MR. C. GODFREY ON THE APPLICATION OF FOURIER’S 
334 
view, a perfectly monochromatic train of waves will, by virtue of diffraction, 
illuminate, not a point, but a small area of the focal plane. The different elements of 
the Fourier integral will not be distinguished separately ; they will to some extent 
be superposed and recombine. The result will be, at each point of the focal plane, a 
disturbance not altogether different from the original motion in duration and 
character. We see then that the Fourier analysis may after all have meaning and 
application, and not lead to a paradox such as Poincare anticipated. 
It must be noticed that this recombination of the different elements of the integral 
is essentially connected with the phase-relation which exists between the said simple 
elements. 
§ 8. We have seen that Poincare’s objection will not prevent us from regarding 
the original ether-motion as mathematically equivalent to the Fourier integral. 
Whatever services the Fourier analysis can render us we may safely accept. 
It will be found that the different simple elements of the Fourier integral cannot 
in general be said to have any independent physical existence. On the other hand, 
part of the following essay is an attempt to prove that in certain cases the different 
Fourier elements can be regarded as having such jihysical existence. A special 
case of this nature is that of a steady emission, such as the radiation of an incan¬ 
descent gas. We shall inquire to what extent such radiations are equivalent to 
mixed light, presenting a continuous spectrum of composition determined by Fourier 
analysis. 
The Fundamental Theorem. 
§ 9. We will now introduce a theorem proved by Professor Schuster.* A par¬ 
ticular case was given by Lord Rayleigh, t 
The theorem is as follows :—• 
^ J 0 
f(t)<f)(t)dt = - (A x A 3 + B x BJdu, 
— oo 
-f co 
+ co 
where 
A i = | /(M cos 
+ oo 
cos u\d\, Bj = j f(h) sin u\d\ 
— 00 
+ 00 
A 3 = j <^(\) cos v.XdX, B 3 = j" (f >(\) sin u\d\. 
— 00 — 00 
In other words, if 
00 
f(t) = I R x COS ( ut + l/ljc/tt 
J o 
00 
ifj(t) = Rg cos (ut -f- xjj.-,)du 
J n 
* Professor Schuster, ‘Phil. Mag.,’ vol. 37, p. 533, 1894. 
t Lord Rayleigh, ‘Phil Mag.,’ vol. 27. 1889. 
