DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
343 
deviations from the normal spectrum as taken with a much larger exposure, or as 
observed by the eye. 
§ 30. The extension to aggregates of pulses which are not all similar is obvious. 
Suppose for instance that we have a sequence of pulses of constant displacement 
the lengths of the pulses varying, while at the same time the proportions of 
different lengths are given. The pulses of lengths between x and x + dx are, say, 
f(x) dx, of the whole. They may be taken as equal pulses ; suppose that they give 
an energy function fr(u, x). Then the whole energy of the mixture is 
oo oo 
f(x)cf) 2 (u,x ) dx du. 
J<do 
Rontgen Rays and Ordinary Light. 
§ 31. Professor Thomson* explains Rontgen radiation by supposing it to consist 
of a succession of electro-magnetic pulses. Each pulse is practically a pulse of con¬ 
stant magnetic force, lasting for a short time. The thickness of a pulse is com¬ 
parable with the diameters of the particles composing the cathode stream. Lord 
Rayleigh has pointed outf that these pulses may be regarded as simple waves 
of short wave-length. He did not explicitly consider the properties of a succes¬ 
sion of pulses. Perhaps on account of this insufficiency of statement, Professor 
Thomson! has not fully accepted the above-mentioned view. He has held that the 
Fourier analysis of a pulse has no physical meaning. Now this is a valid objection 
to the identification of the single pulse with ordinary light of any composition what¬ 
ever. The different elements of the integral will possess definite phase-relation ; 
they are in no sense independent. 
On the other hand, it has been proved in the course of the present essay that the 
succession of pulses will actually be equivalent to a spectrum of definite composition. 
The Thomson pulses will certainly possess the property of being brief in comparison 
with the shortest observable interval of time ; there will be a great number of them 
in such an interval; it follows that, as the instrument averages over small ranges of 
wave-length, phase properties will be lost ; the processes of time- and wave-length 
averaging will efface all distinction between the succession of pulses and that 
mixture of light which is determined by the analysis of the single pulse. 
§ 32. We proceed to consider the energy-distribution in the scale of wave-length. 
We must express as a Fourier integral a function of x which is zero from 
— go to — -d ; E from — d to + d ; zero from + d to + co. 
We find 
00 
. , . 2E f sin ud . 
d)(x) = —-cos nx du. 
7T J 0 U 
* Professor Thomson, ‘ Phil. Mag.,’ February, 1898. 
t Lord Rayleigh, ‘ Nature,’ April 28, 1898. 
X Professor J. J. Thomson, ‘ Nature,’ May 5, 1898. 
