362 APPLICATION OF FOURIER'S DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
— 7T 
For nearly homogeneous light, of period , R is small if 
integrand is therefore unimportant, except for values of u 
g — u 
is finite. The 
i. near to q, where we may neglect k, and use 
R-dw, 
n. 
near top, where the emission is practically —„f— 
l r - 2 
1 f R hlu 
A]f) /c 2 + (p — u)~' 
If the light emitted from the vibrator is analysed by a spectroscope, theoretically a 
spectrum of two lines should he revealed; the lines being at frequencies p and q. 
Let d be the half-width of the bright line in the incident light; then d will also be 
the half-width of the q line in the emitted light. 
The total intensity of the q line in the emitted light is therefore of order 
R \d! V \ 
The half-width of the p line will lie k ; the total intensity of the p line is of 
order R;,//cp 2 . 
The ratio of these intensities is 
E£ _ Rp K cl 
Ep — Ih 2 ■ P ' p' 
Now R 2 , R; is great, the incident light at p being by hypothesis invisible. On the 
other hand k/j) and d/p are both small. It therefore appears that, so far as the 
theory of the vibrator carries us, the natural vibration may be as prominent as the 
forced vibrator; the natural vibration varying inversely as the index of damping. 
Whether or no it is strong enough to be visible, depends upon the spectrum of the 
incident light and the constants of the vibrator. 
