DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
335 
where R 1 , IU, t/q, \fj. 2 are functions of u 
then 
oo 
j" = 7r | RjIU cos (\p 1 — xf/ 2 )du. 
The integrated value of f[t)<f>(t) depends therefore upon the distribution of energy 
in the separate elements, and upon the difference of phases of corresponding elements 
in the integrals. 
§ 10. Professor Schuster needs the theorem in order to prove that, in case of a 
bifurcated beam of light interfering with itself, “ the amount of interference depends 
on the distribution of energy only, and not on any assumption respecting the 
regularity or irregularity of vibration.’’ 
The proposition has, however, very much wider consequences. 
§ 11. For the present, the discussion will be confined to the particular case of 
constant light , i.e. light which does not present any perceptible fluctuations or other 
time-features. 
§ 12. All our cognizance of radiation other than the long waves of Hertz is by 
average effects. We average over a length of time great compared with the periods 
of vibration. This is true whatever be the means used to perceive and register the 
radiations, whether by direct visual perception, or by chemical effect, photographic or 
other, or by heating effect (bolometric), or by luminescence which the radiation 
excites. For Hertzian waves, on the other hand, the features of a single wave can 
be discovered. 
In discussing the qualitative effect of constant light, with a view to discriminating 
between different wave-lengths, we are concerned solely with the integral effect over 
a certain interval of time. 
It is doubtful how far we are at liberty to consider the molecule as a simple 
vibrator. But, in so far as this assumption is justified, we may prove that the 
observed effects of constant light- will depend on nothing but the partition of energy 
among the different elements of the equivalent Fourier integral. The phases will of 
course determine whether the light shall be “ constant ” or no ; but, if that condition 
is fulfilled, their further influence will not be perceived. 
+ oe 
§ 13. Energy. —The whole energy of the light motion fit) depends upon j f(t)dt 
— oo 
By the theorem of Schuster and Rayleigh, this is equal to 
f \t) = R cos ( ut -fi \p)do. 
1 o 
where 
