DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
333 
It seems unnecessary to restrict the function by defining it for a finite interval alone. 
We may at once discuss amotion given for all values of the time, from — oo to -f- oo. 
By the theorem of Fourier’s double integrals* 
f(t) = 
(C cos ut 4 S sin ut)du 
J 0 
If- i r 
where C=- f(v) cos uvdu, S = - f(v) sin uvdv. 
— oo — oo 
The disturbance is thus analysed into a sum of elementary simple vibrations, of 
which 
du [0 cos ut -f S sin ut) 
is typical. 
Each of these is a simple circular function ; the results of the undulatory theory 
are directly applicable to it. 
The periods of the elementary vibrations have all values from zero to infinity. 
Now, if the disturbance could be analysed as a number of simple circular functions 
with distinct periods, the separate elements would have meaning, as in the familiar 
harmonic analyses of tides, vibrations of musical instruments, &c. The question we 
have to answer is, have the simple elements meaning in the limit, when their number 
is infinite, and the sum becomes an integral ? 
§ 7. We will at once notice an obvious criticism ; this was, in fact, offered by 
Poincare, f soon after Gouy’s article appeared. Each of the component vibrations 
du(C cos ut + S sin ut) exists unchanged through all time. This is true whatever 
be the nature of the disturbance we are analysing. But this disturbance may, 
for instance, be zero, except within a certain definite interval of time. Take the case 
of a flash of light. Now a spectroscope, says M. Poincare, will separate the 
component vibrations laterally ; they may lie examined separately. Hence a spectro¬ 
scope will enable us to see the light for an infinite time before it is kindled, and for 
an infinite time after it is extinguished. The analysis must therefore lie fallacious. 
The answer to this objection is as follows. No spectroscope possesses infinite 
analysing power. A given point at the focus of the telescope will be illuminated by 
light of a whole range of periods. Or, to look at the matter from another point of 
* On the Applicability of Fourier’s Double Integral to Functions occurring in Physical Problems. —In pure 
mathematics the applicability of Fourier’s theorems to functions is subject to certain limitations. These 
limitations exist when the functions possess infinite sets of discontinuities, or infinite sets of fluctuations, or 
infinities of certain types. Now, in concrete physical cases, we find neither infinities nor discontinuities. 
It is true that infinities and discontinuities may occur in functions commonly used to represent physical 
quantities. But the presence of such features is due to the abstract character of the method; a function 
more closely realising the properties of the physical quantity in question would be without infinite or 
discontinuous features. 
t Poincare, “Spectres Canneles,” ‘C. R.,’ 120, pp. 757-762, 1895 ; also see Schuster, ‘ C. R.,’ 120, 
pp. 987-989, 1895. 
