substitute for mathematically homogeneous 
light proceeding from each molecule, a radiation giving a certain continuous spectrum. 
We could hardly have foreseen without a complete analysis that, for zero pressure, 
the integrated effect of all these spectra gives an intensity curve for the total spectral 
line steeper than before. 
2 z 
VOL. CXCV.—A. 
