DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
359 
In the actual radiation the two causes coexist; the joint effect is of the same order ; 
the radiation will attain independence after every 10 5 periods. In accordance with 
this fact, we have interference up to path-difference of some 10 5 wave-lengths; while 
the width of the lines corresponds to a fraction 1/10° of the frequency of the light. 
Effect of a Natural Light on a Vibration. 
§ 63. We have already proved that the complete solution of 
oo 
x + 2 kx -f ffx — fit) ( It cos (ut + \fj)du 
Jo 
00 
f Tldu 
( 2 _ v?) o + 4/eV \{p° - u ~) cos (ut + if,) + 2 KU sin (ut r f)}, 
f(t ) being such a function of time as can actually represent a natural radiation (see 
footnote, page 336). 
Now we have already pointed out the hypothetical character of this treatment of 
the molecule as a simple vibrator. Nevertheless the method has a historical interest, 
and may be regarded as a foreshadowing of the truth; it may be worth while to 
sketch the result of applying our analysis. 
The composition of the exciting light is 
oo 
( Vrdu, 
Jo 
of the light emitted by the molecule 
® Ii~du 
I (f - v?f + ’ 
J 0 
while the rate of absorption is dependent on 
2 ku~FPcIu 
I (f - uf + 4 Hhfi ' 
Jo 
We will take the light to be constant. Its effect on the vibrator is seen to depend 
entirely upon its spectrum. 
§ 64. This is equally true for nearly homogeneous light. The effect of the irregu¬ 
larities in the light may be deduced from the observed widening of the spectrum line. 
Now Sellmeier treated of this problem in the paper which laid the foundations 
of the modern theory of dispersion.* He recognised that no natural radiation is a 
perfect train of simple waves, and he investigated the effect of the irregularities upon 
a vibrator. The period of the light was to differ from that of the natural vibrations 
of the molecule. He came to the conclusion that the irregularities in the light would 
* Seloieier, ‘ Pogg. Ann,,’ 1872, vol. 145, p. 520. 
